[a. G. Vitushkin] Several Complex Variables I Int(BookFi.org)

Contents

I. Remarkable Facts of Complex Analysis A. G. Vitushkin

1 I ,

II. The Method of Integral Representations in Complex Analysis 9 G. M. Khenkin

19 li

III. Complex Analytic Sets E. M. Chirka

117’ . .

IV. Holomorpbic Mappings and the Geometry of Hypersurfaces A. G. Vitushkin

159

I V. General Tbqory of Multidimensional I&&dues

P. Dolbeault ” 215

c Author’kdex - 243 (

Subjedt Index . 246

I. Facts jof Complex Analysis 1 t

,

A.G. Vitushkin . :

Translated from. the Russian by-PM. Gauthier

‘ Contents /’

1’1 ,

( ,

I

Introduction . . . . . : . .‘. . . ; . . . . . . . . . . . ..a.**. . . . . . . . ..i.*.... ‘1. The Continuation Phenomenon . . . . . . . . . . i , . . . . . . . . . . . . . ‘. . . . 1, Domains of Holomorphy . . . ‘. . . . ,‘. . . . , . . . . . . .t..;......*.. 3 Hoiomorphic Mappings., Classification Problems. . . , . . . . . . . . , . . 5 Integral Representations of Fun&ions -’ . . . . . . . . ...*...\ . . . . . . L. ‘6 Approximation of Functions. . :. . . . . . . . . , . . ; . . . . . . . . . . . . . .I 8 Isolating the Non-Holomoiphic Part of a Fynction. . , , , . . . . . . . . 10 ’ Construc‘tion of Functions with Given Zeros :, ‘. . . . . . . . . . . . . . . . 1’2 Stein ‘Manifolds. . . . . . . . . . . . . . . . . . . . , . . . . . . . * . . . .,:... :.. . * 14 Deformations of Complex Structure.. , . . . . . . . . . . . , . . . , . . . . . . . 16

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lrrirodubtion b- . ‘*

The present article gives a short survey of results in contemporary complex analysis and its applications. The material presented is conccnt’rated around several pivotal facts whose understanding enables one to have a general view of this area of analysis.

.*. ,.

§ 1. The Continyatik Phenqm@xa “. . , : ,

The most impressive f&t from cmiplex analysis is the phenomenon of the ’ continuation of. functions (Hat-togs; E&6; Poinkr& 1907) We elucidate its

,2 A.G. Vitushkin I

an example. If a function f is defined and holomorp iI ic on the l3 in n-dimensional complex space C”(n 1 2), then it turns out

continued to a function holomorphic on the yhole ball B.: Analogousliifor an arbitrary bounded dofniin whose complement is connected, any functiob

f morphic cant olomorphic on the boundaqy of such a domain]dmits a holo-!

puation to the domain-itself. Let us emphasize thatthis holds only for n z 2. Inkhe one $me,nsional case, this phenomenon clearly does not

,occur. Ind&$ for each set’ E c @’ atid each pohii z,i@\E,~.the function l/(z -i$) is%opmorphic on E but cannot be holomotphically continued to the point zo. ‘i,

1 This$sco\;e y TIlrl&d the beginning o/f-the systematic study of functions of s&era1 complF$ v&rlables. Two fundamental notions,‘orig+ting in connection with this pro&y of holomorljhic fun&ions,, are “envelope bf holomorphy” and “domairi :of .h?$omorphy”. Let D be k domain or a comp& set in C”. The envelo& I$ hokk~~7,+~ d of the set D is the laqgest set to which all function& holomorphic OIJ D extend’ holomorphically. The envelope of holomorphy of a domain in c is a domain which in general “cannot fit” into C”, but rather is a multi-sheqed domain over C” (Thullen, 1932). A domain D c C” iS called a domain ofholomorphy if d =‘D, i.e. if there exists a holomorihic function on D which cannot be continued to any larger domain. Domains of holomorphy are also sometimes called holoyorphically conuex domains.

The theorem on discs (Hartogs, 1909) gives an idea helpful idcqnstructing the envelope of holomorphy of a domain: if,a sequence of aqalptic discs, lying in the domain D, converges towards a disc whose boundary lies in D, then this entire limit disc lies in the envelope of hdlomorphy of 4. An analytic disc is the biholomorphic image of a closed disc. The technique of constructioc pf the envelope of holomorphjl of a dompact set and in particular of a surfac&rePes on

. ” conglomeration of “attached” discs whose boundaries lie bn,the given surface (Bishop, 1965).

Closely related tgthe &ion of envelope of holomorphy is the notion of hull with respect to son@ cl&s or other of functions, for example, the polynomial hull, the rational,hull, etc. The polynomial hull of a set D c C” is the set of all ZEC” for which the following condition holds: for each polynomial P(c),

“. P@N s ;s IJWI. ’ .

/’

: . Every, smooth curve is ‘holomorphica~ly coma, i.e. its envelope of ho]& morphy coincides with the curve itself. The polynomial hull of a curve is in gei~ral non-trivial. For example, if a smooth curve is closed and without self- intemcetions, then its polynomial hull is either trivial or it is a one-dimensional complex analytic sebwhqst boundary coincides with the given curve (Wermer, 195s; Bishop, 1962). We recall that a set in C” is called analytic provided thit id

*

l tk vicinity of each O[ its p&s it is defined by a finite bystem of equations

{/r(c) - O}, ~hwe Ifif. are holomorphiti functions..

I. Remarkable Facts of Compkx Adydd 3

-. An uqohed pml~+m. Is a set in C” consisting of a finite number of pairwe,

disjoint b;tlls polynomially convex? If the number of balls is at, most 3, then the! answer is boqitive; their union is polynomially convex (FaRin, /964).

.Another ’ varlint of the continuatiqn phenomenon 4 the theorem d Bogolyubo;;,lnicknamedlnicknamed the edge-of-thi-wedge theorem (S,Ni Berqstein, 14% N.N, Bogoly+ov, I956; . . . , , V.V. Zharinov, 170). Let Cf be ti acute convy cone in Wt cogsisting of rays e,tianating fropl the origin. Let ,C- be the symmetric io C,’ with respect to the origin. Let Q be a dosp R”, and D) D- two wedge& i.e. domains in 4=” of the type’ i / . . - D+ = {~+tzC”: Rezen, Im zEC+} / ’

and - b- ‘= {zEC”:Rez~R, ImzcC-).

‘Suppose f is a function ho!omorphic on D+ u,D- and suppose the functions flo+ arid &- have boundary values which agree in the sense of distributions along the edge of these cones, i.e. on the set Do = {z E C”: Re z E Q, Im z = 0). Then, f has a holomorphic extension to some neighbourhood of the set Do. The theorem on C&onvex hull (V.S. Vladimirov, 1961) ‘ves an estimate on the size of this’ neighbourhopd. For example, if Q = U?“,’ hen (D+ u Do u D-)-= Q=* r (Bochner, 1937). ’ /”

The theorem df Bogolyubov has been used to establish several relations in axiomatic quantum field theory. This theorem’ also laid the foundations of the theory of hyperftmctions (Sato, .1959; Martineau, 1964; . . . , V.V. Napalkov, 1974). For-&ore details, see artiicl~ II, III and volume 8,.article IV.

/ /”

92. Demains of Holoeg@Y

Domains of holomorphy are of interest because in suqh domains oqe can solve traditional problems of a&y&. In certain of these domains holomorphic functions have integral representations ‘and admit approximation by polynomials. In domains of holomorphy the Cauchy-Riemann equations are solvable; it turns out to be possible to interpolate functions; the problem of division is solvable; etc.

Two of the simplest types of domains of holomorphy are polynomial polyhedra and strictly pseudoconvex domains. A polynomicrl polytion is a domain givenbyasystemofthetypeIp’(z)l<l, jT1,2,...,,whereeachP’(z)isa . pdynomiid in z. Polynomial polyhedra were introduced by Weil(l932) and are also calkd Weil polyhedra. A domain is &led strictly pseuducorwex if in the neighbourhood of each of its boundary points the domain is strictly coayex for a suitable choice of cooidinates. Suppose the hypersurface bounding a domain is given by an equation p(z, 5) = 0. If in each point of the h-u&ice the Levi

‘\

4 A.G. Vitushkin 1 , I

form of the hypersurface is positive definite, then the domain in q$stion is.

strictly pseudoconvex (E. Levi, 1910). The Levi form is the for& & & dzi. &

restricted to the compIex tangent space to the hypersurface at ‘thepomt zo. The solution of various forms of the ‘problem of Levi concern g the holo-

morphic convexity of strictly pseudocqnvex domains remain i the central problem of complex analysis for several decades. Oka (1942) showed that each strictly pseudoconvex domain is holomorphically convex and conversely ‘each domain of holomorphy can be exhausted from the interior by domains of this type. Polynomial polyhedra are easily seen to be polynomially convex and consequently holomorphically convex.

Boundary points of a domain of holomorphy are not equivalent. A particularly important role is played by that part of the boundary which is called the distinguished boundary or the Shilov boundary. The Shiloo bound&y of a bounded domam is the smallest closed subset S(D) of the boundary of D ,such that, for each function f continuous on the closure of D and holomorphic in D

and for each point z E D the inequality 1 .&)I s max { f(Q[ holds. For a ball the

Shilov boundary coincides with its topological c:idary. The Shilov boundary of the polydisc lzjl < 1, j = 1,2, . . . , j-1,2,...,

n, is the n-dimensional torus lz,l = 1, n. For domains whose boundary is C*, the Shilov boundary is the

closure of the set of strictly pseudoconvex points (Basener, 1973). For domains of holomorphy, a strong maximum principle holds. If D is a

domairrof holomorphy and’f is non-constant, continuous on the closure of D, holomorphic in D, and attains a local maximum at some point, then that point lies in S(D) (Rossi, 196 has analytic structurti

simple cases the non-Shilov part of the boundary folliates into analytic sets. This was shown, for

example, for domains in @* having C’. boundary (N.V. Shcherbina, 1982). Concerning the topology of domains of holomorphy, it is known that the

homology groups Hk of order k are trivial for all k > rr. For polynomially convex domains, the n-th homology group is also trivial (Serre, 1953; Andreotti and Narasimhan, 1962).

Several classical problems of analysis are solvable only ‘for domains of holomorphy. For example a domain is a domain of holomorphy if and only if each function holomorphic on a complex submanifold of the domain is the restriction of some function holomorphic on the whole domain (Oka, H. Cartan, 1950). Analogously, a dqmain is a domain of holomorphy if and only if the problem of division is @vable’(Oka, H.,Cartan, 1950). The problem ofdivision is said to be solvable in the domain D if for any functionsf,? . . . ,& holomorphic in D, and any holomorph{c function f in D whose zero set contains (taking into account m’ult@icities) the set of common zeros of the,functions.Ji , . . . ,A, there exist ~fnnctions ai, . :, , gk; holomorphic in D, such that Chgi =$ We recall

, . . _..‘( ; j +

I. Remarkable Facts of Complex Analysis 5

that on accobnt of the Weierstrass preparation Theorem (188S), the local problem of d&&ion is always solvable. .

One can define the notion of holomorphic convexity in terms of plurisubharmonic functions. A function is called ‘plurisubharmonic if its restriction to each complex lind; is a subharmonic function. A domain D is a domain of holomorphy if an only if. the function -In&) is plurisubharmonic on D,

& where p(z) is the di rice from the point z to.the boundary of D (Lelong, 1945). For further details see article II and Volume 8, article II.

$3. Hdomorphk Mappings. Classification Problems

By the Riemann Mapping Theorem, in C’ a-ny two proper simply-connected domains are holomorphi&ly equivalent. In the multidimensional case, the situation is substantially different. For example, a ball and a polydisc are not equivalent (Reinhardt, 1921). Moreover, almost any two randomly chosen domains turn out to be non-equivalent (Burns, Shnider, Wells, 1978).

Let us consider the class of strictly pseudoconvex domains having analytic boundary. In this situation any biholomorphic mapping from one domain onto another ,extends to a biholomorphic correspondence between the boundaries (Fefferman, 1974; S.I. Pinchuk, 1975), and by the same token, the classification problem for such domains reduces to that of classifying liypersurfaces. There are two approaches to this problem. The first is geometric; the hypersurface is’ characterized by a system of differential-geometric invariants (E. Cartan, 1934; Tanaka, 1967; Chern, 1974). In the second approach, the characterization is by a special equation, the sa-called normal form (Moser, 1974). Both of these constructions enable one to distinguish the infinite-dimensional-space of pairwise nonequivalent analytic hypersurfaces.

In connection with the classification problem, .a description of mappings- realizing the equivalence between two surfaces has been obtained The results for mappings are described as for the’ case of functions by properties of continuation. In the case of mappings a new variant of this phenomenon appears. For example, it turns out that a holomorphic mapping of a sphere to itself given in a small neighbourhood of some point of the sphere can be holomorphically extended to the entire sphere and moreover, is in fact a fractional linear transformation (Poincare, 1907; Alexander, 1974). If the surface is not spherical, Le. cannot, by a local change of coordinates, be transformed into the equation of a sphere, then the germ of such a mapping of the surface into itself can be continued, not only along the surface, but also, in a direction normal to the surface. Namely, if a strictly pseudoconvex analytic hypersurface is not spherical, then the germ of any holomorphic mapping of this surface into itself has a holomorphic continuation (with. an estimate on the norm) to a “large”

6 A.G. Vitushkin

neighbourhood of the center of the germ. Moreover, a the neighbourhood as well as for the constant esti mined by the two characteristics of the surface, analyticity of the .surface and its constant of non-spherici 1985). In particular, a surface of the indicated type has which all automorphisms of the surface extend. It is wort both examples we have presented, the mappings, in contr not only to the envelope ,of hofomorphy of the domai defined,,but also to some domain lying outside the domain of holomorphy. The theorem on germs of mappings concludes a lengthy chain of works on holomorphic mappings of surfaces (Alexander, 1974; Bums and Shnider, f976; S.I. Pinchuk, 1978; V.K. Beloshapka and A.V. .tOboda, 1980; V.V. Ezhov and N.G. Kruzhilin, 1982).

From the Theorem on Germs, it follows that a stability group of a surface (group of its automorphisms which leave a certain point fixed) is compact. _ Hence, by Bochner’s theorem on the .linearization of a compact group of automorphisms (1945), one obtains that a stability group of a non-spherical surface can bc linearized, i.e. by choosing appropriate coordinates, every automorphism can be written as a’ linear transformation (N.G. Kruzhilin and A.V. Loboda, 1983):Tdgether with the theorem of Poincare, this means that for each pair of locally given strictly pseudoconvex analytic hypersurf&s, every map ping sending one hypersurface into the other can be written as a fractionai-fine& transformation by an appropriate choice of coordinates in the image and preimage. The-problem on the linearization of mappings of surfaces having a non-positive”Levi form remains open. For further details see art@e IV and Volume. 9, articles V and VI.

We have considered here only one aspect of the problem of classification. Large sections of complex analysis are concerned with the study of invariant metrics (Klhier, 1933; Caratheodory, 1927 Bergman, 1933; Kobayashi, 1967; Fefferman, 1974, . . . ); classification of manifolds (i-lodge, Kodaira, 1953; Yay Siu, 1980; . . . ); description of singularities of complex surfaces ‘(Milnor, 1968; Brieskom, 1966, Malgrange, 1974; A.N. Varchenlco, 1981; . . . ).

$4. Integral Representations of Functions

A smooth function in a closed domain I!j c C can be expressed using the Cauchy-Green formula

The first term on the righr side is the formula which reproduces a holomorphic


function in a domain in terms of its boundary values. The second term isolates af the non:hqlomorphic part of f and yields a solution to the &equation z = g.

For functions of. several variables, there does not exist such a simple and universal formula, and hence it is suitable to consider the problem of integral formulas for : holomorphic functions and the solvability of the &equations _’ separately.

For some .cl+sses of domains in C”, there are explicit formulas which re- produce a holomorphic function in terms of its boundary values. For polynomial .polyh&a such a formula w& obtained by A. Weil (1932); for strictly pseudoconvex domains, by GM. Khenkin (1968). Such a formula was given for the polydisc’by Cauchy (1841) and for the ball, by Bochner (1943). There is a formula of Bochner-Martinelli (1943) for smooth functions on arbitrary domains having smooth boundary. In this formula, in contrast to the previous ones; the kernel is not holomorphic, and this often makes it difficult to apply, For polynomial polyhedra there is still another formula which distinguishes itself from the Weil formula and other formulas in. that its kernel is not only holomorphic but also integrable (A.G. Vitushkin’, 1968).

I,et us introduce the formulas for the polydisc and the ball. If f is holomorphic on the closure of the polydisc D”, then

I

-,

If f,is holomorphic on the closed ball B: lil 5.1, then inside the ball,

where V is the (2n- l)-dimensional volume of the sphere aB and dV is its element of volume.

All of the formulas which we have mentioned above differ from one another in appearance, The appearance of the formula depends on the type of domain. There is a formula due to Fantappit-Leray (1956) which gives a general scheme for writing such formulas. Let D be a domain in C:, where z = (zl, . ., . , z,) is a

, set of coordinate functions, and let f be holomorphic on the closure of D. Then

fiz) r (n- I)! j f(r) WY 7 CttlKl -z1)+ * * * +tt&-ZJI”

where y is a (2n”- -l)-dimensional cycle in the space @F x CE lying over the boundary of the domain D c @F and covering it once. By choosing suitably the

,1 form of the cycle y, having chosen q as a function of C, one can obtain any of the preceding integral formulas.

8 A.6 Vitushkin

One of the applications of integral formulas is in solving the problem of interpolation with estimates. If a complex submanifold M of the ball B crpsses the boundary of the ball transversally, then every function holom&phic and bounded on M can be continued to a function holomorphic and bounded in the entire ball (G.M. Khenkin, 1971). The extension is constructed ad follows. The functionf(z) for z E M can be written as an integral I(z) off on the boundary of M. Moreover, it turns out that the function I(z) is defined for all z E B, and from the explicit formula for I(z), one obtains that the extended functionf(z) = I(z) is holomorphic and bounded on B. *

The problem on the possibility of division with uniform estimates remains open. Namely, it is not known whether for,each set of functions fr, , , . , &, f holomorphic and bounded in the ball B c C,” and such that :nfffr 1 &(()I # 0,

e = there exist functions gl, . . . , & bounded and holomorphic on B such that vi fig, = l.‘This is a modified formulation of the famous “corona” problem. In :ii one dimensional case, this problem was solved by CarIeson (1962). The answer is” positive: in the maximal ideal space for the algebra of bounded halomorphic ‘functions in the one-dimensional disc, the set of ideals, corresponding to points of the disc, is everywhere dense.

The above enumerated formulas are forbounded domains. In the present time analysis on unbounded domains is also flourishing. In part&rIar, integral formulas have been constructed for such domains: There are explicit formulas for tubular domains over a cone (Bochner, 1944), on Dyson domains (Jost, Lehmann, Dyson, 1958; V.S. Vladimirov) and Siegel domains (S.G. Gindikin, 1964). Weighted integral representations for entire+ functions have also been constructed (Berndtsson, 1983). For further results see’article II and Volume 8, articles I, II and, IV.

$5. Approximation of Functions * ‘_

Let us denote by CH(E) the set of all continuous funchons on the comnact set E c C” which are holomorphic at interior points of E. It is clear that f&&c& which can be uniformly approximated on E with arbitrary accuracy by complex polynomials or by functions holomorphic on E belong to the class CHQ. When we speak of the possibility of approximating functions on the compact set E, we shall mean the following each function in W(E) can be approximated uniformly with arbitrary precision by functions holomor’phic~oir~E.

If a compact set E in C’ has a connected complement, “then each function holomorphic on E can be approximated by polynomials(Rurigc, ,J885). This is , equivalent to a theorem of Hilbert (1897): on each polyno& polyhedron in @*,

I. Remarkable Facts of Complex Analysis 9 4

any holomorphic function can be represented as the sum of a series of polynomials. Runge’s Theorem reduces the question of the possibility of approximating functions by polynomials to that of constructing holomorphic approximations of functions. The criterion forthe possibility of approximation by holomorphic functions (A.G. Vitu’shkin, 1966) is formulated as follows, The assertion that each -function in U-Z(E), where E’? C’, can be uniformly approximated with ‘arbitrary accuracy by functions holomorphic on E is equivalent to the follo.wing condition on the compact set E: for each disc K, a(K\E) = . a(K\E), where E denotes the interior of E, and a(M) is the continuous analytic cbpdcity of a set M. By definition

* a(M) = ,sup’ lim zf(z) .’

I I’. ML;/ z-00 The supremum is taken over all compact sets Iwc i M and all functions f which are everywhere continuous on C’, bounded in modulus by I and holomorphic outside of M*. In particular, approximation is possible if the inner boundary of E is empty, i.e. each boundary point of E belongs to the boundary of some complementary component of E. For example, all compact sets with connected complement belong to this class. The above criterion emerged as a result of a long series of works on approximation (Walsh, 1926; Hartogs and Rosenthal, 1931; M.A. Lavrentiev, 1934; M.V. KeVysh, 1945; S.N. Mergelyan, 1951 and others). l

The notion of analytic capacity is useful not only in’approximation. It appears along with its analogues in integral estimates (M.S. Mel’nikov, 1967). Such capacities are used for depcribing the set of removeable singularities of a function (Ahlfors, 1947; . . . E.P. Dolzhenko, 1962; . . . Mattila, 198s). Among the unsolved problems, we draw attention to the problem of the subadditivity of analytic capacity: is it true that for any two compact sets, the capacity of their union is no greater than the. sum of their capacities?

The integral formula of Weil is a generalization of Hilbert’s construction.. Using this formula, A. Weil (1932) showed that on any polynomially convex compact set in~Q=“, each, holomorphic function can be approximated by polynomial.+? Thus in C” as in C’, polynomial approximation reduces to holomorphrc approximation. The integral formula of G.M. Khenkin emerged as a result of attempting to construct holomorphic approximations on arcs., While developing such approximations, the technique of integral formulas: ,found various applications. Nevertheless, the initial question on the possibility of approximating continuous functions on polynomially convex arcs by polynomials remains open.

The possibility of holomorphic approximation has been established for the following cases: arcs having nowhere dense projection on the coordinate planes (E.M. Chirka, 1965); strictly pseudoconvex domains (G.M. Khenkin, 1968); non’ degenerate Weil polyhedra (A.I. Petrosyan, 1970); and C.R.-manifolds (Baouendi and T&es, 1981). There are several examples of compact sets on

10 A.G. Vitushkin

which approximation is not possible. Diederich and Fornaess (19%)* constructed a domain of holomorphy in C2, with C”S-boundary, whose closure 6 not a compact set of holomorphy, i.e. it cannot be represented as the intersection of a decreasing,sequence of domains of hoiomorphy. Moreover, on this domain, one can define a holomorphic function, infinitely differentiable ‘up to the I? boundary of the domain, which cannot be approximated by functions holomorphic on the cIosure of the domain.

For related results, see papers II and III. . :. .I

Above we discussed only the possibility of approximation. There is a lengthy series of works devoted to the explicit construction of approximating functions. . In recent years in connection with applications, there has been a renewed interest in classical rational approximation (continuous fractions, Pad& approximation, etc.). We mention one example concerning rational approximation in connection with the holomorphic continuation of functions. Letf be holo-

morphic on the ball B c C”, and set rk(f) i inf sup 1 f(z)- cp(z)l, where the

infimum is taken over all rational functions cp of dag;i:k. Then; if for each q > 0,

lim rJf)q-t k-m

= 0, then the global analytic function, generated by the elementf,

turns out to be single-valued, i.e., its domain of existence is single-sheeted over C” (A.A. Gonchar, 1974). See. Vol. 8, paper II.

96: Isolating the Non-Holomorphic Part of a Function

Sometimes in order to construct a holomorphic function with given properties,’ one proceeds as follows. One constructs some smooth function cp

roperties and then one breaks up cp as the sum of two functions holomorphic while the second is insome sense small. In this

situation, the first function may turn out to be the function we require. The second term is sought in the form of a solution to the equation af= g, where

II and g = 53. This scheme is used for constructing

in approximation, etc. Equations of the type af= g are called the Cauchy-Riemann equations or &equations.

Let us consider a more general case of the equation aj= g, namely, we shall fake for Q’ a differential (p, q)-form, i.e., a form, having degree p 2 0 in & and degree 4 2 1 in d2. A necessary condition for the solvability of this equation is that the form g be &losed, i.e. & = 0. This is a necessary compatibility condition and SO it is always assumed to be satisfied. The Cauchy-Riemann equations ‘are solvable on each domain of holomorphy (Grothendieck,

I. Remarka& Facts of Complex Analysis 11 0

Dolbeault, 1953). If the domain is bounded and gE L2, then there exists a ’ solution ‘to- the C.-R. equations which ties in L, and is orthogonal to the subspace of &closed (p,q- I)-forms (Morrey, Kohn, Htirmander, 1965). For sirictly pseudoconvex domains thereare explicit formulas for the solution of these equations and estimates on the solution in the uniform norm and in several other metrics (G.M. Khenkin, Grauert, Lieb, 1969). ’

For some simple domains, the question of the possibility of solving the &equations with uniform estimates remains open. For example there are no such estimates on a Siegel domain, also called a generalized unit disc. This is the domain,‘in the n2dimensional space, of square matrices 2 determined by the condition E-Z-Z* 3~ 0, i.e., consisting of matrices Z, for which the indicated expression is a positive d&&e matrix.

To every complex manifold is associated a system of cohomology groups called the Dolbeuult cohomology (1953). The Dolbeault group of type (p, q) is the quotient of the group of &closed (p, q)-forms by the group of &exact (p, q)-forms. In many cases (for example, for compact Kihler manifolds), these groups can be calculated using de Rham cohomology. However, on domains of holomorphy, the Dolbeault cohomology is trivial while the de Rham cohomology may be non-trivial.

Interest in the &equations is also connected to the phenomenon that there is a wide class of differential equations which by a change of variables are transformed to the &equations, and in many cases this yields the possibility of characterizing the sol‘utions of the initial equations in one form or the other. In the general situation, this change of variables leads to the &equations on a surface (the tangential Cauchy-Riemann equations). In these situations the &quations are to be understood as follows: f is called a solution to the equation @= g on the surface M if this equation is fulfilled for all vectors lying in the complex tangent space to M. Each system of linear differential equations in general position, with analytic coefficients, and one unknown function, can be transformed by an analytic change of coordinates to the z-equations (of type (0,l)) on an analytic surface (Rossi, Andreotti, Hill, 1970). Such equations satisfying the natural compatibility conditions, are locally solvable-@pencer, V.P. Palamodov, 1968). If the right-hand side is not analytic, then, such equations are, generally speaking, not solvable. For example, on the sphere in C2, one can give an infinitely differentiable (0, l)-form such that the equation @ = g turns out to be not lo&lly solvable (H. Lewy, 1957). Explicit integral formulae for solutions to the &quations yield criteria for solvability (G.M. Khenkin 1980). Systems of equations with smooth coefficients are, generally speaking, not reduceable to &equations (Nirenberg, 1971). If we extend the class’ of transformations acting on these equations, namely, by adding homogeneous simpletic transformations in the cotangent bundle, hen almostall linear systems of equations with analytic coefficients can be locally to &equations on standard surfaces (!&to, Kawai, Kashiwara, For further results! see paper II.

I

12 A.G. Vitushkin

$7. Construc.tion of Functions with Given Zeros

Let us consider several examples from which it will be’clear how the problems on the zeros of functions arise. The first example is the use of the Weierstrass Theorem on the representation of an entire function of one variable in the form of an infinite product. This theorem has applications in information theory. The formula, regenerating an entire function from its zeros, is used in the problem of encoding signals having finite spectrum. A signal with finite spectrum is a function of time, whose Fourier transform is a function of compact support, i.e. an entire function of time of finite type. The most economic code for such signals is constructed in the following form. It is necessary to compltxify .time and to calculate the zeros of the function. As a code,, the function takes the coordinates of its zeros. Using the zeros we write the infinite product which gives the function and this is the formula regenerating .the original function. It has been shown’that for such coding systems which don’t increase thedensity of the code, it is possible to broaden without limit the dynamic range of the connecting channel or of the reproducing system (V.I. Buslaev, A.G. Vitushkin, 1974). The dynamic range is the ratio between the maximal and minimal signals which are reproduced with a specified accuracy. Codes which specify a wide dynamic range for the amplitude of a signal are required for example in sound recordings.

It is not known whether there exists an analogous coding system for entire functions .of two variables. In this vein we propose a problem. Let us denote by K, the colle:tiion of all sets in C2 each of which is thkintersection of the zero set of some polynomial of degree at most n with the ball lzl I 1. As a metric on K, we take‘the HausdorfT distance between sets. The problem is to calculate the entropy H,(K,). By definition, H,(K,) = log2N,(K,), where N, is the number of elements for a minimal s-net of the compact space K,. The conjecture is that for

small E and large n, W,(K,) x 5 n2 log, i. .

. The next example is related to differential equations. Suppose we are given a

system,of linear diK&ential equations, with constant coefficients and smooth right hand side, defined on a convex domain. Then,, provided certain necessary conditions (of compatibility type) are satisfied by the right-hand side, the system admits a solution on this domain (Ehrenpreis, V.P. Palamodov, Malgrange, 1963). If, for.example, the right-hand side is of compact support then the Fourier transform carries this system to a system of the type CX = F, where F is a vector of entire functions. Here, C is a matrix of polynomials and X is an unknown vector function, By solving this problem and using the inverse Fourier transform, we obtain from X the solution to the oiiginal system. If the system has a solution of compact support, then’F must be divisible by C and this gives the form of the compatibility conditions for the right-hand side. In order that the

I. .Kcmarkab!e Facts of Compiex Asalyzis 13

inverse Fourier transform be defined, one must solve the division problem with estimates on- the growth of the solution at infinity.

Each meromorphic function on C” can be represented as the quotient of two entire functions. This assertion was proved by Weierstrass (1874) for C:‘, by Poincare (1883) for @‘,.and by Cousin (1895) in general. This was essentially the first series of works on several variables and it Iayed the foundations of several directions in complex analysis. The modern theory of cohomology comes from the work of Cousin, while potential theory and the theory of currents stems from the work of Poinsarb.

The statements of Cousin concerning the solvability of certain probicms have come ‘to be called the first and second- Cousin problems. The first Cceosin problem is said to be solvable on some domain or other if it is possible on this domain to construct a meromorphic function with given poles. The second Cousin problem is said to be solvable if it is possible to construct, on this domain, a holomorphic function withgiven zeros. The first Cousin problem is solvable on each domain of holomorphy (Oka, 1937), and’if the group of second cohomology with integer coefficients for the domain is trivial, then the second problem is also solvable (Oka, 1939; Serre; 1953). In C” the second cohomology is trivial, and so, the second problem of Cousin is solvable. Consequently, any meromorphic function in 6” can be represented as the quotient of entire functions.

If M is the set of zeros of a function f; then ,f satisfies the equation of .

PoincarCLelong: k #lnl f 1 = [M],.where [M] is a certain closed (1, 1)-current

of integration on M, taking into account the multiplicity of the dkisor M. A current is a generalized differential form, i.e. a linear functional on forms of compact support of complementary degree. A current is called closed if it is zero on exact forms of compact support. The current [M] is integration on M of the product of a test function and the multiplicity of the divisor M. The formula of

’ Lelong, giving a solution of this equation in @“, yields a completely accurate estimatebn’the speed of growth of an entire function depending on the density of its set of zeros (1953). For example, dn (n -. 1)-dimensional closed analytic subset of @” 1s algebraic (i.e. is the zero set of a polynomial) if and only if the (2n -2) dimensional measure of the intersection of this set with an arbitrary ball of radius r can be estimated from above by the quantity C*;2n-2, where C is independent of j (Rutishiiuser, Lelong, Stoll, 1953).

Currents were introduced by de Rham. The Lelong theory of closed currents was one of the fundamental tools in the research on analytic sets (Griffiths, 1973;..., E.M. Chirka, 1982). and plurisubharmonic functions (Josefson, 1978; Bedford, 1979; A. Sadullaev, 1981). Currents are practical in that they allow one to carry delicate problems on analytic sets over to standard estimates on integrals. Using this technique, Marvey and Lawson (1974) showed that if a (2k + l)-dimensional smooth submanifold M c C” has at each point a complex

J.G. Yang

J.G. Yang

J.G. Yang

J.G. Yang

J.G. Yang

14 A.G. Vitushkin

tangent space of maximal possible dimension, ie. dimension UC, and if M is pseudo-convex, then M is the boundary of a (2k + 2)dimensional analytic subset which (together with M) is the envelope of holomorphy of M.

For related results, see articles II, III; Vol. 8, article II; and Vol. 9, articles’& II! ‘- I

$8. Stein’Manifolds

A great deal of what we have discussed on domains of holomorphy carries over to manifolds which are called Stein manifolds (195 1). A complex manifold M is called a Stein manijbld ic first of all, it is holomorphicaliy convex, i.e. if the holomorphically convex hull of each compact set in M is compact in M, and secondly, if there exists on M a’finite set of holomorphic functions such that each point of M has a neighbourhood in which these functions separate points. Each domain of holomorphy is clearly a Stein manifold. Closed complex submanifolds of C” are also Stein manifolds. Conversely, each Stein manifold M can be realized as such a submanifold, i.e., M can be imbeddcd into C” by a proper holomorphic ‘mapping (Remmert, 1957).

It is not hard to see that a bounded Weil polyhedron can be imbedded in the polydisc in such a way that its boundary lies on the boundary of the polydisc. It turns out that the ball can also be realized as a closed complex submanifold of a polydisc. (A.B. Aleksandrov, 1984). However, there exists a bounded domain with smooth boundary not admitting such a realization (&bony, 1985).

A great achievement in complex analysis was the solution of Whitney’,s problem. It has been shown that any real analytic manifold can be analytically imbedded in a real Euclidean space of sufficiently high dimension (Morrey, Grauert, 1958). On a given analytic manifold S we fix an atlas After complex-, ifying the charts of this atlas, that is, allowing the coordinates to take not only real but also complex values with small imaginary part, we may consider S as a submanifold of some con$ex manifold M. For an appropriate choice of metric on M, it turns out that Oneighbourhoods .M, of S are strictly pseudoconvex domains for small E. The crucial moment in the construction is the general, ization of Oka’s theorem. Namely it is shown that on a complex

I: @fold, each

strictly pseudoconvex domain is holomorphically eonvex. From he hc$omorphic convexity of M,, it follows easily that for small s, M, turns out to be a Stein manifold. By Remmert’s theorem, Me-can be imbe-dded in C”, and by the same token, S turns out to be imbedded m R*“.

On Stein manifolds just as on domains of holomorphy the‘ problems 4 interpolation and division are solvable, the &problem is solvable for arbitrae type (p, $); the.first Cousin problem is solvable and the second Cousin problem is also solvable provided the second integer cohomology group is trivial. A.U of these problems are solved by one and the same scheme. Let us look, at this

ic :


6 theme for the case of solving the a-problem of type (0,l). First of all one solves the problem locally, i.e. we fix a covering of the manifold such that the equations are solvable on each set of the covering. If an element of the covering is, for example, a ball or a polydisc, then one can give an explicit formula for the solution. On the intersection of two elements of the cover, the local solutions may not agree, i.e., their difference may not be zero. The next stage in constructing a solution consists in determining “correcting factors”, in this particular situation, holdmorphic functions, defined on elements of the cover and who&difference on any intersection of elements of the cover is the same as for the local solutions constructed above. If such “correcting factors” exist, then subtracting these correcting factors from the corresponding local solutions we obtain new local solutions which agree on intersecting elements of the cover and hence yield a global solution to the equation. The difference of two local solutions on the intersection of two elements of the cover is a one dimensional closed cocycle of holomorphic functions. The existence of the desired correctind factors amounts to the exactness of this cocycle. Thus, on a given manifold, the &equation is solvable for any choice of the right hand side if and only if the one- dimensional cohomology with holomorphic coefficients (or, as they say, with coefficients in the sheaf,of germs of holomorphic functions) is trivial.

The triviality of this group as well as that of other one-dimensional cohomology groups, corresponding to the above enumerated problems, follows from a theorem of H. Cartan (1953): a complex manifold is a Stein manifold if and only if its one dimensional cohomology group with coefficients in an arbitrary

, coherent analytic sheaf is always trivial. Locally a coherent analytic sheaf is a special type of subspace of the space of germs of holpmorphic vector-valued functions on the given manifold or on some submanifold thereof. It can be, for example, the space of germs of vector-valued holomorphic functions itself, the subspace of germs having a given set of zeros, the quotient space of the first sheaf by the second, etc. The theorem of Cartan systematizes the material on domains Rf, holomorphy accumulated till the early 50’s. &tan’s theorem successfully combines the results and techniques of Oka with Leray’s theory of analytic sheaves (1945). The next step in the development of cohomology theory was the . theorem of Grauert (1958) which has come to be called the Oka-Grauert principle.

Let M be a complex manifold. Let o be a collection of domains in M forming a cover of M. For each two intersecting elements of this cover c(, p E o, let C,, be a non-singular square matrix of degree n consisting of functions defined and holomorphic on the intersection a 6 /?. We suppose that the collection {Car)i forms a cocycle; in other words, they are compatible on a triple intersection, i.e.’

Caa c,, cw = I on an /? n y. Such a collection of matrices of functions is calle initial data for the Cousin problem for.matrices. We say that the second Cousi i problem with initi’al data {Cma) is solvable if one cab. find a collection (C,} of matrices of functions defined on o and such that on each non-empty intersection an/3,wehaveC;Cg’ = Car. The Oka-Grauert printiple states that on a Stein

/

16 A.G. Vitushkin

manifold, the second Cousin problem with given initial data has a solution {CA}, where the C, are matrices of holomorphic functions, provided it has a solution where the C, are matrices of continuous functions. This was proved by Oka (1939) in the case where n = 1 and M is a,domain of holomarphy. He made use of this construction in order to find functions with given zeros.

. Grauert’s theorem has various applications. For example, Griffiths (1975) while working on a problem of Hodge obtained from this theorem that on a Stein’ manifold every class of even-dimensional cohomology with rational coefficients can be realized as a closed complex submanifold. For related results, see Vol. 10, articles I and II.

59. Deformations of Complek Structure

To specify a complex &&re on a manifold means to specify an atlas with holomorphic transition functions. A deformation of complex structure is a new complex structure obtained from the given one by modifying the trapsition functions. For example, the extended complex plane can be considered as the two-dimensional sphere with a complex structure. This complex structure does not admit*deformation, i.e. on S, the complex structure is unique. On the. Zn-dimensional sphere S2,,, for n # 1,3, it is in general not possible to introduce a complex structure (Borel, Serre, 1951). It remains unknown, whether one can introduce a complex structure on the six-dimensional sphere.

The various structures of a compact complex one-dimensional manifold’ of genus g(g > 1) form a manifold of real dimension 6g - 6 (Riemann, 1857). This manifold of structures has itself acomplex structure which can be obtained by factoring by a discrete group on a bounded domain of holomorphy in CJBs3 (Ahlfors, 1953).

Small deformations of the structure of a compact complex manifold of arbitrary dimension can also be parametrized as the points of a complex space which can be realized as an analytic subset of C” (Kuranishi, 1964). Compact manifolds, obtained by factoring some simple domain (for example, a ball) by a discrete subgroup, have a rigid structure, i.e. do not admit small deformations of structure. Moreover, if two such spaces are topologically equivalent, then they .are also holomorphically equivalent (Mostov, 1973). For non-compact manifolds, the space of structures is as a ‘rule infinite dimensional.

In order ‘to discuss deformations of bundles, we recall that an n-dimensional vector bundle is a bundle for which th.e fibre is the space C”. The structure of such a bundle is ,g+en by the cocycle of transition matrices {C., f (of dimension n), defined on the intersections {a n R} of elements of a cover of the base. If the base is a complex manifold and the.transition matrices { Caa} are holomorphtc, then the bundle is called holomorphic. For example, the tangent bundle of a complex manifold M is a holomorphic vector bundle of rank n. The space of

I. Remarkable Facts of Complex Andysis 17

tuples zo, . . . , z, on n-dimensional projective space CP” cai be considered as a one-dimensional holomorphic bundle over CP”. In this case a fibre is the collection of all tuples which can be obtained from each other by.multiplication by a complex number. Two bundles X and X*..on one and the same base M are called equivalent if one can find a homeomorphic mapping of X into X*, carrying fibres to fibres, acting linearly on each fibre, and fixing each point of M. If there exists a holomorphic mapping having the above properties, then the bundles are called holomorphically equivalent. A deformation of structure of a .bundle is a new bundle not holomorphically equivalent to the given one but , obtained from it by varying the transition matrices. From the Oka-Grauert i principle it follows that a bundle whose base is Stein has a rigid structure and moreover from the topological%equivalence of such bundles follows the holomorphic equivalence (Grauert, 1959). On Stein manifolds there are, so to speak, as many holomorphic bundles as continuous ones. If the base is not Stein, then it i is usually not so; and this is good..Sometimes a complicated manifold can be interpreted as a space of deformations of a bundle thus yielding significant - ’ information concerning the initial object. I I

The twistor theory of Penrose(l967) is founded on such reductions. The ideal of the genera1 plan of Penrose can be taken as follows. Several notions 04 mathematical physics can be interpreted in terms of complex structure. For’ example, the metric on Minkowski space satisfying the Einstein equations, i.e. the gravitational field can be interpreted as a holomorphic structure on a domain in CP3. More precisely, there is a one-to-one correspondence between conformal classes of autodual solutions of the Einstein equations and ‘deformations of structure of domains in @P” (Penrose, 1976). A structure of a domain is the same as a choice of functions of three variables. The Penrose trans- 1 formation, associating to each choice of functions a solution of the Einstein equations, has a sufficiently simple form and hence allows one to write down many solutions to these equations.

There is an analogous correspondence between autodual solutions of the Yang-Mills equations and bundles of rank two over domains in CP’ (Ward, 1977). In this direction the class of so-called instanton and monopole solutions

___ ~_ -

to the Yang-Mills’equations have been obtained which present interest for theoretical physics (Ward, Hitchin, Atiyah, . . . , Ju. I. Manin, 1978). Because of this series of works, this area of mathematics, which Penrose calls “the complex geometry of the real world”; has become very popular. For related results see Vol. 9, article VII and Vol. 10, articles II and III. %

Within the limits of this article, we have restricted ourselves to discussing only a few of the outstanding facts from complex analysis. Unfortunately, we have not touched upon several major areas: the theory of residues (see article V and Vol. 8, article l), the theory of singularities (see Vol. 10, article III), and value distribution (see Vol. 9, articles II-IV). A broad overview with the corresponding bibliographies for the various areas of multidimensional complex analysis is given in the series.of articles in volumes 7-10 of this series.

.

I

Translated from the Russian ’ by P.M. Gauthier

, Cotitents *

$0. Introduction. ............................................ 9.1. Fundame& Problems. ............................... 0.2. ASurveyofResults..; ..:. ...........................

.. The Bochner-Martinelli Formulas and Their Applications ....... 1 .l. The Bochner-Martinelli Formula and the Hartogs Theorem . . 1.2. The Integral Representations of Bochner and Hua Loo-keng

on Classical Domains .................... , .:‘: . : ....... $2. The Weil Formula and the Oka-Cartan Theory ...............

2.1. Integral Representations in Analytic Polyhedra ............ 2 2. Solution of “Fundamental Problems” in Domains of

Holomorphy ....................................... 93. Integral Formulae and the Problem’of E. Levi ................

3.1. Pseudoconvex Domains. Theorems of E. Levi and H. Lewy ... 3.2. Oka’s Solution to the Levi Problem ...................... 3.3. Applications and Generalizations of Oka’s Theorems. .......

94. The Cauchy-Fantappie Formulas .......................... 4.1. The Formulas of Cauchy-Leray and Cauchy-Waelbroeck ....

. 4.2. Multidimensional Analogues of the Cauchy-Green Formula . . 4.3. Integral Representations in Strictly Pseudoconvex Domains. .. 4.4. The Theorem of FantappiiLMartineau on Analytic Function& 4.5. The Cauchy-Fantappie Formula in Domains with Piecewise-

Smooth Boundaries. .................................. 4.6. Integral Representations in Pseudoconvex Polyhedra and

Siegel Domains .....................................

20 20 24 27 27

31 35

‘35

38 41 41 44 47 49 49. 51 52 53

55

47

20 GM. Khenkin

$5. In&ral Representations in Problems from the Theory of Functions on Pseudoconvex Domains . . . . . . . . . . . . . . . . . . . . . . . . . ..a... 5.1. Estimates for Integrals of Cauchy-Fantappit Type and

r --. Asymptotics of Szegii and Bergman Kernels in Strictly Pseudo- convex Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , .

5.2. Localization of Singularities and Uniform Approximation- of Bounded Holomorphic Functions. . . . . . . . . . . . . . . . . . . . . , .

5.3. Interpolation and Division with Uniform Estimates . . . . . . . , , $6. Formulas for Solving &Equations in Pseudoconvex Domains and

, Their. Applications . . . . . . . . . . . . . . . . . . . . . . . ,. . . . . . . . . . , . . . . a.% 6.1. Th&&Equations. The Theorem of Dolbeault . . . . .‘. . , . . . . , ,

6.2. Problems of Cousin and PoincarC as &Equations. Currents of Lelong and Schwartz. . . . . . . . . . . . . . . . . . . . . . . , . . , , . . .

6.3. The &Problem of Neuman-Spencer . . . . . . , . . : . . . . . . . . . . . 6.4. Formulas for Solving &Equations. . . . . . . . . . . . . . . . . . . . . . . 6.5. The PoincarC-Lelong Equation. Construction of Holomorphic

Functions with Given Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . , $7. Integral Representations in the Theory of CR-Functions. . . . , 1 . . ,

7.1. Approximation and Analytic Representation of CR-Functions. 7.2. CR-Functions and the “Edge-of-the-Wedge” Theorems . . . . . , 7.3. Holomorphic Continuation of CR-Functions Given on Concave

CR-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.. The Phenomena of Hartogs-Bochnerand H. Lewy on l-Concave

CR-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. Analytic Discs and the Holomorphic Hufl of a CR-Manifold. .

$8. The %Cohomology of p-Convex and q-Concave Manifolds and the Radon-Penrose Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. I’he &Cohomology. Theorems of Andreotti and Grauert , . . . . 8.2. I_ntegral Representations .of Differential Forms and

%Cohomology with Uniform Estimates . , . . . . . : . . . . . . . . . 8.3. The Cauchy-Riemann Equations on q-Concave CR-Manifolds 8.4. The Radon-Penrose Transform. . . . . . . . . . . , . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

61

64 66’

69 69

70 72 74

77 80 81 85

87

91 94

96 96

101 104 IO? 112

\ 60. Introduction

0.1. Fundamental Problems. Let D be a domain,in the complex plane Cl with rectifiable boundary JD ahd f a complex valued function, continuous on fi together with its Cauchy-Riemann derivative:’

$=t(g+i$),’ 2=x+@.

1: The Method of Intcgd m in Camp&x Andyi 21

The fofmula Or~Cauchy-Graen-Pompu (1904, rae [33]! [40], [453) has the form t

This formula becomes the classical Cauchy formula in case j’ is’ a Momorphic fin&on, that is when df/i% = 0 on D.

The &why- and Cauchy-Green formulae ate fundam&tal technical tools in ‘the theory of functions of one complex vstiable. Examples of profound applications of these formulas are given ‘in the works of Carlesoa [26] and Vitushkin [SO]. In the first of these, the famous “Corona” problem fOF. the disc in C’ is solvedAt the second is solved the problem, going back to Weierstrass and Runge, on the uniform approximation by holomorphi functions on compact sets in Cl.

Till the beginning of the thirties, the only multidimensional integral formula was the Cauchy formula for a polydomain D = D1 x . . , x D, in C”, where each D, is a bounded domain in C’ with rectifiable boundary.

Let f be a function continuous on D and holomorphic in 0; i.e., 8f = 0, where

i?=~y&dZ,

is the CauchyrRiemann operator. Then the Cauchy formula (1841, see [4 3) holds

This formula allows one to prove the fundamental properties of hdomorphti functions of several variables, for example, ‘the-local representation of holomorphic functions by power series, the property of uniqueness of analytic continuation, etc.

Using the classical Cauchy formula,‘Hartogs (1906, see [Sl J, [23]) showed that in C”, for n > 1, there is a domain D such that each function holomorphic on D' necessarily has a holomorphic cxmtinuation to some large domain n 3 Dl

For example, any function .$ holomorphic in the domain

12”,u~~={z:~z,~<1+&,1-~c~Z~~<1+E}u

u{z:(z,~<l+&,~Z1-ll<E

contained ‘in an s-neighbourhood of the boundary of the bidisc @ = {z:lz, I < 1, lzzl < 1), has a holomorphic continuation to the bidisc fY = ( z: I zl 1 < 1 + E, lzz I -z 1 + s}, given by the ,formula

.

$2 _. .; _ .‘. .,,G.M,:~:. i i 1 . -f

-_

. ~.fPQi~~i~bhOhMQ’p&c iaj)l hi! k?ogshd*agreeg with f in Rci by the &U&Y formula, &p”fi~Ily,. agrees tit&i& G1 by & uniqueness &tom,. 1

Paid (1907, se% 1831) using ihe expansion pf a function on a sphere by spherical harmonics, sho%%l that ‘qh funetion, ,holomorphic in a neighbour- bdod of the boundary of-a ball in @,exte,@ hoiamd;rph~~to’the interior of this balL

. . (’ -_

The more geperal theoT,of Hartogs gssertsrhqt XD is a.hoW domajo in C”(n 2 2) with connected complement C”.\D, then any function.h&morp&h a neighbourhood *of Jo, uniquely ~&ends to a holomo&ii function m D. VW Hqyever, a ngour+s proof of this: asseffjon was, obtained,only ip 1936 by

, Brown.(see 1233). ’ ./ .;.I‘: “:, (-St, .- f& ” .‘J .:;, “. .- These phenon?eaa Iwi Hartogs to.the,&#o~ng definitioni A.domah Din 6” is called a domain of bti@y if there goes mt.exist .a larger domain Q ? D to which eve+ holomorp& fuoction oa:D ex&nds hofomorphically.

Using the cla@cal Cauchy formula as in the above examples, Hartogsshowed that for any domain- qf holomorphy in C”, the follow& contin& ppi@ple by= cwi, c233).

vrn 0.1 (Hartogs, 1$9). Let fZbe a domain ofholomorphy. Zf(D j ban arbitrary sequence bf analytic discs whose closures ‘are ‘contained in R a& such

that liin D, = D,,, 80, r-m

c 0,‘ where Do is an analytic disc, then D, c R.

Here, an amdytie disc means the holomorphic imbeding in C” of the unit disc in Cl:

8 : :. E Levi (1910) introdud a notion of domain of meromorphy. He proved that the Hartogs continuity principle is valid for domains of meromorphy as well (see E23 3, [ 10 3). Following Hartogs.and Levi a domain in C” is calkd.pseudoco~x if tl+ continuity principle from theorem 0.1 is valid.for it. .

j ,E Levi (1911) formulated a natural problem: to show that ea& mudocoovex domain is a domain of holomoehy. This problem turned out to be eSze&ingly dii%cult and tias solved by Oka only in 1942 (+!e 53).

The .other principal problem of multidimensional complex analysis was the problem, going back to Weierstrass and Poincarb, of representing an arbitrary

I *meromorphic function in a domain Q c CR as a quotient of functions holomorphic in C”. It is known (see [65]) that the solution to this problem would haie been most welcome to Weierstrass in constructing the theory of Abelian functions. Id order to solve this problem, Weierstrass and Poincati thought it

, was necessary to have a complete description for the structure of the zeros and poles 6f meromorphic functionS in C”. . . ).

The local properties of the zero sets of holomorphic functions are described by tb following result of.We$strti (see C331).

Tbeo& 0.2 (Weierstrass, 1879). Letf 4 f(z,, . . . , z,) be afunction holomorphic in a neighbourhood of zero in C” and suppose the function f(0, z,) .I;’ is

II. The Method of Integral Representations in Complex Analysis

holomorphic and different from zero at zero. Then, the function f can be rep resented uniquely in the form f = g * h, where h is holomorphic in a neighbourhood of zero, h(0) 2 0, and the function g (a Weierstrass polynomial) is of the for&

k-l

g(Z) = Z”, + jgo Sj(z’)’ z!t

where the functions gj(z’), z’ = (zl, . . . , z, _ i ), are holokorphic in a neighbour-

hood of zero and g,(O) = O.-

Let Q be a domain in @“? f a function meromorphic in-R, and M + and M _ the sets of zeros and poles respectively off in Q. From Theorem 0.2 it follows that the sets M * are the closure of the set of their regular points and the regular points split into irreducible complex (n - 1 wimensional components { M:,}.

, Let us denote by y-$ the multiplicity of the zeros (y:.> 0) or poles (if; < 0) of the function f on a component Mf (see Article III of the present tilume).

From Theorem 0.2 follows the following integral representation of Poincark (1883, 1898) for log 1 f I:

loglfl=G+H, (0.3)

where H is a harmonic functions on 0 and G is a potential of the form \

G=C,xy,f j’ I~-z12-2ndh-,(C), , Y - Mf

where d V, is the k-dimensiona! vollime form on k-dimensional submanifolds of’ @” (see [73], [J2]).

The representation of a meromorphic function f in C” as a quotient qf entire functions was accomplished by Weierstrass for n = 1 (in 1876), by Poincarcfor l

n’= 2 (in 1883), and in the genfral case by Cousin (in 1895) as a consequence of the following result.

Theo&m 0.3 (Weierstrass, Poinck, Cousin, 1895, see-[23], [32], [33]). Let M; be an (n - l)-dimensional analytic set in C”, {M: } its regular (n 7 l)- dimensional components, and (7: } the inultiplicities of these components. Then, ~ there exists an entire function f+ whose zeros are on M+ with multiplicity y:,v=1,2 ,....

The approach of Poincark, which allowed him to obtain only the case n = 2, consisted in constructing an entire function satisfying equation (0.3).

‘The original approach of Cousin consisted in the following successful generalization of the Mittag-Leffler Theorem (1877).

Theorem 0.4 (Cousin 1895, see [23], [32], c-333). Let (‘klj}, jEJ, be a locany

finite cover of a polydomain D = D, x . . . x D, c C”, where each domain Dk, with the possible exception of one, is simply connected in C.l. L+ ( pj} be a family of functions meromorphic on { Q,} and such that the functions { cpi - ‘pi > (respectively ‘pi ‘p; ’ ) are holomorphic on ( Q n Q, >, i, jEJ. Then there exists a meromorphic

function cp on D such that all functions ( rp - cp,) (respectively (cp/cp,)* ’ are holomorphic on { fi,}, je J. .

For the proof of Theorem 0.4, using the classical Cauchy formula, Cousin established the following fundamental result on the splitting of singularities of holomorphic functions of several complex variables.

Theorem O.!i (Cousin, 1895, see [23], [32], [33]). C” containing the origin. Set

Let D be a polydomain in .

’

fJ,=(zeD:Imz,>-s};

R, = {zcD: Imz, <E}

‘S.7 { zeD: Imz, = 0}, E > 0.

Then, any function f, holomorphic in the domain R, A$& can be represented on S in the form $ = fi - fz, where fl and f2 are functim holomorphic in the domains R, end& respectively.

.--/ The problems of extending the assertions of Theorems 0.3 and 0.4 to general domains of holomorphy in C” came to be called the problems of PoincarP and Cousin respectively (see [lo]). Just as in the case of the Levi problem, no

--progress was made on these problems until the works of A. Wejl, H. Cartan and Oka,(see [lo]).*

.~_ In the end;tht principal difficulties arose (see [ 101) in attempts to extend to several complex variables, the result of Runge (1885) on the representation of an

-arbitrary analytic function o? a domain R c Cl as the limit of a sequence of rational functions uniformly converging in R.

0.2. A Survey of Results. The reason for difficulties in proving the general Hartogs Theorem, solving the Levi problem, the Cousin and Poincare problems, obtaining a multidimensional analog of Runge’s approximation theorem, and

- ‘solving ‘a series of other problems in the theory of functions of several complex -variables was tied to the absence of a natural analog in several variables of the

<Cauchy formu1.a for a more general class of domains than polydiscs. Finally, in the, thirties two sufficiently general formulas were found which

lplayed a remarkable role in the development of multidimensional complex snalysis’(A. Weil, 1932; Bergman 1934; Martinelli, 1938; Bochner, 1943).

A simple formula with non-holomorphic kernel, found by Martinelli and Bochner, allowed one to give a simple proof of the Hartogs-Brown Theorem and to obtain a generalization of this theorem leading to the theory of CR- functions. These results are dealt with in section $1,

A deeper formula, with holomorphic kernal, found by A. Weil and Bergman ‘for functions holomorphic in analytic polyhedra allowed one: first of all, to obtain a multidimensional Runge approximation the.orem (A. Weil, 1932); secondly, to solve the problems of Cousin, Poincare and Levi (R Cartan, 1934;

. .

II. The Method of Integral Representations in Complex Analysis ’

25

Oka, 1936,~1953; Norguet, 1954; Bremermann, 1954); and thirdly, to give a fundamental formulation for the Oka and H. Cartan’s, theory of ideals of analytic functions, 1940-1950.

In this-connection Oka (1953) remarked that in the theory of functions of several complex variables, for a long time problems were only accumulated, A. Weil made the first step in the opposite direction, i.e. in the direction of solvingthese problems. We remark that the majority of the following steps were made by Oka himself. An account of the results related to Weil’s formula is given in sections $2 and-43.

In the fifties H. Cartan (1951) and Grauert (1958) using the sheaf theory of Leray not only obtained far reaching generalizations of Oka’s theorems but also completely banished the constructive method of integral representations from multidimensional complex analysis (see [ 333).

In the sixties further developments in the Oka-Cartan-Grauert theory took place essentially thanks to the methods of a priori L*-estimates for the &Neuman-Spencer problems, developed in the works of Morrey (1958), Kohn (1963)‘and Hiirmander (1965) (see [22J, [45], and below 86).

The constructive method;‘of integral representations was revived anew essentially in the seventies. Here the source of fundamehtal new ideas was the works of Leray (1956) and Lelong (1953).

In particular, Leray starting from an idea of Fantappie found a completely ’ general integral formula for functions holomorphic on arbitrary domains, which as became clear later, contains in itself the cases of the formulas of Bochner, Martinelli, Weil, and others.

.’

The formula of Cauchy-FantappibLeray allowed one to construct (Grauert, G.M. Khenkin, Lieb, Ramirex, 1969) a good integ?al formula, in strictly pseudoconvex domains D of C”, for holomorphic functions and for solutions f of the non-homogeneous Cauchy-Riemann equation: 8f = 8, where (J is a (0, l)-form’in D (= Cl91, C-U. .

These integral representations led to the theorem on uniform ap&oximation by functions holomorphic on the compact set D (G.M. Khenkin, 1969,1974); to the formula for the continuation, .with uniform estimates, of functions holomorphic on submanifolds af D (GM. Khenkin, Leiterer, Amar, 1972, 1980); to decompositions of functions in an ideal in terms of generat6rs (Berndtsson, 1982). etc. (see [ 11 J and [40]). The& results are explained in #4,5,6.

Lelong transformed the Poincare equation (0.3) to a &equation of the form:

adf - = i[A4], f

called the PoincarbLelong equation, and he found for it a fundamental integral formula, analogous to the canonical Weierstrass product, for entire functions with finite order of growth and having zeros on a given (n - 1)dimensional analytic set in C”. A similar construction was obtained independently by Stoll (1953), based. on the work of Kneser (1938) (see [73], [32] and [78]).

26 G.M. Khenkin

In the elaboration of these works certain integral representations were found (G.M. Khenkin, Skoda, Sh.A. Dautov, P:L. Polyakov, Charpentier, 1975-1984) for solutions of the &equations on strictly pseudoconvex domains and analytic polyhedra which’allowed a complete description of the zero sets of functions of I finite order on these domains (see [49] and [41]). A survey .of these results is given in $6,

Integral representation is strictly pseudoconvex domains were also.successfully applied (Kerzman, Stein, Ligocka, Lieb, Range, 1978-1984) towards con-

1 strutting a “parametrix” for the &Neuman-Spencer problem and to obtain the asympi’otic expansion of Fefferman (1974) forthe Bergman and Szego kernels in such domains (see $6).

The method of integral representatians turned out to be particularly fruitful in the theory of CR-functions, i.e., functions on real submanifolds of C” satisfying the tangential Cauchy-Riemann ‘equations. This. theory, which has various applications to differentiarequations and mathematical physics, previously contained (see 6821) disparate although effective results (S.N. “Bernstein, Bochner, H. Dewy, Bishop, Rossi, Harvey, Wells, Lawson, Sibony, ,1912-1977)

In recent. years, thanks to the construction of integral representations forCR- functions (Bawdi, GM Khenkin, T&es, 1980, 1981), the theory .of CR- functions has received significant growth: holomorphic approximation [8]; analytic representation [38];’ uniqueness theorems [2], [8]; propagation of singularities [34]; holomorphic continuation [39], [2], [lS], [7]; and.theorems on. the “‘edge of the wedge: for CR-functions [2], [3].

A survey of these results is given in $7 of the present article (see also [3]). . In sections l-7 of the present work, then, the met.hod of integral represen-

tations is set forth only in applications to problems of the theory of functions on domains in the space @” and on submanifolds thereof. In this connection primary attention is given to those classical problems of the theory of functions which in the first place were solv$d or arose in the context of integral representations of holomorphic functions and which once again have developed in recent years.

However, the method of integral representations works effectively also outside the framework,of classical functions theory in C”.

First of all, the above mentioned results on holomorphic functions and CR- functions have meaningful generalizations to the case of differential forms of arbitrary type.

In particular, explicit formulae were found (Lieb, Plvrelid, P.L. Polyakov, ;:,,Range, Siu, Sh.A. Dautov, A.B. Romanov, 1970-1975) for solutions of.the

equation ?jf= g, where f and g are differential forms of respective types (0, q) and (0,‘q-f 1) on piecewise pseudoconvex domains, analytic polyhedra, .or their

: Soundaries. An essential role here was played by the analog, for differential forms, of the Bochner-Martinelli formula, constructed earlier by Koppelman ($967) (see [l J, [54], [49], [4O],.and &8 of the present article). r

I . . \

II. The Method of Integral Representations in Complex Analysis 27 .

I Certain problems in complex integral geometry (Andreotti, Norguet, Penrose,

‘, 1967-1976) and. in the theory of tangential Cauchy-Riemann equations (H. Lewy, Kohn,.Andreotti, Hill, Naruki, 1957-1972) required integral representations for (0, q)-forms not on’ pseudoconvex domains in C” but rather on

/ q-pseudoconcave domains in Cp” and on q-pseudoconcave CR-manifolds.

The construction of integral formulae on such domains and manifolds ’ (G.M. Khenkin, S.G. Gindikin, P.L. Polyakov, 1977-1984) led to an. exact .cr.iterion for the local and global solvability of the inhomogeneous Cauchy-

: x Ri&ann equations on q-concave domains and on q-concave CR-manifolds, and also led to the description of the Radon-Penrose transform and its kernel (see [49], [28], [52], [49] and [51]). Section 8 of the present article is devoted to these questions.

_ .

The author extends his profound gratitude to P. Lelong, A.G. Vitushkin, S.G. Gindikin, and P.L. Polyakov for their interest in the present work and their help in its realization.

5 1. The Bochner-Martinelli Formulas and Their -Applicatiqns _’

1.1. The Bochner-MartinelK Formula and the Hartogs Theomm. Let D be a

bounded domain in C” with “rectifiable” boundary aD (i.e., the (2n- l)-dimensional Hausdorff measure of aD is finite). We choose the orientation of the space C” such that

( - i)” (I, pJi0 * 40 > 09 where o(c) = dc, A . . . A dc,. ’ . .

For vectors i, q E C”, we set

In most integral representations, a differential form o’ on C” is used which has the form +

.d(tjj = i (.-I>“-’ tjkdql /\ . . ../\ dqk-l A dqk+l /\ . . .,A dq,, k=l

and satisfies the equation do’(q) = nap. We consider the spaces LP(i3D) and Z,“(D), consisting of functions whose pth

powers are integrable with respect to the respective Lebesgue measures on dD and D.

Let W’(8D) (respectively W(D)) be the subspaces of LP(aD)) (respectively LJ’(Dj)having a holomorphic continuation to (respectively, which are holomorphic in) D.

_. ‘- .f

.

2d I .

GM. Khcnkin L 4

:’

Let s(D) denote the space of all holimorphic functions on D and C(D) the snace of all continuous functions on D.

By C@‘(D) we denote the space of all continuous functions on d which have continuous derivatives up to order [ct] inclusively, satisfying a Lipschitz condition of order a - [a].

We set d@‘(D) F C(“)(d) A X(D) &d A@) = C(a) n X(D).

Theorem 1.1 (Martinelli, Bochner, 1943). able

For any bounded domain D with boundary and $01 every function f such that f and 8f are continuous

m we have the equality

1 , (1.1) This formula was proved, first of all, by Martinelli (1938) for the ball in @”

and then for general domains, by Martinelli and Bochner (1943), for 8f = 0, and by tippelman (1967), when af # 0 (see [4] and [40]).

the casd

To prove formula 1.f it is sufficient to apply the Cauchy-Green formula (0.1) to f on each complex line passing through z, and then to average the equalities so obtained (see [313).

! Nf is a holomorphic function on D, then formula (1.1) becomes the classical integral representation of Martinelli-Bochner

ZED.

I

(1.2)

We shall consider that a domain\D has a smooth bound&y if

D = {zE@“:P(z) < 0}, . (1.3)

where p is a function of class C’(P) with the.property grad p ,# 0 on dD. For domains with smooth boundary, Bochner obtained the equality (1.2)

irom the Green formula for a function F harmonic on D:

qz) - @- l)! d

F(C)- It” b IC-zl’”

. where

-$=($,..+, ZED. ’

From this formula, it follows that the Martinelli-Bochner formula (1.2) holds’ if and only if F is harmonic on D and satisfies the boundary &condition of

II. The Method of Integral Representations in Complex Analysis

Neuman-Spencer

($(g),$g)) =OonaD.

29

(1.4)

The &condition of Neuman (1.4) forces an arbitrary harmonic function to be holomorphic; since

(in this connection, see [22]). ) As a consequence we have the following (see [4]).

Theorem 1.1’ (A.M. Aronov, A.M. Kytmanov, L.A. Ajzenberg, 1975). If for

some function fe C(D), the Martinelli-Bochnner formula (1.2) is satisfied, then fH@.

The Martinelli-Bochner formula tias used by the authors to prove a more general and sharper version of the Hartogs Theorem (see [4] and [19]).

Theorem 1.2 (Bochner, 1943; Weinstock, 1970). Let D be a bounded domain

with rectifiabIe boundary and with connected complement in @“, n > 1, and let fE L’(aD). Then a necessary and suficient condition in order that f extends holomorphically to D is that f satisfy the tangential Cauchy-Reimann equations: &f=O.

In case f is a smooth function and the domain 0 has a smooth boundary, then the equation &f = 0 signifies the equation

~~~$=OonaD, (l-5)

where Jis any smooth extension off to a neighbourhood of aD. In the general situation, the equation 8.. f = 0 is to be understood in the

generalized sense: (1.6)

for any smooth differential form cp of order (2n- 2). Theorem 1.2 is proved by the following scheme. Let f be a ‘fixed integrable

function on aD and set

gJDf(O 463 A w(t;) =

IC - z12” I

f+, Y’zED, f-, ifz&“\D.

(1.7)

We have the following analog of the classical theorem of Sokhotskij bee C41, C491, E351).

) f =f+,m-f-IMP (W

30

,

GM. Khenkin

Moreover, from the condition g= 0 on aD it follows that the functionsfk . are holomorphic in z despite the non-holomorphicity of the Martinelli-Bochner

_ kernel. Indeed, -

‘Finally, to complete the proof of sufficiency in Theorem 1.2, we must verify that f- 3 0 in @“\D. Because of the connectedness of the domain C”\D, it is ‘ sufficient to establish this equality for ZE@“\R, where n is some polydisc containing the domain D. .-

Noting that for z E @“\Q, the form w’(f- Z)/}C - zlz) A o(f) is &exact in the polydisc R, i.e. has the form &cp,(c), where cp,(C) is some (2n-2)-form, we deduce. from equations (1.6) and (1.7) that f- (z) = 0. The assertion of Theorem 1.2 follows from this and from equation 1.8.

Theorem 1.2 has an explicitly geometrical reformulation: under the hypotheses of this theorem, the (2n - l)-dimensional graph

is the boundary of an n-dimensipnal. manifojd

G,= {(w,z)E@"+':w =f+(z),z~D} if and only if aZ f = 0 on aD.

A very careful geometric analysis together with the Bochner-Martinelli formula allowed Harvey and Lawson to reformulate the above version of the Bochner-Hartogs Theorem as the following brilliant result on bounding complex manifolds by odd-dimensional cycles (see [3’5]).

Theorem 1.2’ (Harvey, Lawson, 1974). Let M be Q compact (2k- l)-dimensional (k > 1) smouth manifold in C”. Then M is the boundary of some k-dimensional analytic set ifand only ifM is a maximally-complex manifold, i.e. fdr each point USE M, the real tangent space TC M contains a (k - I)-dimensional complex subspace.

- A more expanded formulation of this result can be found in thb article III by Chirka.

We have enunciated the most classical applications of the Martinelli-Bochner formula. In recent years the Martinelli-Bochner formula has been-successfully used in a series of questions in the theory of multidimensional residues. On this basis, for example, one can give an elementary proof for the multidimensional

.logarithmic residue formula (see [4]) and for the Lefschetz formula for the number of fixed points of a holomorphic mapping (see [31]), (see also article V

II. The Method of Integral Representations in Complex Analysis 31 ’

by Dolbeadt and Vol. 8, article 1. by L.A. Ajzenberg, A.P. Yuzhakov, and A.K. Tsikh on the theory of multidimensional residues).

Although the formula of Martinelli-Bochner is completely universal and-for n = 1 coincides exactly with the Cauchy-Green formula, it has an essential draw-back compared to the Cauchy formula: the kernel in ( 1. I ), (1.2) for n > 1, is not .holomorphic in the z variable.

Nevertheless, for the ball in C”, Bochner deduced immediately from (1.2) the follow:ng formula with holomorphic kernel (see [22]).

Theorem 1.3 (Bochner, 1943). Let D be the unit ball in C”, i.e. D = {zE@: (z( < l}, and let f~ A(@. Then, ,for any z ED, we have the eyuaiities

(W

(1.10)

For the proof of formula (t.9) let us consider a function of the type

holomorphic on the product of balls {(z, W)E C*“: lzl < 1, IwI < 1). On account of the Martineili-Bochner formula (1.2), this function coincides wtth the functionf(z) on the real sub&e w = Z.

From an elementary uniqueness theqrem, we have F(z, w) -f(z), for ]zI < 1, IwI < 1. In particular, F(z, 0) = f(i), for z E D. The last equality is precisely (1.9). Formuh (1.10) follows from (1.9) with the help of Stokes formula.

Bochner and Bergman showed that the operatorsf++ K’fandfH K’J given by the integrals (1.9) and (l.lO), are orthogonal projections of the spaces L*(aD) and L*(D) respectively onto the spaces H*(dD) and H*(D) (see [4] and [23]).

1.2. The Integral Representations oiBochner and Hua Loo-keng on blassical Domaid Let D be an arbitrary bounded domain in C”. For a point z e D we consider the functional z-f(z) on H*(D). By the Riesz Theorem we have

’ f(z) = j$)B((, z)dV&) = Eft (1.1 i) .

where &,a, Z)E H2(D) and B(i, z) __-

-Btz, 0. Bergman (1933, see [24] and [SS])

showed that the operator f t+ Bf is the orthogonal projection of L'(D) onto H*(D). The operator B is called the Bergman projection and the kernel B([, z) is called the:Bergman kernel function for the domain D.

For a bounded domain D in C” we denote by S(D) the smallest closed subset on aD on which each function holomorphic in D and continuous in D assumes its maximum (the Bergman-Shiloo boundary).

32 G.M. Khenkin

If D is the ball in C”, then S(D) = dD: If D = D, x . . . i x D, is a polydomait, in C”, then II

S(D) = (aD,,) x , . . x (aD,)-c aD.

Let L2(S(D), &) denote the space of functions f square integrable with respect to the measure dp on S(D) and If2 (S( D), &) the subspace of functions in L2 (S( D), &),admitting a‘holomorphic continuation to D.

The kernel in the Cauchy formula (0.2) for the polydisc D = (zE~~:]z,] < 1, j=l,..., n > gives the orthogonal Szegii projection from L2 (S( D), d V,( 0) onto H2(S(D), dV,([)), where d V, is Lebesgue measure on S(D). Gleason (1962, see [4]) showed that for any domain D, there exists a positive measure dp on S(D), such that the kernel K([, z) of ,the orthogonal projection of L2(S(b), dp) onto H2(S( D), dp) is holomorphic in ZE D and p-integrable in ~ES( D). Thus,

f(z) = i,;,D.fWW, z)&(C),

for any function ~EI-Z’ (S( D), dp).

ZED, (1.12)

In applications, the Cauchy formulas of type (O-2), (1.9) or (1.10) work effectively, not so much because their kernels give Szegii or Bergman projections, but rather because these kernels are holomorphic in the outer variable and have an explicit formula allowing a precise analysis of its singularities.

In the abstract formulas of Gleason (1.12) and Bergman (l.ll), there is not enough information about the singularities of the kernels and so it is very difficult to apply these formulas.

For classical homogeneous domains of holomorphy, including the ball and fhe polydisc, the kernels of Szega and Bergman have been explicitly calculated ‘by Bochner (1944) and,Hua Loo-keng (1958, see [46]).

A bounded domain D c C” is said to be Classical if the entire group of its analytic automorphisms (one-to-one mappings of the domain onto itself) is a classical Lie group,acting transitively on D.

According to the classification of E. Cartan (1936) there are four types of irredu’cible classical .domains (see [71]).

A classical domain of the first type is a domain 0:. ~ in CM, consisiing of all complex p x q matrices 2 such that the matrix I - Z*Z is positive definite (a 0), where p 2 q 2 1, I is the unit matrix, and Z* = z is the conjugate matrix of 2. The Bergman-Shilov boundary of a domain f$, has real dimension q(2p - 1) and consists of matrices of the form

s(n;*,)=(z:z*z= l}.

A classical domiin of the second (respectively third) type is a domain R’ respectively 0;. P) in the space W’+ 1)‘2 (respectively Cpfp- 1)‘2), consisting’$ complex symmetric (respectively skew-symmetric) matrices of order p, satisfying the condition I- Z* Z % 0 (respectively I + Z* 2 b 0). The manifold S( g p) has real dimension p(p + 1)/2 and consists of all symmetric unitary matrices’of

II. The Method of Integral Representations in Complex Analysis 33

order p. The manifold S( 0,;. ,, ) has dimension (p - l)(p + 1)/2 if p is even and consists of all skew-symmetric unitajy matrices of order p. If p is odd, then S(@, ) has dimension p(p - 1)/2 and consists of matrices of the form UDU’, whe;e’U is an arbitrary unitary matrix and

0=(-y “,,@‘. . . @(-; ;)@O. i

Finally, a classical domain elf the fourth type a,*- consists of vectors 2= (Zl,..., z,)E~“, n > 2, satisfying the condition

l(z,z)I2+1-2(2,z) >o;)(z,z))<l.

The manifold S(Rf ) has dimension n and consists of vectors of the form eie. x, with ~E[O, ~I],xES,,, where S, is the unit sphere in W.

Theorem 1.4 (Bochner, 1944; Hua Loo-keng, 1958). Let R be a simple classical domain and f~@(S(n), d V), where d V is the Haar measure on S(n). Then, in the Cauchy-SzegZ formula (1.12), the kernel K([, z) has the following

form:

for domains Q:, 4

for domains ai, p

K([, Z) = [det(Z - c.Z)]-p;

for domains Qi, p

for domains 0:

K([,Z)=[det(Z-r*Z)]-(P+1)/2;

’

K(i,Z)=[det(l+~.~)1[~‘2-1/4-(-1)~i41;

where [ = eie. x.

Theorem 1.5 (Hua Loo-keng, 1958). * Ler R be an irreducible classical domain andfEH2(n). Then the kernel B([, Z) in the Cauchy-Bergman formula (1.11) has

the following form:

for domains f$, 4, *

B((, 2.) = V-‘(Ri,,)*[det(1- c.Z)]-(p+4);

for domains a;,,,

B(b,Z)=.V-i(R:,p)[det(‘l-i;Z)]-‘P+l);

s or domains Szj, P,

B([,Z) = v-‘(n;,,)[det(r + ,r.z,]-(p-l);

0

34 GM. Khenkin .

for domains Q,4,

B(i, z) = v-‘(fg)( 1.+ I(<, z>12 L2(cz))-“,

.where V(Q) is the volume of$2 c C” in the Bergman metric- c bj,k(z)dzj*djk, with j> k

I r b, =d2Wz,z) J. k s=jr

The formulas of Bochner-Hua Loo-keng allow, one to prove the following significantly more precise form of the Hartogs-Bochner Theorem for classical domains (see [75]).

Theorem 1.6 (Schmid, 1970; Naruki, 1970; Rossi, Vergne, 1976). Lec’R be an irreducible classical domain in Q=” with the property dimn S(Q) > n. In order for a .fitnction f in L’(S(Q)) t o satisfy the tangential Cauchy-Riemann equations

. d,f= 0 on S(Q), ‘t I is necessary and suficient that fc H’ (S(Q)).

We remark that if a classical domain R is such that dimn S(n) = n, then S(Q) has no complex tangent vectors and consequ&tly, there are also no tangential Cauchy--Riemann equations. Theorems 1.4 and 1.5 were used by Hua Loo-keng [46] in order to solve the Dirichlet problem on an’ arbitrary classical domain f2 for functions satisfying the Laplace equation with respect to the Bergman metric on $2.

Theorem 1.7 (Hua Loo-ken& 1958). Let R be an irreducible classical domain in C” and A, the Laplace operator on iz for the Bergman metric, i.e.

A, = c N.L(z)&-, -’ j. k J k

where (bj* ‘) is the inverse matrix of {b, k }. Let K ({.z) be the Cauchy-Szegii kernel jar R. Then, if.f is any continuous function on S(Q), there exists a function (unique moreover) FEC( fi) such that A,F = 0 and FI,,,, =f: In addition, the jiinction I; can be represented, by the faflowi?g Poisson formula:

F(z),= j ,f(i)P(L =)dV(i), i’s(R) .

where P(C, 2) = (K(z, z))-’ lK([, z)12. . .

For further properties of Cauchy and Poisson type integrals, see the papers of volume 8, III, IV.

I There are fundamental problems of multi-dimensional complex analysis connected with non-homogeneous domains or manifolds. In this situation it is possible to construct effective analogs of the Cauchy formula which as a rule Et go of the requirement that the kernel yield an orthogohal projection of Szego or Bergman.

II. The Methed of Integrd Abpr~btions in Complex Analysis * 35

. -.:‘- 1 &?:: TheMeithrmula ad t&2. Oka-Carthn! Theory : .,:.’ _/‘.,> . ,) -. _ ..‘ : t -; ,,i _‘? ,_x ,.~ ._ II -1 .. _, .,,,.,I. ‘,-. c1 ..--,, ‘I.

s.~~2T1, &#$gral .Reprcaentntio~ 4; .Analytic Polyhedra. :; A&ang (non-hot& ge&ous) domains of holomorphy, analytic polyhedra ‘-present particular interest. -, ‘: :+ .;:; ‘: .:,.

A domain D in @” is called an analytic polyhedron ifit in be’.‘&presented in j the form r.l , _ : .’

‘b = (2d2: IFj(& ii 1, j= 1,2,. .‘. , Iv), v ‘. (2.1) * where {F,} are fun&ions holomorphic in some domain R 3 2 D.

The role. of analytic polyhedra is explained by the following important assertion (see [23], [33] and [40]).’ ’ ’ ’

Tbeorem 2.1 (H. Cartan, Thullen, 1932; Behnlce, Stein, 1938). A domain Kt in Q=” is a domain of holomorphy if and only if n can be approximated from the inte?+or by analytic polyhedra. .

The principal advance in multidimensional complex analysis was the construction& Weil(l932. 1935), Bergman (1934; 1936)) of an authentic analog to the Cauchyformula for analytic polyhedra (see [Sl], [23] and[40]).

The boundary of an analytic polyhedron of the type (2.1) is the union of the hypersurfaces

a~=(z~iTD:lF~l=i}, j=1,2 ,..., N.

An analytic polyhedron is called a Weil Polyhedron if the intersection of any k hypersurfaces bJ = Ujl n Up i7 . . . u,; has dimension no greater than 2n - k. -

We give to the real analytic manifold (possibly with singularities) Uj the orientation induced by the orientation of the domain D. Further, by induction, we give to the manifold a, the orientation induced by the orientation of ui. Weil (1935)‘considered the hypothesis that an expansion .

F,(c) - F,(Z) = (P,(& z), c - z), (2.2)

holds, where Pj = (P,‘, . . . , P;) is a holomorphic vector function of the variables c, ZEQ.

Theorem 2.2 (A. Weil, 1935). Let D be an analytic polyhedron of the form (2.1) with the property (2.2). Then any function f E A’(D) can be represented in the form

(2.3)

where I’+~ h(z),

R

Mz) = jfW,iL zkJ.40,

*J((,az)J-$+‘)~2 ;;;y;;;;), ni ’

r=1

the sum 2:’ in (2.3) being taken over strictly monotonic multiindices~ J of length n.

36 Ci.M. Khenkin .

Bergman obtained a result which is similar but ,less effective. We remark that the integration in (2.3) is not over the entire boundary of the polyhedron D, but only on the n-dimensional part S c aD consisting of the union of those bj on which the form o(5) # 0.

The set S coincides (Hoffman, Rossi, 1962) with the Bergman-Shilov bqund- ary S(D) of the polyhedron D (see [ 251). ‘,

A Iveil polyhedron (2.1) is said to.ije complex non-degenerate if for each monotone multi-index J = (ji,‘. . , , j,); we have

‘? dF,I A . . . -. hdF/.ZOona,. . i. (2.4) If D is a complex non-degenerate polyhedron; then

S(D) =‘ u’ o;, -1JI P” x .

We call the d-extension of the distinguished boundary S(D) the set ’

W”) = u’ o;,~, where

A.G. Vit&hkin (1968) suggested the following,effective modification of the Weil formula for non-degenerate polyhedra (see [69 3).

Theore; 2 2 . . Let D be a non-degenerate Weil polyhedron of type (2. l), ‘(2.2) (2.4), and f a continuous function, holomorphic on D, andiwith support @I the domain

- D,=(zEQ: I~j(Z)l<l+S, j=l,2;...,N}, ’

where 6 > 0 is su@ciently small. Then, for ZED the formula

holds, where

.

f(z)= ‘C fi..(z) IJI=n

(2.3’)

Formula (2.3’) is obtained’from-formula (2.3) with the help of the Stokes Formula.

The following result on the separation of singularities of functions holomorphic in Weil polyhedra is an immediate consequence of the Weil formula.

Proposition 2.3 (A. Weil, 1935). Any function holomorphic in a polyhedron D of type (2. l), (2.2) can be represented as a sum offunctions fJ holomorphic in the big

II. The M&d of Integral Representations in Complex Analysis 37

domains - DJ = {z&: IF,(z)1 < l&J), I

I where 1 J I = n>.

A less-simple consequence of the Weil formula is the following stronger version for analytic polyhedra of the Hartogs Theorem on the automatic continuation of holomorphic functions.

Propositidn 2.4 ([40]). Any function f, holomorphic in the neighbourhood of the (n + l)-dimensional distinguished bout&try

IJI = n + I

of a Weil polyhedron, has a single-valued continuation to a holomorphic function on the domain D. .

With respect to the necessity of the decomposition (2.2) for Theorem 2.2, A. Weil (1935) remarked that if the functions (F,} are rational, then the decomposition clearly holds and stated the conjecture that the decomposition 2.2 is verified for any function F holomorphic in a domain of holomorphy Q.

It seems that this technical problem led to the fundamental results of Oka and H. Cartan on the theory of ideals of.analytic functions.

In its original form, this problem was solved through the efforts of dka, Hefer and H. Cartan (see Theorem 2.11).

The original proof of A. Weil for Theorem 2.2 is based on the use ‘of the classical Cauchy formula and the fact that the kernels @J appearing in (2.3) form a holomorphic cocycle. More precisely, for any multi:index I = (it, . . . , i, + i ) of length n -C 1, we have

n+1

,C1(-lY@ir . . . . . &l.i.+~ ,... .in+1(~9z)=oF, (2.5)

where [, z are such that F,(c) # F,(Z), ViEI. Formulas (2.3) and (2.5) essentially give an analytic representation for the

evaluation functional at ZEQ on the space &‘( 6) in terms of a holomorphic cocycle on Q\D. Besides, these. formulas allow “us to obtain an analogous analytic representation for an arbitrary functional on X(D).

Proposition 2.5 (Martineau, 1962). Let D be an-analytic polyhedron in the

domain of holomorphy R c C”, n > 1; Then, the Fantappib transformation

.

where

cd, = {(dk IFi > 1, jeJ>, ’

realizes a cannonical isomorphism between the space offuncti&als H(D)* and the space of holomorphic cocycles { aJ},. 1 JI = n, defined on the domains (fiJ.>,

. - .

satisfying (2.5) andfactored by the coboundary, that is, by the subspake of cocyctes (aJ) ofthefirm : . . : ’ ‘.

@.f ,....,j. ' *tl (- l lkyjl,, . . I i '._ jk-,.jk+I....*jn9 ,. .,< ' ::.:. ,: :. .'_

id&e the $&lions ‘( +,; ) are ,ho$morphic in the d&&in { $ ), 1 J’ 1 -= n ‘- 1. Here, the holomorphii cocycle (qJ) corresponds to the functional qgiven by ’

,_ :. ((PA = ,,g ~/cr,pm4a fG@>. .‘.I’ ‘,

The idea of Proposition 2.5 goes back to Fantappie whopssentially obtained it for the case whe,n D Js ,a polydisc From Proposition 2.5 Martineau (see [28]) deduced the following

Corollaj, 2.5;’ Let K be a hoiomorphically convex compact subset of a domain of ho?omorphy R c @“, n > 1. Then, the space offunctionals (.W( K ))* iscanoni- ealy Isomorphic to H” - l (R\K, @I), the n - l-dimensional cohomology group on the domain Q\K with coeficients in the sheaf 8 of germs of holomorphic functions.

I

2.2. Solution of “Fundamental Problems” in Domains of Holomorphy. A. Weil (1932), used his own formula to obtain a multi-dimensional analog of Runge’s approximation theorem (see [24]). . _

A compact ‘set K in a domain R is said to be holomorphically (respectively polynomiahy) convex with respect to this domain-if for each point z&\K there exists a function holomorphic in R (respectively a holomorphic polynomial)

such that max 1 f(()l.< 1 f(z)l. SEK

Theorem 2.6 (A. Weil, 1932; Oka, 1937). Let K be a holomorphically (respectively polynomially) convex compact subset of a domain 0. Then, each function f holomorphic in some neighbourhood U(K) of the compact set K can be uniformly approximated on K by funciions holomorphic (respectively holomorphic polynomials) on R.

This theorem was proved first of all by A. Weil for polynomially convex compacta and’later by Oka in the general case. .

The proof of Theorem~2.6 starts ut by showing that because of the holo- YL morphic convexity of the compact, t K with respect to the domain Q, there

exists a Weil polyhedron D of type (2.1)‘such that Kc D c U(K). Then we represent the function f by the Weil formula (2.3) in which we mike

the substitution

II. The Method of Integral Representations m Complex Analysis 39

where the series converges for ZE D. We obtain the formula

.fiz) = 1 C .L.,(r)iFj,(Z))” . , . iFj'j,iZ))k", (2.6) i, < <j, k ,,.... k.20

where .-

The series in the right member of (2.6) consists of functions holomorphic on the . domain R and converges uniformly on the compact set K c D to the functionf: After the appearance of Weil’s work, it was remarked (H. Cartan, 1934; Oka, 19361, that if in the proof of Theorem 0.4 (Cousin,’ 1895) we use an integral of Weil type (compare Proposition 2.3 and Theorem 0.5) instead of a Cauchy-type integral, w.e ,obtain a generalization to analytic polyhedra on which the Weil formula holds. In particular we have (see [24] j the following.

Proposition 2.7 (H. Cartan, 1934; Oka, 1936). The first additive Cousin problem is solvable on any polynomially convex domain in C”.

in order to show the solvability of the first Cousin problem in an arbitrary analytic polyhedron D, Oka represented such a polyhedron as an analytic submanifold MD of a polydisc G in C” + N:

M,={(z,w)~G:w~=F~(z), j=1,2 ,..., N 1, (2.7)

where

Moreover, Oka succeeded in proving (see 1241 and [33])

Proposition 2.8 (Oka, 1937). Any analyric submanifold M of type (2.7) in (1 polydist~ G is polynomially convex.

We,limit ourselves here to the formulation of only those criteria which guided Oka in the proof of Proposition 2.8.(see [24]).

Proposition 2.9 (Oka, 1937). A compact set K c c” is polynomiall~ COIII:C.Y l/ and only tffor each .point ZEQ=“\ K, we cun construct u ,family of holomorphic polynomials P,(z), dependi ,g continuously on the parameter t E[O, ,x5), such thlrt none of these polynomials x ave zeros on K, P,(z) = 0. and the distance-from K to the surface { P,(z) = 0} tends to infinity as t --) cr3.

From Propositions 2.7 and 2.8 follows the following fundamental result (see [24], [333 and [403).

Theorem 2.10 (Oka, 1937). Thejrst Cousin problem is solvable on any domain .

of holomorphy in @“.

40 . 9.M. Khenkin’

We remark that for domains in C2, H. Cartan showed the converse: if in a domain D c C2, the first Cousin problem is solvable then D is a domain of holomorphy (see [24]).

The solution of the first Cousin problem with the help of the Weil formula made it possible in turn to solve the problem of the Weil factorization (2.2) on arbitrary analytic polyhedra (see [24] and [40]).

Theorem 2.11 (Oka, Hefer, N. Cartan, 1941-1944). holomorphy in @” and gl, . . , ,

Let R be a domain of gr ho!omorphic functions in Q, 1 I k <n, with the

property dg, A . . . A dg, # 0 on the set of their corn&on zeros

M={zER:g,(z)=... =g,(z)=O}.

Then, for any function cp, holomorphic in R and zero on M, there exists functions (Pl,...? q+ holomorphic in R such that

The following converse also holds: if a domain R in C” is such that for any holomorphic functions gr, . . . , gn having no common zeros in R, there exist

holomorphic functions cpl, . . . , (P” in n such that 1 qjgj = 1, then R is a

domain of holomorphy (see’[33]). J‘

To obtain the factorization (2.2) from Theorem 2.11, it is sufficient to consider the functions gj = rj - zj;i = 1, . . . , n, in the domain R x R and to apply Theorem 2.11 to the function cp(z, [) = f(z) -f(C). Theorem 2.11 is proved in

_ parallel with the following result which is no less important (see [24], [40]).

Theorem 2.12 (Oka, Hefer, H. Cartan, 1941-1944). Under the hypotheses of Theorem 2.11, for any functi_on f holomorphic on M, there exists a function 7 holomorphic on Q such that f = f on M.

If the domain R is not a domain of holomorphy, then the assertion of Theorem 2.12 also fails.

Theorems 2.11 and 2.12 are proved by induction on k. Let k = 1 and M, = {zER: g,(z) = 01. I n order to obtain Theorem 2.11 in this case, it is s,ufficient to set cpr = cp/gl. To obtain Theorem 2.12, we consider a neighbourhood R,, of the manifold M, in Q, which admits a holomorphic retraction z-R(z) onto M,. The function F(z)=f(R(z)) is well- defined in R,. Set R 2 = a\ MI and consider the first Cousin problem in R with Cousin data F/g1 in R, and zero in R,. By the Theorem of Oka 2.10, there exists a meromorphic function 0 in ? such that @ is holo_morphic in Q2 and (0 - F/g, ) is holomorphic in 0,. Consider the fun&ion f = @*gr . By construction, this function is holomorphic in &I and coincides with f on M .

In order to prove Theorem 2.12 in the gem&l case, it is necessary to consider manifolds M, = {z&2: g1 (z) = . . . = g,(z) = 0) and, using the solvability. of

.

II. The Method of Integral Represcnthtions in Complex Analysis 41

the Cousin .problem on such manifolds, succe_ssively continue the function f, given on M = Mk, to a holomorphic function fk- 1 on M,- I;_‘hec to continue X _ 1 to a functionjj, _ 2 on Mk _ z etc. until we obtain a function f = fO on M0 = R.

To prov.e Theorem 2.11 in the general case we suppose it has been verified for

: the case of (k - 1) functions gz, . . . , gk on the manifold M r; that is,

J

: Then, by Theorem 2.12, there exist holomorphic functions ( ‘pj > on R agreeing with (Gj} on Mr,j=2,.. . . , k. To complete the proof of Theorem 2.11, it is sufficient now to set

cP1 (Z) = (Cp(Z) - ,k2 gj(z)' cPj(z))lgl tz), Zd-2. .

Subsequently, H. Cartan using the works of Leray (1945) and Oka (1950) significantly strengthened the assertions of Theorems 2.10,2.11 and 2.12 getting rid, in particular, of the condition that M be a complete intersection in fi (see [33] and [40]).

Theorem 2.13 (H. Cartan, 1951). Let M be a closed analytic submanifold of codimension k in a domain of holomorphy Q. Then,

a) any holomorphic function f on M can be continued to a holomorphic function on 0, .

‘b) in any subdomain D c R, the manifold M can be represented in the form l

DnM = {zeD:gt(z)=. . . =gn(x)=O},’ -(2.8)

ii;h~i;e the rank of the matrix (grad g1 , , , . , grad gn) is k everywhere on M n D; c) for any function cp holomo?+ . . . _ tan in D and equal to zero on the mangold M n D

of the form (2.8), there exist holomorphic function3 in D, qp1, . *. . (in such that ,

,

q(z) = i rpj(z)lgi(z), ZEQ j=l

d). on M the first Cousin problem is solvable.

Part a) of Theorem 2.13 was stated earlier as a conjecture by A. Weil. The works of Oka and H. Cartan, arising in connection with the Cauchy-Weil formula, thus led to the solution of several long-standing problems of multidimensional complex analysis.

t

$3. Integral Formulae and the Problem of E. Levi

3,l. Pseudoconvex Domains. eorems of E. Levi and H. Lewy. E. Levi (1911) showed that applying Hartog’s pseudoconvexity (see Theorem 0.1) to

42 G.M. Khenkin

domains of holomorphy with smooth boundary leads to the more intmtivl! pseudoconvexity in the sense of Levi (see [24], [33] and [40-J),

Let D be 9 domain in C” with smooth boundary. That is,

D - (~4: k(z) < 0}, * (3.1)

\ where p is a real-valued fufiction in the class C’(Q) in some neighbourhood Q of the compact set d and

ddz) f 4 z~i?D. 8’ The Levi form of a reai:valued function PEC’(~Z) at a point [cQ is the

Hermitian form .

WE@” (3.2)

We denote by q( 80) the complex tangent space to 8D at the point [EJD, i.e.

q(aD)= {zEC”:~ $(c)(Z,-[j)=O).

A domain D given by (3.1) is called weakly (respectively strongly) pseudoconvex at the point C&D, if it’s Levi form L&w) is non-negative (respectively positive) for each no&zero vector

’ wET;(aD)‘- c.

The domain D is called Levi pseudoconvex if it is weakly pseudoconvex at each point C E 8D. ‘(I

me domain D is called.strongly pseudoconvex if it,is strongly pseudoconvejt at each point [E a.D.

If fbr a domain D, given by (3.1), and some point {* E?D, the.E. Levi foti at c* has at least one non-positive (respectiveiy negative) eigen-value, then the domain D is said to be weakly (respe&ely strongly) concave at the point C*.

Theorem 3.1 (E. Levi, 1911, see [33]), .Zf a domain D is Levi pseudoconvex, then it is Hartogs pseudoconvex. Conversely,’ if D given by (3.1) is not Levi pseudoconvex, i.e. is strongly co&hoe at some point c*~aD, then for some neighbourhood UP ofthe point C*, every function holomorphic in U, n D contjnues holomorphically to UP. * ‘*,j

H. Lewy significantly strengthened Theorem j.1 (see [45)) in the following way. .

Theorem 3+’ (H. Dewy, 1956). :Suppose ibe domain D is given by (3.1) and is strongly concave at some point (*&D. Then there exists a neighlkurhood U,* of the point C* such that any ficnction f of class C’ and satisfying the tangential Cauchy-Riemann equations af A 8p = 0 on U, n dD extends holomorphically to the domain Urn(Cn\D)..

.

.____ :r i -> . s: ?j, V~‘, FB

II. The hktbod cd Intqgd Rcprcsentationa ifi compIcx AnaIysis 43

‘ i,;., -3, *\ ,cr.~:-ti- ,; : i::. - ,> ̂I / :‘f <,+ ,:y”.ff; ‘i’ ‘.I :,‘. L y:̂ ” -- r-4: & ,;qGs ^~,, ;_ .. -= i PjL&+Qz+&

Rossi (1%&j general&d the assertion of Theorem 3.1’ to the case dktc@dM functions s$sfying the @%F$$, ~~uchy$i~%~t&?s in, $q, yse of distributl’ons &i’ur h%b. ‘i.” :‘. (r*?-:r

The ij+pItst~~prty3$. of- Theorem %&is to re&ce ,h to; ThetSuP3.C ‘%jk representing (see -Thearem .SS &low)‘-f asthe &B&ncc ,of fittctfbns$+~&d$G hol~arphic..r&#&dy in:& do&& U, n D. ad .up:n (ef+D$. ?;-: i: : : 6*~ 2:

For a real function p E C2 (a), we introduce the so-called Levi polynoniial f- 1 1’ . . . . . : .:’ z :.t’. 1 ** i a,? \ < ;‘.:~,+x.. 1 :.-a,,+;?‘;? I f

,; : Fp‘(t; 33 ..& ;. I.. ,.,_ ‘: :f. .,, pp... ;, ,2.,, ;:: ._ T-.i

T .+H?~.T C,) + 2 &&&Oh - Qh..+ G).. ‘.- :jg,,

.: ._ ~,‘\:-’ ,, ” ‘-i .“,“..:. ;: .._ ,. ‘,. I:‘ .:.: ,: ,-3,.:

E; -I;evi; in.proving’ Theor&3;1 iand in--p&@ -theqro%m bf,whetber.iach: pseudoconvex domain in c” is a domain of hol&orphy , based himself oti tk following, result (see [40-J). : ., . . ,’ .’

Theorem 3.2 (E. L&i, 1911). Let B be a strongly pseudoconvei domak The* for any ball B of su@iently small diameter, the dopain D n B is a domain of holomorphy. In addition-foj each point CG B n aD, the null-set of the Lzevi polynomial F,,( C, z) is a Strong. barrier for the doinain Bn D; that is,. . .j ‘c .,.

(z:F,({,z)=O}nBn6={[). .’ ’ ,: $3.4)

rf, on the othei hand, the do&in D is strongiypseudocancave at a point C*fi?LJ, then for- some neighbourhood Ur of the point c* and some two-dimensional plane Me passing through C*, the holomorphic curve

{++nVr: F,(I,,z)‘y 0) ,,:

lies in the domqin Q & is tangent to i?D from within at the point C?. :” For <he proof of Theorem 3.2 we use the Taylor formula to write

P(Z) = p(6) + 2Rc F,(C, 4 + L,,(& z - C)‘+ o(lC - ~1~). (3.5) d ,

If the domain D is strongly ps&doconv&x, then ,

&x, z - C) 2 yl( - 212 on q(aD), .

where y > 0. From this and fr6m (3.5); it .follows that if the diameter of the ball B. is suf&iently small, then (3.4) holds, -’

To complete the proof of the first part of the theorem, it is su@cient now to notice that for aqy point CE B n aD, the function [ F,,(& z)]- 1 is holomorphic in B n D and does not continue holomoiphically to the point c.

Thi second-part of the theorem is proved analogously. For weakly pseudoconvex (ptidoconvex, but not strongly) domains D (even

with an&tic boundary), there are difficulties, related to the construction of barriei functions for 80, which are well illustrated by the following surprising result (see [SS]).

4 GM. Kbcn&in

l%somm $2’ (Kohn, Nirenberg, ,1973). Let D be the fblbwdg pseudoconvex porrairi*c2;

- .

D = ((ti.ts)~C2: Rez2 + Izl~z2~2 + (zlle + 15/71z;12=Reg < 0).

Lcthbca~.kolomorpMcinane~~~ofthcpoint(O,O)adDand qwl to 239-o at this point. Then the set ((z,, z2): h(zl, z2) = 0) necessarily has both mm& paints in the interior as well as in the exterior of the domain D.

$2, Oka’s %Iatiea to the Levi Problem. From TheQmn 3.2 it follows that the solution of the Levi problem for strongly pseudoconvex domains rests on ckarlng up the conditions. under which the union of two domains of holomorphy. is again a domain of holomorphy. The following principle for the joining of domains of holomorphy was brought forth by Oka in 1942 to solve the Levi problem in C2 (see (241).

I’knvan 33(Oka). Let Q be a domain in C” which is the union of two domains qf holomorphy of the type

*

D, = {zED: Imz, > - e}, D2 = (zED: Imz, <a).

l%en, the domain. D 3: D, v D2 is also a.domain of holomorphy. In order to prove Theorem 3.3, it is sufficient to show that the first Cousin

problem is solvable in the domain D = D, u D2. The proof of the latter reduces to the following assertion of Oka, which is a profound generalization of Theorem 0.5 (see [24]).

Ropo&ion 3.4 (Oka). Under the hypotheses of Theorem 3.3, any&&on f holomorphic in a neighbourhood of the set S = (zcD: Imz, = 0} can be represented on S in the form f = f+ -f- , where f* are functions holomorphic respectively in the closures of the domains

D* = {zED: f Imz, > 01.

This proposition was proved by Oka once more by making use of the Weil. . integral in an interesting way. The original reasoning of,Oka (see [24]) is as follows.

By Theorem 2.1, without loss of generality, we may assume that the domain Do = D, n D2 is an analytic polyhedron, that is

D,~{z;&(z)l<l, j=1,2,:..,$

where the functions ( Fj} are holomorphic in a neighbourhood of &. By Theorem 2.11 we have

F,(C)-F,(Z)=~~~~*((,_~~)~. P

where P$EH( &, x D,,). We, consider, in the domains Dof = D, n Dk ;’ the

folh&ng functions, given by We&type integrals *

- ,.- -f*(z) = ,,, 5 * ~fK)(P)(L MCX (3.6) . I- ‘. I wherea,~~~~a~:i~,i~i,ie~~~, ,..., j,)j, . ,

. det14*1:,,-2

(F,(C) _ F,tz)j

’ - ZdDf ’ 0. (3.1)

n . . . The kernels 9~; are meromorphic functions of the variables (c, z) in a neighbourhood V(o, x D,f) of the compact set a, x 0’;. By the Sokhotskij formula we have

f(z)=f~(z)-f~(z~~ zes. (3.8 From Oka’s Theorem 2.10, there exist meromorphic functions @f (C, a) in the domains .U(a, x o’* ) which are holomorphic for zoU * \D,’ and which for ZE Dof have the same poles as the functions gf . Moreover, by the Oka-Well Theorem 2.6, for each e > 0, there are hmctions Gf holomorphic in the domain U(a, x b*) such that .

I@$ -.cpf -GfI<s ’ * ,‘; ,‘. I for(c,z)oa, xb

.‘. (’ 0’. Consider the kernel @f = #f + & . Then, on a, x D#‘tie

have )I. I ++=cp,‘+Rf,

‘, ,,; ,. . _ * (3.9) _ *

where 1 Rf (C, z)/ < e for (C, Z)E a, x Dof . Let us de6ne holomorphic ftmctiona~m the domain D* by the formula “,i ,‘#‘,’ ,,,.’

f*(z) a c f f (o~p(~,z)w(~~~ : ‘.;‘+ (3;iq .’ ‘: IJI-nil a, .

We set . . . ,. ,. ,, . -:,, ; fiz, -5 (z) -E(z).

L,. “’ (3.H)~ i

From (3.8), (3.9) and (3.10) follows the equality

where j:f+Rf, * (3:12)

Rf= c j f(f)(R; - R;)dC). IJI=n-1 0, *

Since the operator R has smail norm, equation- (3.12), seen as an integral equation for f l M(g), can be solved with the help of iteration, for any&%‘($). The assertion of Proposition 3.4 follows from this, and from (3.10) and (3.11).

From Theorems 3.2 and 3.3 it follows that any strongly pseudoconvex domain in C” is a ‘domain of holomorphy.

In order to solve the Levi problem in complete gcderality, it &&es, by Theorem 2.1 to show that any pseudoconvex domain can be approximated from

w&e {fi: joJ} is a finite eo&&ion of functions~hojomorphic on D. . ..).’

A fumti~~‘,p~C*(fl) is called strictly plurisubharmon~~ in $2 if the Levi form &;,Li(y)-is po&j+~&jj~~~ fdryF&KtEfi. - “’ 1 . -: \-’ I’ -’ ’

:. . ,. . ~.T~HNWI & &~long, &94>; Ql&,’ i953).

. . . . ‘&:order~& a, domain i k C? to be

&rtogs pset&Conve$?* ti is nee6sSor~ and suficbnt that there .sxist’ a plurisubharmonic function p which is’an exhausting function for &that is, for any atz W, we have _’

. j _, DO={eD:&)<a)eCD. . (3.13)

In addition, ‘g D # C”, then a criterion of pseudoconvexity is the plurisub- hartiwnicity of the function - In dist (;z,‘aD). Moreover, a criterion for the pseudo- cont.&y ofD G C is the existmce in-D of astrictly plurisubharmonicfunction in D with the property (3.13).

For bounded domainswith smooth boundary, the following stronger version of.Theorem 3.5 (see [54]) has,a@o be proven. ’

Theorem 35’ (Diederich, Fornaess, 1977). Let D be any bounded .pseudoconvex domain with boundary of class Cf in @“. Then, there exists a dejning fin@@ p for D of class C2(C”) (Q = {z p(z) -z 8)) such that the fun&ion j? = - ( - p)‘lN is not only @austing for D but also strictly. plurisubharmonic in D for all s@iciently lvge N. .’ ‘.

From Theorem 3.5 it follows in patiicular that any pseudoconvex domain can be approximated from the interior by domains of the,form .

where p is a strictly plurisubharmonic function i the domain’ R I> 5. Byi an elementary lemma of Kohn (see [32]), a domain in ic” with smooth

boundary is sttiCtly pseudoconvex if and only ifit can be represented in the form (3.14). For this reason, in the sequel, we shall tail a domain in C” of type (3.14) strongly (or strictly) .pseudoco~vex irrespective of whether it has a smooth’ boundary or not. 7,

From Theorems 3.2, 3.3 and 3.5 we have &he fo&nvfng fundamental result.


Theorem 3.6 (Oka, Norguet, Bremermann, 1953, see [33], [40], and [45)). Every pseudoconvex domain in @” is a domain of Holomorphy.

3.3. Applications and Generalizations of Oka’s Theorems. An n-dimensional compkx manifold SJ is called a Riemannian domain if on n there is a system of n holomorphic furictions which form a local coordinate system in the neighhourhood of each point of R. Oka (1953) showed that the Levi .problem and the first Cousin problem are solvable for any pseudoconvex Riemannian domain. Oka’s solution to the Levi problem immediately led to several remarkable results. One of them was the description of the envelope of holomorphy of an arbitrary domain in C” (see [33] and [45]).

‘I. Theorem 3.7 (Oka). The envelope of holomorf;hy of an arbitrary domain n in

C” (or even of an arbitrary Riemannian domain over C”) &T a Riemannian doW?t # holomorphy 6 over C”.

This fact, that the domain of holomorphy of a domain in C” can be many- sheeted, was first discovered by Thullen (1932), (w [81] and [23])-

The proof of Theorem 3.7 -is based on the fact that, with the help of the Hartogs Continuity Principle (Theorem O.l), it is relatively simple to cowtr&Ct a Riemannian domain fi to which all functions holomorphic in Q extend holomorphically. That there is no larger domain, to which all functions holomorphic on fi extend, follows from Theorem 3.6. .

Another application of the solution of the Levi problem on pseudoconvex , Riemannian domains is the solution of the Poincar& problem for arbitrary domains in C”.

Theorem 3.8 (Oka). For an arbitrary domain Q in’ C, any mesomorphic’ function f in D is the quotient of two holbmorphicfunctions irt D.

We recall that for the case R = C”, the assertion of Theorem 3.8 is a classical result of Poincare and Cousin.

To prove Theorem 3,8, we extend the function $, using the continuity principle of Levi, to a function f meromorphic on a pseudoconvex Rjemannian domain d. Now, using the solvability of the first Cousin problem on 0, we represent 7 as the quotient of two holomorphic functions on Q.

One more beautiful consequence of Theorem 3.6 is the following’ result concerning the representation of a plurisubharmonic function in terms of modules of holomorphic functions (see [73]).

Theorem 3.9 (Lelong (1941), n = 1; Rremermann .(1954), n 82 1). For any continuous plurisubharmonic junction p in a pseudoconvex don&n D c P, there exist holomorphic funtions (fn} in D such @

p(z) = lim sup L ,-aD no, ,,,WWl. AX

48 GM. Khenkin -

For the proof of Theorem 3.9, we notice that the domain

0” = ((2, w): ZED, WECl, Ip( < lnlwl}

is pseudoconvex in @” + I. Thus, by Theorem 3.6, there exists a holomorphic functionf(z, w) in 5 which cannot be holomorphically continued to a larger domain. Now we may take, as functions {f,}, the coefficients in the power series:

For further results on plurisubharmonic functions and their role in complex analysis see C811, C733, WI, C241, [32],‘[333, [45] and the article Vol. 8, II.

From Oka’s Theorems 2.10, 2.11 and 3.5 one can deduce the existence of global holomorphic barriers for strongly pseudoconvex domains, thus generalizing Theorem 3.2 where such barriers are constructed locally. 4

Theorem 3.10 (G.M. Khenkin, 1969, 1974; Ramires, 1970; IZlvrelid, 1971; Fornaess, 1974). Let D be a strongly pseudoconvex domain given by (3.14). Then, for some neighbourhood U(D) of the compact set 0, there exists a smooth function cf, = @(I& z) of the-variables ([, Z)E U(D) x U(6) such that <D is a holomorphic function of zGU(D),

~Re~(;,z)‘~p(r)-~(z)+yli-z12 (3: 15)

for some y >:O, and

(3.16)

where P=(P1,..., Pn) is a smooth Vector-function of the vpriables (i, z) MU x U(D), holomorphic in z~U(6).

Theorem 3.10 improves results of Bremermann (1959) and I&ossi (1961, see [33]) stating that fo; any strongly pseudoconvex domain D, the Bergman-Shilov bqundaq S(D) = aD .and each point of aD is a peak point for the algebra A( 0):

The barrier functions constructed in,Theorem 3.10 allow one (see [39] and here $4) to obtain analogues .of the. Cauchy ,formula for arbitrary strongly pseudoconvex domains and these formulas are as convenient as the Cauchy- Bochner formulas (i-.9) +qd (1.10) for the ball in C”.

On the basis of Oka’s theorems, the following description of the Bergman- Sh-ilov boundary for an arbitrary pseudoconvex domain with smooth boundary .vas also obtajned (see [SS]). r

Theorem 3.11 (Rossi, 1961; Hakim, Sibony, 1975; .Pflug, 1975; Debiard, Gaveau, 1976; Basener, 1977). Lit D be a pseudoconr~es domain in c” with .smooth boundary of class C2. Then the Bergman-Shilov boundary S(D) is the closure of the points of strong pseudocontlexity on ?D.

II. The Method of Integral Representations in Complex A&sir 49

Grau&t obtained the fundamental generalization of Oka’s Theorem 3.6 to the case of arbitrary strongly pseudoconvex complex manifolds (see [33), [40] and [451)*

An abstract complex manifold D is called strongly pseudoconvex if there exists on D an exhausting function (in the sense of (3.13) p-of class Cz and a cpmpact set K c D such that p is strongly plurisubharmonic on the domain D\K.

A complex manifold D is catled holomorphically convex if for each qmpact set K, its holomorphic hull

i = {=D: lAdIs :qlfWl, W’(M} e

is also compact in D.

Theorem 3.12 (Grauert, 1958). Every strongly pseudoconvex complex manifold is holomorphically convex.

In contrast to thesituation in C”, the assertion of Theorem 3.12does not carry o’ver to arbitrary weakly pseudoconvex complex manifolds. Counterexamples were constructed by Grauert and Narasimhan (1963).

Theorem 3.12 was fiqt.proved by Grauert based on the cohomology theory of coherent analytic sheaves (see [33]). Another proof pf Theorem 3.12 foHows from the fundamental works of Morrey (1958) and Kohn (1963). on the b problem of Neumann-Spencer (se& [22]).

However, the most elementary proof of Theorem 3;12 was obtained in [49] based on an elaboration of the original proof of Theorem 3.6 by Oka and using . only elementary integral formulas in strongly pseudoFonvex domains (see also

I [4]; [40] and c72]).

$4. The Cakhy-Fantappie Formulas .

41. The Formulas of Caucby-Leray and Caucby-Waeboeck. pray (see [59] and [60]), while developing the theory of residues on complex analytic manifolds, found a general method for constructing integral representations for functions of several complex variables.

Let 0 be a domain in C” and z a fixed point in 0. Consider, ip the domain Q = C” x Q with coordinates q = (ql, . . . , q;) E C” and C =. (11, ., . . , C,) ~0, the hypersurface

Pz = ((MEQ: (SC -z> = 0).

Let hZ be a (2n - I)-dimensional cycle in the domain Q 2 pZ whose prq jection on n\(z) is homologous t? aSI. Also, lei HZ be the class of compact homologies, on the domain Q - Pb containing the cycle hZ.

.,’ .,’

“\

1 - I so G.M. Khenkin

Theorem 4.1 (Leray, 1956, see CSS]). : domain f& we have

For ani holomorphic function in the.

(4.1)

For thk proof of Theorem.4,1, we notice that the form

, I f(g) o’(tl) * MC) C&C-2)“. ’

is closed in the do&+ Q - Pz. Thus, it is sufficient to prove .(4.1) f2r any one cycle h, E Hz. Let D be a’neighbourhood of the point z with rectifiable botindary, where d c Q. As jzz. we take the graph of the mapping CH~(C) = f; - 5, for [E JD. .For this choice of h,, formula (4.1) becomes plccisely the Martinelli- Bochnei formula (1.2). As an imniediate corollary of Theorem 4.1 we obtain (see [60]) the folkwing formula of Ca&hy-Waelbroeck, generalizing tke Bergman trpi integral tepreseritation.

.’ . ~

’ Tbeoretn 4.2 (Waelbroeck, 1960; Leray, 1961).‘,. &t 0’ be a do&in with req@abfe boundary in @” and zf;ls. Let q = q(C; z) = (ql, . .l, q,) be a smooth’ C”-valuedfunction of the variable [E iJ such da (q(t;, z), C - z) 7 1, for ( ;aD. Then, for any function J; holqmorphic in D and continuous on 6, at&i any integer s 2 0, we have

f(z) = KY(z), L, * where

(4.2)

’ (4.3) .ifsrl,and

. KOf(z) ;- (n - l)! (2ni)” ( J,f (C)@‘(rl(C, z)) h O(C), ’

E for s = 0. .

We remark that Waell&eck used formula (4.2) for the construction of a ’ i@c$onal calculus on elements of linear topological algebras (see [25]). %

For the proof of (4.2) with s =. we have

1, we notice that by (4.4) and Stokes formula,

f@ = - J d[f(C)o’(& z)) A o(()] *

(21cir, leD


- - Further, f&m Stokes formula and the equalities

4U - ML z), C- z>,s-‘f (OMr;9 z)) A w(C)3 = _’ .

=(s+n- lN1 - <qK, z), 5 - wIfwM9 z)) * 40 - .

- (s.- 1x1 - <rt(l - z), r; - W-“f (Wo(C, z)) h N), I.’

we deduce that the right side of (4.2) is independent of s. Thus, (4.2) follows ,. from (4.5).

~I

4.2. Multidimensional Analogues of the Catihy-Green Formula. The for-, ’ “hiulas of Cauchy-Fantappih and Cauchy-Waelbroeck (4.1) and (4.2) togethe’r

with the Martjnelli-Bochner formula (1.2) ,allow (see [2OJ) us to obtain the following multi-dimensional arialogues of the Cauchy-Green formulas.

%. Theorem 4.3 (G.M. Khenkin, i970; s = 0; G.M. Khenkin, A.V. Romanov,

1971; s = 1; G.M. Kbenkin, 1978, s 2 1). Under the colfpition of Theorem 4.2, for any function f such that both f and af are continuous on D, the foftiiwing integral representation holds:

f(z) = K”f (z) - H%fk), (4.6) -

where

(n - l)! H”@-f)C4 = o”

-G noa( * 40 + s df(C) -’ A d(S) A o(i)], (4.7)

r ifs =0, and

(n + s - l)!

,H”(af)(z) = (s- 1)!(2ni)“(I,10jED x co,,3 I

Zff(gjx' * *

x CJOU - ML z), r - ms-’ x c-2

xQ4(l-M,c~z,z ____ + ~,rl(L 2)) * 40, (4.8)

ifsrl.

The following extension of Theorems 4.2 and 4.3 was obtained in the decent works [ll] and [12].

;-Theorem 4.4 (Berndtsson, Andersson; 1982). Under the hypotheses of Theorems 4.2 and 4.3, let Q([, z) = (Q1([, z), . . . , Q&I, z)) be an arbitrary smooth C”-valued fur&ion of the complex variable CE D, and G(t) 4 function of te C’, analytic in a domain containing the image of D x D by the mapping (l, Z)H (Q(c, z), [ - t ) and such that G(0) = 1. Then, for anyfunction f holomor-

52 GM. Khenkin

phic in D and continuous in 6, we have )

f=K’f--HG@,

where (4.9)

KGf(Z) = C(n) j f($$ - 1y.b G”‘((Q, l- z))o,(Q, q) - dD

(- 1r - ,!~fK)@‘KQ, C - z>b,(Q) 1 , (4.10)

l$K,A ;$; (- 1)’ ; G”‘((Q, C - z))o k( 9 ,i’--lfz) Q -+

o,(Q) = @Q A dS)“, G”‘(t) = $, tin) = (- lJ-1”2* (2niY

In the particular case when Q = q and Git) = (1 - t)“+“-*, the assertion of Theorem 4.4 reduces to Theorem 4.3.

43. Integral Representations in Strictly Pseudoconvex Domains. For concrete domains D in C”, the formulas (4.1), (4.2), (4.6) and (4.9) yield not one integral representation but a, whole class of such representations. The choice of an appropriate representatton with holomorphic’ kernel among these is a problem requiring special consideration. For the case of a convex domain with smooth boundary: D = (C E CR: p(C) < 01, already Leray himself (1956, see [59]) found an appropriate realization of formula (4.1). Namely, in this case, for the cycle h, in formula (4.1) (or (4.2) with s = 0) we may choose the graph of the mapping 1

Then, formula (4.1) takes on the form

f(z)

for&A(d), ZED.

(4.12)

11. The Method of Integral Representations in Complex Analysis 53

LA. Ajzenberg (1967, see [l]) remarked that formula (4.12) holds also for arbitrary linearly convex domqins D with smooth boundary, i.e. domains such that T@D) c &‘\D for each point CE~D.

The most appropriate realization of formulas (4.1) and (4.2) for strictly pseudoconvex domains (G.M. Khenkin, 1969, 1970; Ramirez, 1970; G.M. Khenkin, A.V. Romanov, 1971, see [40]) of the form (3.14) is obtained by setting

(4.13)

in (4.2), where @(c, zI= cP([, z), C - z) is the barrier function from Theorem .3.10. In this case, formula (4.2) acquires the form

O’(p(t, Z)) A O(c) I?%-, z)l” .

(4.14)

for s = 0, and

for s > 0, where fo A(d). The more general formulas (4.6) and (4.9) also turn out to be most useful in

applications to strictly pseudoconvex domains when the vector-function rl has the form (4.13). Namely, Theorems 4.3 and 4.4 were first formulated for this case in the cited works.

We remark that in the particular case where D is the ball ([: I (: I < 1 > in C” and

r1Gz) = f f (~c-z>-((m-l) = l-<w

the operator K” is precisely the SzegGBochner projection (1.9); the operator K’f is the Bergman projection (1.10); and the operators K’J s 2 1, are the Bergman projections in the L2-spaces with weighted measures (1 - lc12)‘- l d V,,(C) according to the work of Rudin and Forelli (1974, see [76]).

4A. The Theorem of Fantappi&Martineau on Analytic Functionals. Having obtained formulas (4.1) and (4.12), Leray made the following remark [59]: “FantappiC a, plus generalement, exprimt f(z) comme, somme de puissances p-idmes de fonctions lineaires de z (p: entier negative); d’oti l’un des tisultats. essentiels de sa thCorie des fonctionnelles lineaires, analytiques: une telle fonctionnelle F[S] est connue quand on connait les valeurs qu’$le prend lorsquef est la puissance p-i&ne d’une fonction lineaire.”

.

54 G.M:Khenkin : ’

For a linearly .convex domain D in C”, containing the origin’ and having smooth boundary, the more general Cauchy-FantappiC’ formula, to which Leray refers, can be expressed (S.G. Gindikin, G.M. Khenkin, [29]) in the form

. . ‘. f(z) - (p - I)!’

(2@fl (pP)l-qfo 0’ g to A o(C) ( >

(~to,,_l)p( ~,,,~-p’ (4*15) where

DPf = Pf + $& [It $; ’ fE C'"-p)(B) n S’(D). h

In the special case when D is a convex and circular domain with smooth boundary, a formula equivalent to (4;15) was obtained first by A.A. Temlyakov (1954, see [4]).

We now formulate a result from the theory of analytic functionals which is ‘easily proved with the help of formulas (4.12) and (4.15).

Let QY” be n-dimensional projective space with homogeneous coordinates z=(zo,z1,..., z,). A domain D in @P” is called strictly linearZy concave if for each point z E D there exists an (n - l)-dimensional projective plane [s, depending continuously on z, with .the property z E [s c D.

The set of all (n - l)-dimensional hy@erplanes {z E @Pn: (5; z> =, 0}, con- I tained in D, forms a domain D* in the dual projective space with homogeneous

coordinates (co, [i, . . , , [,). A compact set G in CP” is called strictly linqrly convex if the domain @P\G

is strictly linearly concave. Let E+(G) denote the space of linear functionals on the space X(G). Let hex*(G). The Fantappit, indicator F,,;,h, p < 0, of the functional h is a

holomorphic function of the variable [ = (Co, Ci,. . . ,c.) c (CP”\G)* defined by F,h([) = (h, (c, z)“). We have F,h(n[) = /PFph([). Functions with this property are called sections of the line bundle G(p) over (CP)*. The space of all holomorphic sections .of O(p) over D* is denoted by #‘(D*, O(p)). r

Theorem 4.5 (Martineau, 1962, 1966; L.A. Ajzenberg, 1966; S.G. Gindikin, G.M. Khenkin,. 1978; S.V. Znamenskij, 1979). Let G be a strictly linearly convex compfct set in CP”. Then for any integer p < 0, the Fantappie’ mapping her F,h jrields an isoiorphism between the space S’*(G) and the space .WW’\G)*, O(P)).

In the original work of Fantappit (1943, see [68]) this result. was proved essentially for the case when G is a polydisc. The case of convex compacta was examined in the works of A. Martineau and L.A. Ajzenberg; the general case was considered in [28].

II. ‘l’he Method of Integral Representations in, Complex Analysis 55

4.5. The Cauchy-Fantappi& Formula in Domains with PiecewiseSmooth Boundaries. The Cauch)l-FantappiC-Leray formula (4.1) allows one also to obtain an effective integral representation in domains with piecewise smooth boundary and with several barrier functions, for example, in Weil polyhedra or in classical domains.

Thegeneral scheme for obtaining such representations is as follows.- Suppose the boundary dG of a bounded domain G in Q=” has a regular decomposition into smooth oriented (2n - l)-dimensional pieces r,, j = 1,2, . . . , N, such that

dG = ,a1 r,, dr) = ,fil r,,j, . (4.16) t , I

. where J = (jr, . , . , jr) c (1, 1 . . , N}, and each I-,,, is an oriented (?n - r. - l)-

dimensional (smooth) pie&of the boundary of the (2n - r)-dimenwonal manifolds r,. Here, the manifolds r, and l”, , are disjoint if J is not a permutation of J’ and coincide in the opposite case. The orientations of the manifolds TJ are, on account of (4.16), antisymmetric with the respect to permutations of the multi- indices J.

Now let us consider the simplex ” .

endowed with the standard orientation. For, each ordered subset J = 0’1,.** ,jr)of(O,l ,..., N)weset -

A, = (tcA:zt,= 11.

The orientation of each subsimplex is chosen such that

whereJg=(j; ,..., jv-!,jv+i,.:.,jt). For each ordered subset I = (ii, . * * * i,)of(l, . * . ) N) with strictIy increasing

components, let th = q,(C, z, t) be a given Cnrvalued vector-function of class Lip 1 in the variables CE I’,, z E G, t E A, with the proljerties

(rl,(C,z,t~C-.z> 20 (4.17)

for (c, z, t)e r, x G x A,; and. ‘tlli * fl,(ts, 29 tl (4.177

for. e. (C, z, t)c r, x G ii A,,;

For fixed ZEG and i c (1,2,‘. . . , N) we denote by hf the manifolg

,i {(C, rl): CErI, rl = tc, 29 t>>

in C” x C” with orientatidn induced by the orientation of the manifold r, x A,.

56 ‘ G.M. Khenkin

By (4.17) the chain

h.=,$,h: i

is a cycle in the domain ~

K~W” x C1:<rl,I--2) a>

,

and satisfies the conditions of Theorem 4.1. Formula (4.1) leads to the following integral representation.

Theorem 4.6 (Norguet, 1961; G.M. Khenkin, 1971; R.A. Ajrapetyan, G.M. Khenkin [3]). Let G be a domain with boundary oftype (4.16) and (v,} vector functions with the properties (4.17). 7’henjki any z E G and any functio! f such that f and af are continuous on C, we hatkthe equality:

f=Kf-Haf,

where

Kf = G,JF, K,f; e

K,f iz) = s f(W’ (

rl,(C, z, t) <rl,(L z, t), c - 2) > A wK)* (C.oEr, x 4

Waf) = $$j ,;$‘,HJ(@-, + Ho@i’l, I

H,@f)(z) = ( - l)lJ’ j af(o A (c.b.nEPAo,r

OJ’

p-2 tlJ(t, z, t)

(l - to)([ - 42 + fo(q5([, z, t), C-z) > *“(f3,

H,(af) (z) = CjDaf (r) A w’ * Qm

(4.18)

(4.19)

(4.20)

*

The assertion of Theorem 4.6 was first obtained by Norguet (1961) for holomorphic functions and by G.M. Khenkin (1971) for’smooth functions with the following additional assumption:

‘IJ(C, Z, t) = ,rJ tj4’j(h Z)* (4.21)

In [4] it is remarked that the hypothesis (4.21) can be successfully replaced by 3 , the more general hypothesis (4.17).


In the situation when (4.21) holds it is possible in formulas (4.19) and (4.20) to integrate explicitly with respect to the parameter t E Ap

Proposition 4.7 ([74], [3]). F or a vector-function q, of type (4.21) we have for fixed i and z

(4.22)

where the sum X:’ runs over all monotone multi-indices BE J.

Formula (4.22) generalizes an assertion of Fantappik (1943, see [63]): The ,kernel in the Cauchy formula (0.2) for the polydisc can be expressed in the form

1

(it - zt) . . . (i. - Z”) (4.23)

4.6. Integral Representations in Pseudoconvex Polyhedra and Siegel Domains. In order to obtain, for example, from formula (4.18) the Weil formula (2.3) for analytic polyhedra of type’(2.1) (2.2) it is sufficient in (4.18) to set

(4.24)

and to notice that by Proposition 4.7, in this situatiori the summands K,fin (4.18) are zero if / .I I< II, and .’

n(n- 1)

K,f(z) = ( ; YTj f(h) det(f’j,(i,z),. . ~pj,(i~z))w(i) ,

n * fi n!= t (Fj,,(i) - Fjy (Z)) ’

iflJI=n. ’ The integral representation (4:18) works particularly well in pseudoconvex

polyhedra, a natural class of domains which includes both analyticpolyhedra and piecewise-strictly pseudoconvex domains. : .

We call a bounded domain D in @” (see [19], [54]) a strictly pseudoconuex polyhedron if there are given holomorphic mappings %j, j = 1,2, : . . , N, from some domain R 3 d onto a domain SIj in the space @“J of dimension mj( _< n) and strictly plurisubharmonic functions pj E C*(n) such. that the domain D has

) the’form

D= ii~n:bj(xj(r;))<.O,~-’ j= I,2 ,..., N). (4.25)

If each xj is the identity mapping of D onto itself, then definition (4.15) yields an arbitrary piecewise stiictly pseudoconvex domain. If all jnj are 1. then definition (4.25) yields arbitrary analytic polyhedra.

58 G.M. Khenkin

The boundary of a pseudoconvex polyhedron (4.25) consists of pieces

rj = {t: E aD: pj(Xj.(C)) = O}, j = 1,2,. . . ,N,

joined together by the sets r,= r,,, n . . . nTj,, where J= { j,, . . . ,j,} c (1,2, . . . N}.

Let us denote by Dj the domain in Cm’ of the form

Dj = {<EQj:pp,(<) < O}* Let

be a barrier function for the domain Dj with the properties in Theorem 3.10, where <E U(aD,) and ZE V(~j).

By Theorem 2.11 the holomorphic mappings Xj=(Xj,1, . . . , xj,m.,) admit the representations

where c,z~R,cr=l,..., mj, and the functions Qt,= are holomorphic in the domain R x R.

We may take the barrier functions Qj(t;. z), for the surfaces rj, of the form

We have

where

cpi(C, z, = i pjstlP z, tC@ - ‘#A (4.26) 0=1

For pseudoconvex polyhedra of type (4.25), it is,natural to take the vector- functions {q,(C, z, t)} of the form

s.AL z, t) = jxJ t,P,tL 4, * (4.27) 6

where teA,,[~r~,z~D. These vector-functions satisfy conditions (4.17) and (4.21). From this and

Theorem 4.6 and Proposition 4.7, it follows that in a pseudoconvex polyhedron of type (4.25) satisfying the non-degeneracy condition:

dPj,(Xjl(C)) h * * * h-dPjk(XjktO) f O On L

where J=(j,, . . . , j,), 1 J 1~ n, the integral formula (4.18) for holomorphic func-

where J=(j,,. . . ?j&. , ,i. we now show how the integral repres&ation of Cauchy-Bochner-Gindikin

f@lo*s troti urn&+ c4.19 f&~a~‘impbrt*$nt class oi convex, .d,o&iiins. whi&, geieliert$i @&citi& ‘have noh-s&th b$un&& naiely, && domai&. For eli~~,p~~?‘b~‘a:fh~oreiii. ofti. Pia~~ts~~~sha~iro.(i46t) ev& bou&d.ho& gt&&$d~mti iii e: &an be re@ize$ as ,a Siegei ddeain (see [7,13 ,.

tit’ V &‘an bpen c@e hi .O?’ with .i;ertei at the origin and not containing k single entire line, .i.e. V is An. acute’done. tit ‘ip: @“+x b-L-C be an Hermitian V-positive form, that is?

@(z, w) = 5 , 6,

(pol,p z, ’ bi); a$,@ 6L 6,,= E c”,

@(w,w)EVforall WEC”-~ and@(w,w)=O, onlyifw=O. A Sie& domain.is a domain in C” of :the form .

D = (z = (z’, z”) E Ck x c-t: p(z) = lmz’ - @(z”, z”) E V}. (4.29) * Gl ’ ;

We ‘introduce. for consideration ‘the du?l .cope V*; ,this is ihe cone in (Rk)* given by

v+ = (AE(4?k)~:‘(AJc) > 0; vxe V).

In tetis of V* the Siegel domain (4.29) has the form:

D = {z = (i’, 2”): <A p (z’, zi’)> > 0; VAE V?}? .I h, If k?in, then the domain (4.29) is called a Siegel domain of.Jhe first-kind or a tubular domaih over thi acute cone-V. Ifk > n, then the domain (4.29) is called a

. Siegel domain of the second kind. The manifold

s = {(Z’,Z”)E C”: hnz’ = iJ(zrr,ztt)) (4.30)

is called the skeleton of the Siegel domain (4.29). We shall say that a h&lo- morphic function $ on a Siegel domain belongs to the class .Hp(D), if

Ilf ll?fP =supIlf~z~+ix,-z”)(li,,,<.~. XSV

-

60 GM. Kknkin , .

. Theorem 4.7 (see [3]).

have’ Let D be a Siegel domain (4.29) and f~ H’(D). Then, we

where,p([)=&‘,r”), l’:={n~l’*:[nJ=l}~

For Siegel domains of the first kind, formula (4.31) is equivalent to the Bochner formula (see article III in Vol. 8). For Siegel domains o{ the second kind, formula (4.31) is equivalent to a formula of S.G. Gindikin and Koranyi- Stein [57], which in turn generalizes the formulas of Bochner and Hua Loo- keng introduced in 01 for classical domains (see also [63], (741). !n order to obtain formula (4.31) from formula (4.18), it is sufficient to set

,

_ ((d$O)‘PZ>

in formula (4.19), where c E aD,k {A E V:: (n, p) (&O), and to convince oneself that the integral Kfof type (4.19) is equal to (4.31) on S and equal to zero on ao\s.

An effective integral representation for smooth functions in classical domains was obtained recently by Lu Qi-Keng (Lu Qi-Keng, On the Cauchy-Fantappi& Integral and the &Problem, Preprint, Institute of Mathematics Academia Sinica, 1985).

-.

$5. Integral Rkpresentations in-Problems from the Theory of Functions on Pseudoconvex Domains

The integral representations of Cauchy-Fantappit from $4 allow us to obtain, anew and in a more constructive manner, fundamental facts from the theory of

. functions in pseudoconvex domains introduced in $02, 3. Particularly complete results are obtained for strictly pseudoconvex domains.

In fact, for strictly pseudoconvex domains, the kernels of Cauchy-type operators are not only holomorphic in z’ but .also. allow. estimates analogous to the estimates for the Cauchy kernel in one variable. We introduce here typical examples of such estimates. .

II. Tke Method of Integral Reprcocntati~ in Complex Analyris 61

51. Estim&@ for Intqpds of Ciud~y-F~ Type and hymptotics Of

szegii and Bygman KWIH?S in scricdy P-W% Domrin~.

theorem 5.1 (G.M.Xhenkin 1968, 1974, see’ [19],‘[40]). L.et D be a strict& peudoconvex domain of type (3.14k g a function in C@(D), a > 0; and K” a Cauchy- type operator of the form (4.14), s 20. Then, the Hanhel operators given by ’

.are completely continuous operators from .H”(D) to the space C(b), while the’ Top&z operators ft+KS(g-f) are bounded operators in the space H”(D) with a closed range in H”(D) and finite dimensional kernel: Moreover,

g *f - y&f I= Wf*~g),

where HS ii given by (4.7), (4.8), (4.13).

(5.1)

Theorem 5.2 (Ahern, Schneider, Phong, Stein, 1977, see [76], s=O; Ligotcka, 1982, see [62], s = 1). Let D be a strictly pseudoconvex domain with boundary of class C” and let K” be a Cauchy operator given by (4.14). Then for any integer s 2 0 and any non-integer a > 0, the operator K” maps the space C”(D) continuously into the space CqD) A X(D).

Of course a rather direct proof of Theorems 5.1 and 5.2 can be based on the estimate (3.15) for the barrier function @(&z)=<P (J, z), c-z). Formula (5.1) is an immediate consequence of formula (4.6).

More complete information on estimates of integrals of Cauchy-Szegii type for the ball or for strictly pseudoconvex domains with smooth boundary are set forth in the paper by A.B. Aleksandrov Vol. 8, II.

For a strictly pseudoconvex domain D with smooth boundary, we may represent the operators K” and K’ given by (4.14) in the form

. K”f (4 = j f(W”(L z)d Vzn - 1Cl

dD

(5.2)

.

K’ftz)= ~f(W1tLW’,,t~), l . I

D

where dl/, is the element of k-dimensional volume in C”. The estimates of the projections K”, contained in Theorems 5.1 and 5.2 allow

US to show that for an a’iibitrary strictly pseudoconvex domain, the Cauchy projections’ given by (5.2) differ respectively from the orthogonal Szegii Projection S: L2@D)+H2(aD) and the orthogonal Bergman ejection

q B: L2(D)+H2(D) by completeb continuous operators. More precise y, let us ” t

,: :Aff(z) = If(aA~(g,a)dv,,-,(tQ;J _ ;: :,;. : 4 Y,‘:’ :is~~ :‘ ! : i T : I. : ‘. . . >f,, . ’ ,̂ : ::., :.. -‘: dD , ,.: :, .“.,.‘. ,i !, ’ :. I i : ~ I.. . . - . . ,’ ‘($*. . ! . _ ( , I.., /pf(@&

d j(a&g;,z)~pi(~:’ ;. .‘.‘:-: ! ;:, ; : : :

where A’((, z) = ZP(c, z)- K”(C, z),‘s LO, 1. ’ ’ i . -“.

Theorem 5.3 (K-man, Stein [48], Ligotcka [62}). For any stri@ly .pseudoconvex domain D with .bo&ary pfclass. Cwr the .opeqtors A? ad Al gtven. by (5.3) are comp!etely continuoq and’ the “operators,([ y A!) and (1-A’) arq.continuous and invertible in the spaces L2(aD) and L2(D) respectively. Moreover,

S = KO(f-A+; B =.$(I -,A’)-‘.

in addition, the operators A”, s ~41, map the spaces C”(i?D) ‘or4 C”‘(s> respectively into the spaces Ca+1’2(aD) and C”+ l/2 (D), while the orthogonal projections S and B respectively map continuously into the spaces C’(aD), and cll(a@ for my non-integer a > 0.

Frdm. Theorem 5.3 one can deduce the asympts‘tic representation f&r the Be.rgman and Szegii kerneIs originally obtaiiied by F&feTman (1974, qze [9]) by rather difficult methods.

Theorem 5.4 (Fe&man, 1974; Boutet de Monvel, Sjiistrand, 1976, see [91). Let D=(z~@“:p(z)<d} b e a strictly pieudocon&x dtimain with boundary of class C”. Then the kernels S(z, w) and B(z, w) of thi! brthogonal Szeg6 and Bergman projections respectively have the following asytiaptotic form: ’ .

.S(z, w) = cpokw) ’ (-p(z ,))q Qoiz,wM3(-P(z,w)), ,

cpl(Z, 4

W”= (-p(z,w))n+l + 4h(~~wYog(-P(z, 41,

(5.4)

where P(z, w), (ps(z, w), &iz, w), s=O,l, are functions of class Cm@ x a) with the properties: ’

4 P(z, z) = p(z); b) cpok 4 = 4ofz, 4 h c,, .

onaD, .

.c) the forms 3Z’,cp,, &@,, a,(p, &,&, s =O,l, together with all derivatives, vanish * for z=w. . .’


For analytic polyhedra or more general pseudoconvex polyhedra of type (4.25), one can- also obtain good estimates for the Ca’uchy-type operator (4.28), but under the fbllpwing non-degeneracy conditions: for any ordered choice of indices A = {al,. . . , ar,) and B= {/?,, . . . , &) the real Jacobian of the mapping

(5.5)

is constant in a neighbourhood of the set rAAuB2?D.

Theorem 5.5 (A.I. Pe.trosyan [69], G.M. Khenkin, A.G. Sergeev [54]). Let D be a pseudoconvex polyhedron given by (4.25), satisfying !he non-rlq~enrmcy condition (5.5). Let K be a Cauchy-type operator of thr@rm (4.28). Thrn,,fi~ uny function gE C’(D), a>O, the Tdplitz operatorfw K(g .f) is bounded in rke space HP” P).

Theorem 5.5 was first proved by A.I. Petrosyan [69] for non-degenerate Weil polyhedra using the Weil-Vitushkin formula 2.3’. For general pseudoconvex polyhedra, the assertion of the theorem follows from the main results in [54].

Theorem 5.6 (R.A; Ajrapetyan Cl], Jijricke [473). Suppose that the functions p,(~([)), defining the pseudoconvex polyhedron in Tkeoqm 5.5, are of class C”. Then for any u > E > 0, the Cauchy-type operator f~+ Kf operates continuously from the space C”(@ to the spate CLI-e(6). .

Theo&m’!.6’ was first obtained by .R.A. Ajrapetyan for piecewise strictly pseudoconvex domains, then by Jiiricke for Weil polyhedra. The .general case was formulated in [40]. *

Although the pseudoconvex polyhedra form a rather wide class o/f domains, they are far from exhausting the fundamental domains for which it would ‘be useful to have estimates for Cauchy-type integrals.

In particular, for multidimensional complex harmonic analysis, Siegel domains are very impoitani, but the only ones in the above class are balls and their direct products. I *

A good indication of the Pehaviour of @u&y-type integrals for Siegel domaiqs other than direct products of balls is given by the following assqtioti (= C471).

Theorem 5.7 (Jiiricke, 1983). Let D ,be a tubulqr domain in Q=” over the, sphericalcone V={YEIR”: Y:>Yi+...+ Y,‘, Y,>O}andletSbethe Shilov boundary of this domain. Then, for any a > n/2 - 1, ! > 0, and n 2 2, the Cauchy-Szeg&Bochner type integral f + Kf given by (4.3 1) is a cohtinueus operator from -C’@) ri L2(S) into the space Cc-n’2+1-c(D’) n L’(S). Moreover, the result is sharp with respect. to the exponeni.

More de&led information dn estimates of Cauchy-type integrals for tubular domains can be found in the paper of Vat 8, IV.

64 GM. Khenkin

5.2. La#calization of Singularities and Uniform Approximation of Bounded Holomorphic Functions. We pass now directly to the treatment of those questions from the, theory of functions in pseudoconvex domains for whose solution the most natural apparatus turns out to be the’cauchy-Fantappie formula.

We begin with a significant generalization of the Cousin-Oka Proposition 3.4 (see [35]).

Theorem 5.8 (Andreotti, Hill, 1972; EM. Chirka, 1975). Ler D = (2~0: p(z) < 0) be a strictly pseudoconvex domain and S a smooth hypersurface given by

. s = {zER:pI(Z)*= 01,

wherep,EC’(fQanddp, ~dpZOonSn8D. Set

D, = {zED: + p1 < 0).

Then, for any a > 0, any function of class C” on k?-D, satisfying the tangential Cauchey-Riemann equations on S n D, can be represented in the form

f (4 = f+ (4 L f- (4, zESnD, .

where f* are jitnctions holomorphic in the domains D, and continuous on D, ,

This important theorem was first proved by Andreotti and Hill for, the case when f and .& are functions of class Cm(S n D). The Cauchy-Fantappit formulas (see [38]) allow us to write the desired functions f* with the formulas

where zeD+, @(c, z) = (P(c, z), c - z) is the * barrier function from Theorem 3.10. The equality 8fk (z) = ‘0 for z E D, can be verified by an immediate calculation considering that (7f = 0 on S n D. The estimates fi- E A( D * ) follow from estimates for Cauchy-type integrals and Theorem 5.6.

The following theorem on the “localization of singularities” for bounded functions holomorphic in strictly pseudoconvex domains is a significant generalization to several variables of a well-known result of A.G. Vitushkin (1966, see WI9 I251 1.

*Theorem 5.9 (G.M. Khenkin, 1969, 1974, see [40-J). Let D be an arbitrary strictly pseudoconvex domain given by (3.14) whose boundary is no_t necessarily smooth and let Uj c @” (j = 1,2, . . . ., N) be open sets such that D c uf=, U,. Then, uny bounded holomorphic function f on D can be written in the form

f= jgxy .(5.6)


where each Junction J. is bounded and holomorphic in some neighbourhood of the set D\(aDn Uj). Moreover, if f is continuous on 6, then each function -1. is also continuous on D.

Theorem 5.9 also extends results of A. Weil and Oka on the localization of singularities of functions (without estimates) contained in Proposi!ions 2.3. and 2.4. I

To prove Theorem 5.9.we choose smooth function Xj on C” such that Z xi = 1 on D and Xi = 0 on @” \Vj. Further, we set 4 = K ‘(xi f ), where K ’ is the Cauchy-type operator given by (4.14). The functions (4 1: satisfy formula (5.6)’ by the Cauchy formula (4.14); are holomorphic on b\@D n Ui) by inequality 3.15; and are bounded on d by Theorem 5.1.

Theorems 5.9, 5.1, 5.5 allow us to obtain the following result on uniform approximation which for pseudoconvex polyhedra is a more precise version of Ok&Weil approximation theorem 2.6.“*

Theorem 5.10 (G.M. Khenkin, 1968, 1974, see [40]). Let K be a strictly pseudoconvex compact set given by

K = (zEQ: p(z) I 0), (5.7) where p is a strictly plurisubharmonic function of class C2 defined in neighbourhood iz of the compact set K. Then any function continuous on K and holmorphic on the interior K, of the compact set K can be uniformly approximated on K by functions holomorphic in a neighbourhood of K.

Theorem 5.10 contains two interesting extreme cases. If the set K, of interior points is empty, then the’compact set K is the zero set of a non-degenerate

~ .plurisubharmonic function, and in this case, the theorem was first proved in the works of Hiirmander-Wermer, Nirenberg-Wells, and Harvey-Wells in the years 1968- 1972.

On the other hand if Kc # 4 and grad p # 0 on dK,, then K, is a strictly pseudoconvex domain with smooth boundary, and in this case, the theorem was lirst,proved in the works of G.M. Khenkin, and Lieb and Kerzman in the years 1968-1970 (see 1821).

Theorem 5.11 (AI. Petrosyan, 1970; G.M. Khenkin, 1974). Let D be a pseudoconvex polyhedron given by (4.25) and satisfying the non-degeneracy condition (5.5). Then, any function holomorphic on D and continuous on ,D can. be uniformly approximated by fuHctions holomorphic in the domain R 13 D.

Theorem 5.11 was first obtained by A.I. Petrosyan [69] for non-degenerate Weil polyhedra; for general non-degenerate pseudoconvex- polyhedra, Theorem .5.11 was formulated in [19]. (see also [54]).

For domains in C’, Theorem 5.10 and 5. I 1 are particular cases of the classical approximation theorem of A.G. Vitushkin: for any compact set K t C’ having no inner boundary, any function in A(K) can be uniformly approximated by rational functions (see [80]).

66 GM. Khenkin

5.3. Interpolation and Division with Uniform Estimates. Let us now formulate two typical results on the continuation of holomorphic functions give on submanifolds of pseudoconvex domains. These results sharpen corresponding results of Oka and Cartan 2.12, 2.13 (see [40]). 1

Theorem 5.12 (G.M. Khenkin, 1972; G.M. Khenkin, Leiterer, 1980; Amar; .1980). Let Dbe an arbitrary strictly pseudoconvex domain given by (3.14) and M and arbitrary @ed analytic submanifold of Q domain R 2 8. Then:

a) for each bounded holomorphic function f on M n D, there exists a bounded holomorphic function F on D such that F = f on M n D;

b) if moreover f is continuous on M n D, then there exists. a function F, continuous on D, holomorphic on D and agreeing with f on M n D.

Theorem 5.12 was proved in the first ,place (G.M. Khenkin, 1972) under the assumption that the intersection of M with dD is transversal. We shall indicate here the outline of the proof of Theorem512 for the situation when M is a complete intersection in R, i.e.. when

M = (z&I: g,(z) =.: . . = g,(z) = 0}, (5.8)

where g,, . . . , gr are holomorphic functions in R such that dg, A . . . ‘A dg, it0oi1 M.

Under the condition (5.8), the function F; satisfying the, assertion a) in f’hv 5.12, can be found in the following form (G.M. Khenkin, 1972; Stout, 1975; BQcndtsson, 1983 (see [40],[4], [74])): ‘- _ ,.

F(z) (n - M =o”-*rgJ gf(S)det

n P,,:..,P,,a;P,...,~~)no,(o(5.9)

a.’ d’

where w,(c) is a form on M with the property

dg,(O A . . . A &,(O A q&c) ‘= &i),

is the barrier function from Theorem 3.10; , k are @“-valued functions of the variables [, ztin with the

properttes

Sj(l) - Bj.tz) = Cpj(t, i)9 tl -,z>- (5.10)

Such functions {Pi} exist by Theorem 2.11 (see $1~0 Theorem 5.14 below). In order to verify the equality F 1 M =f, it is sufficient to convince oneself that, for ZE M n D, formula (5.9) yields the Cauchy-Fantappie representation for 1: Indeed, let hs consider the Cauchy-Fantappit formula (4.28) in the polyhedral domain

D,.= CED:supl.gJ(C)l<e . I i I I.

Passing to the limit as E + 0 in this formula, for z E M we obtain formula (5.9). To

. II. The Method of Integral Representations in Complex Analysis t

67

complete the-proof of Theorem 5.12, it is necessary to show (and, this is the hardest part) that

., I’ ;J !Wl 5 Y ,“zllf: If(z

Assertion b) in, Theorem 5.12 is proved on the basis of assertion a) and the approximation Theorem 5.10. For related.results see [39], ’

Theorem 5.13 (G.M: Khenkin, P.L. Polyakov [42]). ’ Lei D = {CEC”: l&l < 1, k = 1, . . .., n}bethepoZydircinC”ahdr,={CfaD:lril=l,jeJ}be a component of aD with multi-index J F ( jI, . . . , jr). L,et M be a closed analytic manifold in a neighbourhood of b satisfying a transversality condition of the form: for each point z E M n 80, there exists a neighbourhood U, :and holomorphic

,’

functions {g,} in U, such that M n Vi = (ce U,: g,(o = 0, r = I, . . . , k} and

&r,(z) A . - . A &&) A 4,, (4 A . . . A dl,$) f 0 (5.11)

@rzEMnr,,q.Sk,p+qSn. ‘. Then, for each bounded holomorphic function f on M n D, there exists a

b&mded holomorphic function F on D such that. F Iy =f

We remark (Rudin, 1969) that without the transversality condition (5.11) for innet points of the boundary components, the assertion of Theorem 5.13 cannot hold (see [42]): _.’

A scheme for proving Theorem 5.13 using a formula for the solution of the ’ ,. &equati@s is discussed in $6. Integral formulas also turn out to be the best tool for the prdblem ,of continuing holomorphic functions from submanifolds of Cc” to entire funktions with optimal estimates (Bemdtsson [ 1 l]), (see also the paper of L.I. Ronkih Vol. 9, I).

,For strictly udoconvex strengthens the d;y”

domains, the’ following &ulf significantly ision theorem of Oka-Hefer 2.11:

Theorem 5.14 (Bmdtsson [ll], Bonneau, Cumenge, Zeriahi [16]). Let D = {zER: p(z) c 0} &e a strictly pseudoconvex domain with boundary of class C” and M a submani$ohKof n given by

M = &h;gl(z) = . . . = gk(z) = 0}, (5.12)

where gjEJt?(R), j=l,...,k, and dg, A . . . hdgk#O onM. Suppose M intersects 8D transversally. I

Then, for any function fcH"O(D) such that If(z)1 = O()@z)l), where lg12 = Z, Igi 12, one can give functions ‘g,&H” (0X-j = 1; 2, . . . , k, using ex#licit formulas, such that .

zeD. .. ) (5.13) ‘. r

If moreover fe t?@(D), then for non-integral 01 x I-, we licive ‘~je c’“-“/z(@, j= l,,.. .,k.

c

68 GM. Khenkin

For the c&e k = n, i.e. when M consists of isolated points with M n aD = 4, the assertion of Theorem 5.14 was obtained earlier (Z.L. Lejbenzon (1966) for convex domains; G.M. Khenkin (1968), Kerzman, Nagel (1971) for strictly pseudoconvex domains) as the solution of a Problem of Gleason (1964, see [76]).

In the genera1 case Theorem 5.14 was proved in [16] on the basis of explicit formulas [I l] for finding the functions { cpj} satisfying (5.13).

Let us invoke formula (4.9) setting therein G(r) = (1 - t)‘,

where cD([, z) = (P([, z), C - z) is the barrier function from Theorem 9.10, and (Pj} are functions satisfying equation (5.10).

In this situation formula (4.9): f = KGf for holomorphic functions in D, acquires the form

A (8cQtL 4 A 4)‘.

If we pass: to the limit as E --) 0 in this’formula, tie obtain the equality (see [ll]):

Jtz) = Ftz) + j$l cPjtz)‘gjtz)9 l ZED, (5.14)

where F is a function of type (5.9) and the functions {‘pi ); holomorphic in D, have the forti

For the proof of Theorem 5.14, we notice that F(z) = 0 in D since, under the hypotheses of the theorem, /= 0 in M. ’

Further, under the condition that If(c)1 = O(lg([)l), one can immediately convince oneself of estimates for the boundedneis of the functions (pi} and also that the (cpj} belong to the class Cci-‘)“(B) iff~C”*@), a > 1.

The formula of Berndtsson (5.14) allows one to obtain an accurate theorem .on division under circumstances more genera1 than in Theorem 5.14. For example, on the basis of (5.14), the following assertion has been proven.

II. The Method of Integral Representations in &#ox Analysis 69

-rem 5.14’ (Passarc [67]). Let gl , . . . , g, & holonwrphic functions in a pse&ocowex domain Q such that the analytic set M of type (5.12) has codimension &. Then, a $mtiOtI f in JV @) can be represented in the form ,(5:14), where cp #

EH(R)j s 1,‘. . . , k, ifatid oirly ifthe current (genertifized (0, k)-forn$@‘i& by f (l/s*) * - - * h J(l/gJ vanishes in R.

L $6. Formulas for Solving &Equations in Pseudoconvex - Domains and Their Applications

6.1 The &tpaths. The Theorem of Do&a&’ The Cauchy-Grcen formula (0.1) allows one to write down a solution of the inhomogeneous CauchyLRiemann equation af/aZ = g in.an explicit form, where g is an integrable functiori in a bounded domain D c Cl. Na&y,ithe function given by

f(z)=& d s dS PDF (6.1).

satisfies the required equation. Moreover, formula (6.1) singles out the unique solution of the equation

af/i% = g which satisfies the boundary condition II mm,..

Kf(z) = -L 21ti AfE = 0; ZED. _I , :' ‘

Let us consider, in a domain D c C”, the system of C&&y.-Riemann aqua-. P tions given by

,

where { gj} are fixed functions in the domain’D.- Grothendieck in 1950, using foimula (6.1);gave an etetnentary proof (see [33jf

that for any polydomain’D =*D1 x . . . x D, in C”, necessary and su#lcicnt conditions for the solvibility of the system (6.2) are ‘the following’ compatibility 1. conditions for the right member of (6.2)

8th asj . ‘. q T -g-’ i, k r I,5 . . . , n. . (6.3)

The equations (6.2) and the conditions (6.3) can be written briefly in the fowl’ I‘. (

8 = g (&equation for. the. function f), * l ,

ag = 0 @closure of the ‘form g); (6.4)

where g is ;he (0, l)-form given by g = g,dZ, + . . . + g.dZ,.

3’,‘ - ; - _ $+I. +enktc ,l’ I’ I

-- 1. ‘I*, _ . . ‘.._ ?r _: ,_ - ;3 .-

,” ,X’T@~ th&&n .ofGFofh&ndieck t&her with@a’s theorems 2.10 and 3.6 lead t6 ttit! solwtility 6f the zwuationif6.4) in ti‘arbiiraxy-mq$eti domain (& @1,@?):.f ‘, 3: ” ._ ..‘:.-e*.-:,: ..;; ::r. i, ..,

..*d& &l (Dol&auft, &3); 1 F& an,jr &&.~phj d & any ’ Zclaizci (0, I)+rm. g’ of class .C&“)(Z$,, @e?& a &t&ion fg C(* O”)(0) ‘jatrsfuing the pequatjon (6.4).in @: 2,; . ,‘bc . - & .+, Y

.- - - Hek, i(- “‘(@Uenott!s jhe.spa& of .@e&&ed &nctiuns of the domain-l). As’s simple corollary bf,ho+ 6J, we’Oobf$ln-the foliotiing’strengthened

. i ‘t&iiht of T%orbm 3.4.09 thG splittin;, $~,sing&rities (SF c32-J). I .; Tbqrem63 (H. &&II, J951). t Let Q be a domain in C” and {flJ} a locally-

+ nrte covgi#of the doqdih fI by domains n,, j = 1,2, . . . . Let &} be a holom&rphic &cycle an Q %\ $Ij ; i.c. a. collection bbf functions, holomor$& in the

- .&mains rZ, A n,,..with the #wqerties -. . . . . .I, ,^ ., . .’ .’ ‘.

; .( .I’, $,j+f;,c +A,1 =o : i 1 (6.5)

on Q n fI, A n, for each i, j, k. If Q is b domain of hoEomorphy, then the cocycle {A,,} is a coboundary, i.e. there exists a collection of functions (fi}, holomorphic in the doniains {a,}, and such #at

.

. 1 ‘, a. : / ._ :,; :.’ c_._

‘on 828 n hj for each i, j. ,. _ :. ..i .. ,_

’ In the language of cotiinology, Theorem 6.2 stated that for any-domain of holomorphy, we have H’(Q, 0) = 0, where H’(Q, 0) is the first cohomology groupbf R with,coeEcients in the’shear 0 of gernis bf holomorphic functioris on Q.

In order to de&& * Theorem 6.2 from Theorem 6.1, we introduce for cunside&oti a pa&on of unity subordinate to the covering {a;}, i:e. smooth

. functions ix,} _with supports in the domains {a,} suc_h @at C xi E 1 O?i R. Further, we set& 5 ZJ;,r~k. On accou@ of (6,5), we havei -fi -ffs. In order to fir@ .holomorphic function& {Xl satisfying (6.6-A. let us consider in R the .&closed form.g ‘ven by g = q for ZEQ,, j = 1,2,. .,. . This is.well-define.d .since

(,. ‘ a = &n fi, A fi . BylTheorem 6.1. there exists a smooth function f such that af = g in a. We setfi =fi -J: By constfuction; the functions (fi) are holomorp- hit in the domains {Q,} and satisfy (6.6).

.’ 6.2. Problems of Cousin and Poincark as &Equations. currents of Belong and

S&wart& Schwartz in 1953 suggested #at !hG fist additive Cou$n problem 6 i&erpreted in terms of solving &equatibns (6.4) with a special generalized

I. (0, l)-form (current) on‘the iight’sidd (see.[Q]).- ’ . ,Namely, let {n,} be a locally finite cover of the. domain R and fi = q,/g, data

for the first additive Cousin problem. Consider a generalized (0, l)-form G given ’ . .

. .~ -. II. The Method of Inttgral Rep~sentations. in Complex Analyak

._ ‘. .- - -- -&“+ \ ;

c i : ; .71,.“3 ’

‘*

j= 1,2,.. . ., (6.7) ’ s- G = af; =:i$Wg,A =fijzi, . Each form qja(l/gj) yields a furictional .on smooth (n, n - l)-forms I/ of

compact support in a, by the formula ,t ‘. 1 1. .

Thk existence of the latter limit is guaranteed by the Theore& of g:;reia- Lieberman-Dolbeault (1971), (see the paper by P! Dolbeault v). . ‘j ”

Since the functionsi.-& are holomorphic on’Q n Sz,, the equaiions (6.?), (6.8) ’ define a &closed (0, l)-current ‘on the whole domain R, which we shall’call t’he ’ , Schwartz current. , , , ‘.

In the language of J-equations,~ the theorem of Oka (1937, 1958) op the (. solvability of,. the. first-. Cousin problem in pseudoconvex domaitis can. be formulated in the following way (see 142-J).

Tlieorein 6.3cfOka, 1953; Schwartz, 1953jl. For any pseudoconvex R&nanriian 1 domain n and any Schwartz current G given by (6.7), there exists’a meiomorphic functidn F on.R such tkat aF = G.

The&em, 6.3 leaves op& the question‘ of characterizing- int&sically ih&e (0, l)-currents &ich c& bi: r&pri%&ted as Schwartz cu&&ts; R&lts ‘in this

,

direction ‘are considered in the paper of Dolbeault. ! . . Theorem 6.3 %llows one (see [:18],, [5], E42]) to reformulate; in terms of ’

solving .&equations, the classical pr&lem of *he continuation 6f a function, ’ holomorphic on a hprsurftice ,M i {z ER: g(z) = 41, t@ a function 9, holo* : morphic. in the domain SE. .Namely, using Theorem .6.3, ,Oka’s. ,$rocedure. for ’ constructing such a faction 0 (se& Theokern .2.12) acquires t,he -form

., , ,:... O=g*F,

whe;e & $A I’ ‘-’ = (6.9) : ,( ,‘.. , B,’ ,... .’ ! I.

Leloni (1953) found an even more fr%&l &i instructiie interpret&on of the ‘

d second (multiplicative) Cousin problem in terms of solving a special &problem; the so-called Poincark-Lelong equation (see [32], [35] and article III).. ’

For a giveri cover (Q,} of a domain R let us consider functionsf; meromorphic in Qj such that for each i and j$ the function X/f is holomorphic in- Q n Qk

These data define a global (n - I)-dimensional analytic subset M = (jYyYMY of Q consisting df the zeros and poles of the functions fi. Here, (My) are the irreducible regular (a - I)-dimensional components of the set M and {y,} &e themultiplicities. on {My} of the zeros (yy > 0) or poles (yy < 0) of the functions ‘8 ifi>.

Let us cons&r the generalized (I,l)-form(Curient).Gi given by ‘-

zd2,; j:= 1,2,. *. . e (6.10) -_

12 GM. Khenkin

Lelong showed that the form G, yields a functional, on the space of smooth tn - 1, n - l)$orms + of compact support in the domain R, by the formula

(GM, $> = i Cr, l 9 = i(CMl, JI). (6.11) Y M”

By a divisor in Q we mean any (1, l)-current [M] defined by the second equation in (6.11), where M is any closed (n - l)-dimensional analytic subset M.= uyyyMy of R.

Lelong established that any divisor is a closed (l,l)-current and hence the classical results of Oka, Stein and Serre on the solvability of the multiplicative Cousin problem as well as the Poincare problem in pseudoconvex domains can be formulated in the following stronger form (see [35]).

Theorem 6.4 (Oka, 1939; Stein, 1951; Serre, Lelong, 1953). tit CM] be any divisor given by (6.11) in a pseudoconvex Riemannian domain Q. Then, in order that there exist a meromorphic function F satisfying the equation

(6.12)

it is necessary and suficient that the two-dimensional cohomology class of the (l,l)-current M uanish in the two-dimensional integral cohomology H*(Q, E).

Theorems 6.3 and .6.4 paved the way to soiving the Cousin and; Poincare problems with estimates as well as related problems. For this of course it is necessary also to have solutions to &equations with suitable estimates.

However, the solution -of the &equations for any pudoconvex domain (Theorem 6.1) was obtained by such a long and difficult path (first trod by Oka) that it was difhcult to pass to the estimates for the &equations in some metric or other necessary in applications, even for such a simple domain as the ball in CR.

6.3. The &Problem of Neuman-Spencer. Exploiting the method of orthogonal projection of H. Weyl, Spencer ‘in the early fifties introduced an original approach to solving the &equations called tke &problem of Neuman-Spencer t= WI). (

Let D = {z E C”: p(z).< 0) be a domain with smooth boundary. We consider 8 as an operator from L’(D) to Li, 1(D) and denote by 8* the adjoint operator which in the Euclidean metric on D has the form

a*g= - i ag, jp, a2,'

where the form g = c gid2, belongs to the domain of definition Dom (a*) of the

operator a*. For a smoothform g, as Spencer remarked, the latter is equivalent to k


the equation --

=OonaD. I

The solution to the &problem of Neuman-Spencer based on a priori L2- estimates for the &operator can be formulated as follows:

Theorem 65 (Kohn [22], [56]). Let D ‘be any strictly pseudoconuex domain with boundary of class C”” or any weakly pseudoconvex’domain with real-analytic boundary in C”.

Then there exists (and moreover uniquely) a completely continuous’operator N: Lg.,(D)-+ L:,,(D) such that ’ ,,

Wk(D)) = Don’@*)

and each function fc L”*‘(D) admits in L’(D) an orthogonal decompositiorrof the f orm

f = Bf+ a*NiTf, (6.13)

where ‘B is the Bergman projection from L’(D) onto H’(D). Moreover, N(Li*: (D)) c L$:+‘(D), where L2qk(D) is the space offunctions L2-integrable in D together with their derivatives up to order k.

The operator N satisfying (6.13) is called the Neuman-Spencer operator. From Theorem 6.5 issues the following corollary.

Theorem 6.6 (Kohn [22], [56]). If D is a domain as in Theorem 6.5 and g a &closed form of class L& l,(D), then

a) there exists a unique solution fc L’(D) to the equation af = g which is orthogonal to the space Hz(D); this solution is given by the equation f = 6* Ng;

b) moreover fc Lzsk(D) gfge L$:(D).

As Hiirmander has shown, assertion a) of Theorem 6.6 is valid for any: bounded pseudoconvex domain D. Moreover, the following result holds.

Theorem 6.7 (Hiirmander [44]). Let D be any pseudoconvex bmain in C” and cp any plurisubharmonic function in D. Then, the equation af = g adyits a solution with the following global L2-estimate

).

d (f(z)12e-+‘(‘)(1 + Iz~2)-2dV2,(z) s Ig(z)12e-*@)dV,,(z). d

(6.14)

Kohn, using Theorems 6.6, 6.7 obtained the following result on smoothness up to the boundary for solutions of the &equations. ,I

Theorem 6.8 (Kohn, 1973, [SS]). Let D be an arbitrary bounded pseudoconvex domain in c” with boundary of class C” and g 2 &closed form of class C$ l(D). Then, the, equation af = g has a solution fe P(@. Y

IT U.M. knenwin I

. , 6.4. Formulas for Solving &Equations. The method of a priori LJ-estimates is not suitable for obtaining the: estimates in the &equations which are’necessary for the applications related to the L” or L’-metrics.

. Moreover, problems from the theory of functions require explicit forniulas for the solutidn of &equations, and also, as a rule, each problem requires its o.wn formula.

,

.The integral’ representatio& of ,iype’ Cauchy-FantappiB prove to be well adapted for obtaining such formulas.

Ntiely, from Theorem 4.3 it is easy to,obtain the following result (see [19]).

Theoiem 6.9 (G.M. Khenkin, 1970, s = 0; G.M. Khenkin, A.V. Romanov, 1971, s = 1: S.A. Dautov, GM. Khenkin,,l978, s > 1). Let D be -:n arbitrary

’ strictli pseudoconvex domain of type (3.14), KS and H” the integral ‘operatoi-s given by (4.7), (4.8), (4.13), and (4.14): where s 2: 0, an4 g Q &losedfirm ofcluss Lh, 1 (D).

, m? the equation 8f = g has a uniquksolution f satisfying the bounddry condition K’f= 0 and f is given by the form&

f = H’g. (6.15)

Hege the operator g + Hag is completely continuous from Lg, 1(@ to Lp(D) for all 1 s p 5 a~. Moreover ifthe domain D has a smooth. boundary, then for s > u + 1,

, the! estimates s .

llfl IPlyco,S Yll Isl*bI” + Is ̂ &wlL-1’211~q~, ~ ,fora>O;tind ” ~’

: -(6.16)

f 0; a i 0, hdld. ,;

ylL(D) y I91 + 19 A mlPl-“211L(D) ’ ‘.

The latter estimate’ was first obtained using other.formulas in the works bi G.M. Khenkin and Skoda in 1975 (see [49]):

Theorem 4.4 allows one to obtain even more general global fotiulas and qstimates for solutions of aequatioris. For example, if in Theorem 4.4 we set

then foi R = @” we obtain the following reiiult. ‘_-

+iem 6.10 (Skoda, 1971; Berndtsson, Andersson, 1982, see [12]). Let g be . Q %closed (0,l) form in d=!’ such that

. . . .I

r-2n ,,11,, ldtXdJ’2.(0 = Ott1 + rJ% Q 2 - 2n.

: ?hen there exists a solution f of the equation af = g wi;h estimate

r-‘” ,,/<, If(C)ldV2.(C) = O((1 + r),+l(l + b(l + r))). ’


This solution is given by the formula

1 x all - 4’ A @N - z12)n-‘-’ A (~QK,,” 1~ - Z\2(- 9

f6 17j

,

where m > a + n 2 m Amin(n - 1, m) - n, and has the property \

where P,,, is the orthogonal projectionfiom

W,(z) (1 + lzl2)“+’ .

Returning’to the assertion of Theorem 6.9, we now remark that for the ball in C”, K 1 = B, and hence in this situation, the solution H 1 g is precisely the Kohn solution a*Ng from Theorem 6.5.

However, the operators H’ and a*N, which agree on &closed forms, are given, nevertheless, by different kernels.

An exact calculation of the kernel of the operator N for the ball was carried out only recently (see [37] and [61]).

Theorems 5.3 and 6.9. allow us to obtain precise LP-estimates (1 5 p < 03) for the Kohn solution.

Theorem 6.11 (Ovrelid, 1976; Greiner, Stein, 1977, ‘see [61]). Let D be a strictly pseudoconvex domain in a=” with boundary of class C”. Then the operator g + f = a*Ng, solving the &equation af = g in D, isfor every PE [ 1, co] and k 2 0 a completely continuous oderator from the space L!;:(D) to the space LP*‘(d).

Further developments in the cited works led (see [9] and [61]) to an asymptotic formula for the Neuman-Spencer operator N in strictly pseudoconvex domains from which the assertion of Theorem 6.11 can be derived immediately.

Obtaining solutions of &equations with L” or L’-estimates in weakly pseudoconvex domains or in pseudoconvex domains with non-smooth boundary turned out to be more difficult problems which up to now have been solved only in the class of pseudoconvex pol)hedra of type (4.25).

Theorem 6.12 (G.M. Khenkin, A.G. Sergeev, 1980, see [54], [19]). Let D be a ‘pseudoconvex polyhedron given by (4.25) and satisfying the non-degeneracy condition 5.5. Let K be the Cauchy-Fantappid operator (4.28) alrd H the operator (4.20), (4.27). Then the solution to the equation i?f = g with the property Kf = 0 is

-.

.

16 . G.M. Khenkin ,

represented by the formula f = Hg. Also

II f II LP(B) 5 Y II 9 II LP(D)’ 1 5 p I m, (6.18) *

for any &closed form g E L,P. 1 (D).

The operator H in this theorem (in contrast to the operator H” in Theorem 6.9), is generally speaking, not continuous from & l(fi) to LP(D); that is, the condition that the form g be &closed is necessary, not only for the solvability of the &equation, but also for the validity of,the LP-estimates (6.18).

The fundamental difficulty in the proof of the estimate 6.18 consists in finding a reformulation of the formula f = Hg such that the estimate becomes immediate.

In practice, for example, the following global formula for solving the &equation in the polydisc

works well. . For ordered multi-indices K, J we set

YX(‘J) = {CeD”l: (J = zJ; lZj,l 2 * * * 2 [Zj-1 I

2 I&l=. - * = IL,l;lrll 5 IC,,l.W~uJ}, wherer=IJ(,p=lKI,

kex (ck - zk)(l - tt-kzkJBk ,, /4&i A dL(l - L&jal-' * I$KUJ (1 - Cz#+ 1 ’

wherefi,>O,k= 1,. ..,n.

Theorem 6.13 (G.M. Khenkin, P.L. Polyakov [65), [40], [41]). Let 07 be the ’ polydisc in @“, and g a &-closed (0,l) form whose coeficients arefinite measures on 07. Then, the function

II-1 f=Hg(z)= - c

: r=O J:(

c (- l)w.P) j s(C) i HJ.x(L 4

K:KnJ = (Iz(} Y&,) >

(6.19)

is integrable on D and satisfies the equation af = g in D. Moreover the operator H is continuous from LP,, 1 (0;) to Lp(DI) for all 1 I p I 00.

Formula 6.19 generalizes a series of formulas obtained earlier for the case of the bidisc (see [1.9], [17J).

Integral representation of Cauchy-Fantappie type allow one also to prove, in an elementary fashion, the following general result on (Y-estimates for /

II. The Method of integral Representations in Complex Analysis 17

solutions of the. &equation in arbitrary convex domains thus complementing Theorem 6.8.

Theorem 6.14 ([52]). Let D be an arbitrary bounded convex or strictly linearly convex domain in @” and g any &closed form of class C$! 1 (D), s > n - 2. Then, for each E > 0, the equation af = g has a solution in D of class Cs-n+2-e(~).

For s = co this theorem was first proved by A. Dufresnoy. (Ann. Inst. Fourier, 29, pp. 229-238, 1979). See also [66].

A description of those pseudoconvex (or even convex) domains D, for which the equation as= g has a solution with a uniform estimate, I/f llL,(n, I y I( g ((Llo(Dj, present itself as a rather difficult problem. We remark that Sibony [77] constructed a pseudoconvex domain D with boundary of class C” and a &closed form g E Lz I(D) such that the equation 8f = g has no bounded solutions in D.

6.5. The Poincark-Lelong Equation. Construction of Holomorphic Functions with Given Zeros. Formulas and estimates for solutions of the &equation yield a rather flexible apparatus for many problems in the theory of functions on domains of C”.

All of the results in $5; for example, can be well interpreted in the language of se-equations. In particular, Theorem 5.13 on the continuation of bounded holomorphic functions from a submanifold to the polydisc was obtained in [42] on the basis of formulas (6.9) and (6.19).

Formulas for solving the &equations turned out to be particularly useful in the problem of constructing holomorphic functions with finite order of growth and having given zeros.

We present now the basic results in this direction respectively for @“, for strictly pseudoconvex domains, and for the polydisc.

Let F be a fun.ctlon holomorphic in @“. By OrdF we denote the infimum of those a > 0 for which

. ,i,” lzl-‘- ’ In+ I WdV2,(i) < co.

If F is a function holomorphic in a strictly pseudoconvex domain D = (z E Q)“:

p(z) < 0} or in the polydisc 0; = (ZE @“: pi(z) = sup In lzjl < 0} then we denote

by Ord F the infimum of those a > 0 for which F belongs respectively to N,(D) or N&D;), where

N,(D) = { FE%(D): J Ip(z)l”-’ In+ I F(z)ld Vzn(z) < co}, zcD

N,(I);) = { FE X(D;): s IPM-’ In+ I Wld~n+I~4 < a>, {zaD;:lt,l = = I&I}

where In+(t) = supfln rt 01.

78 GM. Khenkin

Further, let M = C’J~~, be an (n - I)-dimensional analytic set’respectively

Yin C”, D or D;, wherl {M,.) are the irreducible (n- I)-dimensional components of M and I;!,,} are the multiplicities oi’ these components.

We denote by Ord M the infimum.of those IY > 0 for which

if M c C”;

c yy J I$w’ dV2,-2(z) < ‘;o> Y XM,

if M c D; and finally,

(6.20)

where

M,., = (ZE M,: sup(lnlzjj) 4 y inf (lnlzjl), y 2 0}, j j ’

ifMcD?. c

From the formula of PoincarC-Lelong or the multi-dimensional formula of Jensen-Stoll, it foltows immediately (see [78]) that for any function F holomorphic ,in C”, D or Dl, the inequality

OrdF>OrdM,-

holds, where Mr is the set of zeros (counting multiplicities) of the function F. The construction of a function F with given zeros M such ‘that OrdF

‘= Ord M was carried out, in the case of one cotiplex variable, in classical works: . for entire functions, by Bore1 in 1900, and for functioils in the unit disc, by

M.M. Dzhrbashyan in 1948 (see [21]). For functions of several variables, the first precise results were obtained by Lelong and Stoll.

Theorem 6.15 (Lelong, 1953; Stoll, 1953; see [73], [32], [78]). For any (n - l)-dimensional analytic set M in C”, there exists a.holomorphic function F such chat M = (z E C”: F(z) = 0} and Ord F = Ord M.

Functions b with the property Ord F = Ord M were construetedin the cited works with explicit formulas, which, in the light of Theorem 6.4, can be

’ considered as explicit solutions to the Poincari-Lelong equation: a (dF/F) = i[M]. Par example, we obtain Lelong’s formula if we

’ Theorem 6.10 to solve the PoincarC-Lelong equation. use formula (6.17) of

For strictly pseudoconvex domains, it is possible, with the help of Theorem 6.9, to give (see [20], [49]) a complete characterization of the zero-sets for

II. The Method of Integral Representations in Complex Analysis 79.

functions of the class N,(D), a 2 0, where

N,(D) = FE%‘(D): j ln+IF(z)jdV,,-i(z) < 00 1 ,.

‘. . .aD / ,.

Theorem 6.16 (G.M. Khenkin, 1975; Skoda, 1975; S.A. Dautov, ’ G.M. Khenkin, 1977). Let D be a strictly pseudoconvex domain in 67’ with _. smooth boundary. In order that an (n -‘l)-dimensional set M in D be the set +f zeros of a function ofclass N,(D), a 2 0, it is necessary and sujticient that condition (6.20) hold, if a > 0, and that the Blaschke condition hold

x Yj j Ip(z)ldv2n-2(z) < a, 1 MI 7

klthoigh for the case of the polydisc 07, a. coinplete characterization of the ’ . zero sets‘of functions of class N&D;) has not yet been found, nevertheless, the -, formulas pf,The&cm 6.13 allo-, us to prpve the following. ~

.’ ‘:. Theorem 6.17 .(G.M. l$henkin, P,L. Polyakov, 1984; Charpentier, 1984;

P.L. Polyakov, 1986). For any (n -I)-diken&onal analyticset M in thT,polydisc PT., there eiists a holor@orphk function F whose zeros lie i‘;l M and such that’ * Ord F = Ord M.

This result was obtained for the bidisc in [4‘i], [ 173 and fb’r the general case by P.L. Polyakov ie [70];

One ‘of the central unsolved problem; in the theory of functions in pseudoconvex domains is the “Coromi” problem: let fi , .-. . , fN be bounded holomorphic . functions in a strictly. pseudoconvex. dotnain D such that If,l’ + . . . + IfN12 > S > 0 in D. Do there exist bounded holomorphlc functions h,, . . . , h, in D , such that fihl + . . . + f,h,, 3 1 in D?.

. Making use of formulab(6. !5) and a suitable generalization of the celebrated construction of Carleson and Wolf (see [26]) in the “Corona” theorem for dorpains in Cl, it is possible to prove (see [49], [40]) an asseition which yields< hope that a generalization .of Carleaon’s.Jheorem to strictly pseudoconvex ,domains in @” could be established. i.

Theorem 6.18 (G.M. Khenkin, 1977; Varbpoulos, 1977; Am%r, 1980;. With the hypotheses of the “corona” problem, there exist functi&s h, E n fP(aD), -_ j=1,2 ,I..., N,suchthat.

PC=

< -‘.. N’ >.. . -_ -..

The futidamental difficulty -in the .multi-dimensional “corona” problem consists ii finding precise conditions on a form g in the domain D under$lich the eqtiation af = g possesses a bounded solution in D.

-.

I 80 G.M. Khenkin

The strongest result in this direction was obtained recently by Berndtsson (Berndtsson, B., An L”-estimate for the &equation in the unit ball in @“. Preprint, University of Giiteborg, 1983, 35 pp. and Berndtsson,’ B., &, and -“I Carleson type inequalities, Preprint, University of GGteborg, 1986, 17 pp.).

$7. Integral Representations in the Theory of CR-Functions

Integral representations of Cauchy-Fantappit type allow us not only to make classical results (with estimates on the boundary) more precise in pseudoconvex domains but also to significantly advance the theory of functions on real submanifolds of @“.

at .Let M be a real smboth submanifold of @“, T,(M) the real tangent space to M the point ZE M, and T$(M) the largest complex subspace lying in T(M). The manifold M is called a Cauchy-Riemann manifold (CR-m@& if the

number dime F(M) = CR dim M does not depend on the point ‘5~ M. If CR dim M = 0, then the manfild M is said to be totally real.

A CR-manifold M is said to be generic or of general type if T,(M) @ JTI(M) .-- = @“, where J is the operator in Iw2” obtained by multiplication by J - 1 in the

space @” z (w2”. Such manifolds can be locally represented in the form

M = {z&p,(z) = . . . = p&z) = 0}, (7.1) wherep = {p,, . . . , pk} is a collection of real smooth functions in the domain $2 with the propelety apI A . . . A &I, # 0 on M.

In terms of the representation (7.1), we have

A smooth function f on a CR-manifold M is called a CR-fin&on, in other, words, a function, satisfying the tangential Cauchy-Riemann equations cMj= 0, if for any complex vector field c(z) = (l,(z), . . . , &(z))E E:(M), ZE M, we have

jil i;(z)&) = 0. J

For generic CR-manifolds given by (7.1), this equation is equivalent to the equation

rTTA 5p, A’. . . A apk = 0 on M, L

wherefis any smooth extension of the functionjto the domain R 2 M.


If f is lqcally integrable or even a generalized function on M, then the equation &f = 0 should be understood in the generalized sense: Iwf A 8~ = 0, for any smooth differential form (p of compact support.

If the CR-manifold M is real analytic and f a real analytic function on M, then as Tomassini (1966, see [3]) has shown, the equation &,f= 0 is equivalent to the requirement that the function f be the trace on M of some function holomorphic in a nlighbourhood of M.

However, for a smooth CR-function on M, generally speaking, a holomorphic continuation to a neighbourhood of M does not exist.

i

7.1. Apprqximation ind Analytic Representation of CR-Functions. The theory of CR-functions is rather far advanced for the case of real hypersurfaces in C”. An especially effective’ result here is the analytic representation of CR- functions on hypersurfaces given by Theorem X8.

For a CR-manifold M of arbitrary codimension, it is natural to define an analytic representation of Q CR-function f of the form

(7.2)

where each fv is a .CR-function on M which admits a holomorphic extension to some domain D,, in C” such that M'c aD,. '

Such a representation has been obtained thus far only for isolated cases. First of all, an analytic.-representation holds (see [38]) for functioqs on arbitrary totally real manifolds.

Theorem 7.1 (G.M. Khenkin, 1979). Let M be a C”-smooth generic totally

real submanifold of a d&in Q c Q=” giuen by (7.1). Let {Q, 1 be ajnite covering of the domain Q\M by strictly pseudoconvex domains of the form

R, = {zdl: p:< 0). v = 1,2,. . . , n + 1,

where { py} are strictly plurisubharmoriicfunctiotis on’!3 such that pV = 0 on d and apV, A . . . A apV,, # 0 on Mforeach v, < . : . < v,. Then, for any pseudoconvex subdomain D, g S2 and any functidn fe C”(D, n M), E I s - 2, one can construct functions f,, , . . I ,.,, holomorphic in the domains Don D,, n . . . 6 D,,. and of class Cab” on Do n M such that

n+l f(z)=vp-l)“+vfL. . ,v-l,v+l, . ..+I(49

zeMnD,. (7.3)

In the’ case where M = KS” and a = + 30, the assertion of Theorem 7.1 is contained in the welliknown theorems of Martineau (1970) and Bros-Jaglonitzer (1975) which complement the the,ory of hyperfunctions of Sato (see Vol. 8, article IV).

82 GM. Khenkin .TL . -

It was found [38] that it is possible to write functions satisfying the assertion of Theorem 7.1 in the explicit form

nol- 1)

(- 1)-r

fv1v ” A = - (27ci)” WK, K 4, . . . , p,.(i, .410(l)

J”D”f(cJ a,,,((, 2) . . . @““(C, z) ’ Ve4) where *- .

z~D~nD,,n.. . nDVn; @,K 4 * <py(c,z), i - z)

are the barrier functions for the strictly pseudoconvex hypersurfaces {z E D,: py = 0} with the properties from Theorem 3.10.

The holomorphy of the functionsf,,; . . . , yn in the domains D, n D,,, . . . , -nDVn follows.immediately from the definition (7.4).

To obtain the equality (7.3) it is sufficient to consider the integral representation (4.28) in the pseudoconvex polyhedron, , ’ .

. . DE, = {zED,: p,(z) < E, v = 1‘;2, . . . , n+ l},

and to pass to the limit as E -iO. The smoothness of the functions ji,, . . . , y. on M n D, is a consequence of

the estimates for Cauchy-Fantappie type integrals given by Theorem 5.6. Formulas (7.3) and: (7.4) can be seen as generalizations to totally real mani-

folds of the following Fourier-l&don type transformation

I .

where x, .YE [w”, found in the works of Sat&Kawai-Kashiwara, Bros- Jaglonitzer, and Bony in the years 197%1.976 (see Vol. 8, article IV).

Theorem 7.1 together with formulas ‘(7.4) and (7.5) admit ‘several nice corollaries. First of all, from Theorem 7;l; the weil-knoin- result on ‘global apprcximation of functions on. totally real submanifolds in @“-follows (see [19], C401, C821). I

Theorem 7.2 (Hlirmander, Wermer, 1968; .Nirenberg, .Wells, 1969; Harvey, Wells, 1952; Range, Siu, 1974). On any C’-smooth totally real man$ofold M in C”, anyfunction f ofclass C’-!(M) can be approximated in the C’-‘(K) norm on each compact set K c M by functions holomorphic .in a neighbourhood of M. - .

Theorem 7.2 in turn allows one to prove immediately an important uniqueness theorem for CR-functions on arbitrary CR-manifolds.

Theorem 7.3. (R.A. Ajrapetyan, GM. Khenkin [2], Baounedi-Treves [8]). Let M be any CR-manijbld of theform (7.1) and N a generic CR-stibmanijiild ofM, i.e. N={zEM: pk+j=O, j=b ,... i-m}, where wiSn-k, and {P~+~} yare smooth real functions such that 8p, A . . . . A %pt+,,, # Q on N. Z%en;‘aiily cor& . tinuous CR-function f on M which is equal to zero on N vanishes in some neighbourhood of the manifold N in M.


For the case when M is a hypersurface in @“, the result of Theorem 7.3 was first obtained by S.1: ,Pinchuk,(l974).

. 2- -

Prodof Theo&n-7;3 (see [2]). Without loss of generality, we may assume ’ that_m = n - k, i.e. N is totally real. Let R, be a neighbourhood of the m&ifold N such that N as well as all totally real manifolds NOsufficiently close to N in the C’itopology are holomorphically convex ifi R,.’ _

Let N” be’any totally real submanifold ofM close to N and with the property @V u No,) = 0; and let Q be a manifold in Msuch that aQ = N u No.

Suppose’now thai f satisfies the hypotheses of the theorem. In order to prove that f vanishes on No, it is sufficient, by Theorem 7.2, to verify that

. . * ~f&O. * -., - --

for any holomorphic (n,Oiform. Jndeed, by Stokes’ ‘formula and by the hypotheses of the theorem, we have

, .*,. “gr A cp =pfA’P+p.q+.

.

Baouendi Andy Treves [g] obtained Theorem 7.3 as a corollary* ,of their theorem on the local approximation of CR-functions on arbitrary CR-manifolds by holomorphic functions. .

“? -- Theorem 7.4 (Baouendi-Treves, 1981). -Let M be an arbitrary CR-man$fold

of class C”, s 2 1 in @* and .f ana arbitrary CR-function of class C”- ‘(M). Then, for each point z* E M, there erists a neighbourhood RZ,- such that on M n R,*, the

, function f can be approximated with arbitrary precision in the topology C@-‘)(M n a,,) by pu~ynomials~in z, , . . . , z,.

The result of Theorem 7.4 was known previously only for hypersurfaces (Hire&erg-Wells, E.M. Chirka), tmd also for standard CR-manifolds (Naruki) (see I31).

The proof of Theorem 7.4 in [S] is based on constructing a generalization of the classical proof of the Weierstrass approximation theorem. Namely, let ‘us fix on M a totally real submanifold Mfl passing through the point z* and of the form

Mt. = {zEM:pk+](z) = 0, j = 1,. . . , n - k},

where ( pk +I} are functions of class ,C” such that pk + j(Z* ) = 0 and

&A ..,. /Jp*#O

in some neighbourhood 0,. on M of the point z*. Also, we set ‘. M:, = {zcM: Pk+l = &+..I,. . . ,.p. =.e.}*

84 G.M. Khenkin

I . Consider the entire functionf, given by i

It is a classical fact that for so&e neighbourhood @! of the &nt z* and for any functionfE C’“-“‘(M), we havefIlM,* +flMZ* as v + co in the topology of C’“- l)(MZe n a;.). B, douendi and Treves showed‘ that using the .condition a,&-= 0 on M we may conclude that for any sufficiently imall vector E = (Q+ 1, . . . ..) E,), we also have that

J&. -?flu:. , in the topology of C’“- “(&Ii. n fig.). From this the assertion of Theorem 7.2 follows.

Thus (7.6) can be viewed as a local approximate integral representation bf a GR-function on M.

Although analytic representations of the form (7.2) for arbitrary CR-functions are impossible on arbitrary CR-manifolds (J.M. Trbpreau, 1985), there is a suitable integral representation (generally with a non-holomorphic kernel) which works well in practice and which generalizes’ the representations (7.3) and (7.4). A P-valued vector function P = (P’“, . . . , P’“)) of tilass C’(R x 12) is called a strictly regular barrier for the level function p E C2@) if

and -

. 2RWLz) 2 ~(0 - P(Z) + rll - zl’, _ where ([, z) E R x 0, @((r, z) = (P(c, z), [ - z>, and y ib a positive constant.

Further, let M be an arbitrary CR-manifold of the form (7.1),

k

Pk+lcmjzl@j; D, = {zEQ: pO(z) < 0}

a relatively cdmpact subdomain of 0, MO = M n D,; and

Dj = (z~ D,: pi(Z) < O}; j=l;&...,k. .’

Let (Pi} be arbitrary strictly regular barriers for the level functions pJ E C2(Q, j = 41, . . . . ) k + 1.

For each multi-index J = ( j,, . . . , j,.), we set


where zgDJ z Dj,n.. . nDj,.

Theorem 7.5 (G.M. Khenkin, 1979), ‘For any integrable CR-function f on

MO, the functionsfJ, defined in the domains D J by equation (7.7), have generalized

boundary values on MO. Moreover, we have

lL+i j f(z) * cP(Z)= v~ll(-l)k+y ,,Lfk.. . 3 .-l.v+l....k+l(z) A 4’(z), (7.8)

EM,

where cp is an arbitrary (n, n - k)-form of compact support in Do and of class C”, a 7 0.

Recently, Baouendi,“Rothschild and Treves (Invent. Math. 82,359-396, 1985) have obtained analytic representations of CR-functions on standard (or rigid) CR-manifolds of the form (7.9). This result can be obtained also as a consequence of formulas (7.7), (7.8) for CR-functions on such CR-manifolds.

7.2: CR-Functioris and the ‘*Edge-of-the-Wedge” Theorems. The following result on analytic continuation of CRifunctions through the “edge-of-the- wedge” was-obtained in [2] and C33 on the basis of the integral representations (7.7) and (7.8).

“Theorem 7.6 (R.A. Ajrapetyan, G.M. Khenkin, 1981). Let us-view,a generic.. CR-manifold M given by (7.1) as the common boundary of CR-man$olds given by

Mj= {Zig Pj(Z) <O, P,(Z)=O, V ~fi>.

Then, any function f continuous on the compact set

tj ~j

b .

j=l

and CR on the manifold

has a continuous extension in 62 to a.function holomorphic in a domain R’ such that for any E 7-0, we have

$?n+~ zEU,(M):Pj<e

Mjr\ U,(M) c iif, where US(M) is some neighbourhood of the manifold-M.

In the case where all‘of the functions (p,,} which determine the CR-manifold M given by (7.1) are pluriharmonic, then the assertion of Theorem 7.6 is true also for E = 0.

Theorem 7.6’ (R.A. Ajrapetyan, G.M. Khenkin, 1981). Consider the CR- manifold

Mo={~~~ReFo>0,ReFj=0,j=1.2..- ,n},

86 GM. Khenkin _. -

where { Fj> are functions holomorphic in the domain R c ~2”. T~U& M, c R aruk tiO is-the common. boundary of the CR-manifolds given .by . 1.

Mj=(zCR:ReF,>O,RkFj>O,ReF,=O;yfj}, ,,

jyl,..., n. Then, any continuous CR-function f on ’ ’

has a continuous extension F-which is holomorphic in the analytic polyh&ron _

_ Inthecasewher~‘FO=l,F~=zj,.j=l,;.. , n, i.e. when M = W’and

..

the result of Theorem 7.6’ was obtained first by Malgrange-Zerner (1961) whose resul! in turn extended the classical theorems. on,. separate analyticity (S.N. Bernstein) a@ on the “edge-bf-the-wedge” (N.N. Bogolyubqv;‘Bremerman- Oehme-J.G. ,Taylor, V.!$ Vladimirov) see [al]. .,

From Theorems 7.6 and 7.6’ follow not only a list of classical rksults but also a series of gt%eralizations‘theresf (Epstein, Browder, S.I. Pinchuk, E.M. Chirka, B&ford) (see [3])p~,~-m -’ ~

The prdof of The&em 7:6 ,tinsists in verifying that the desired holomorphic cqntinuation F off can be given by the formiila of A. Weil

.’ X

detPX,z), . . . I pv- 1 (6, d9 P,G a, P,+ l(C, z), . . . P.(<, z)]

j&tFjCO - Fj(Z))

o(C) 9

. L 1 ,_ * ,

where Fj(c) - Fj(z) = (P~([,z), [ - z), j = 0, 1, . . . , n, and {Pi) are holo- i morphic functions of the variables c’ ZER.

In applications, Theorem 7.6 works well in conjunction with the uniqueness Theo&n 7.3 and-the strong princi#e of continuity of Hat-togs called the strong theorem on discs (see [Sl]).

Theorem’ 7.7 (Bremerman, 1954). Let .‘z I = z(t), t E CO, 11 be a real-analytic I curue in C: and D(t) = {(z,w): z = z(t), Iwl < l} a continuous family of discs in

@: x C,.. Thea, if a function f--f(z, ~0) is holomorphic in o <v< lo(t) imd at least


in one poinf ofrhe limit disc D(O),.then f is holomorphic at all points of the limit disc

D(O).

We remark that, without the assumption that the curve z = z(t) .be real analytic, the assertion of Theorem 7.7 is in general not true, (see S. Favorov, Funct. Anal. Appl. 12, 90-91, 1978).

Theorem 7.7 is a corollary of a classical lemma of Oka: if z(t) is a continuous curve in d=’ and u(z) is a subharmonic function in a neighbourhood of the curve, then

pj u(z(t)) = u(z(0)).

The’ following result [34] can be seen as a deep generalization of both the strong theorem on discs as well as the Bogolyubov theo.rem on the “edge-of-the-wedge”. i

Theorem 7.8 (Hanges, Treves, 1983). Let M be a generic CR-manifold in @“; N a connected complex analyt.ic submani,fold of--M; and f P (generalized) CR-function on M. Then, iff has a holomorphic continuation’in the neighbourhood of some point z* E N c. M, then r extends holomorphically in the neighbourhood of each point ZEN.

. The original and rather difficult proof of..Theorem 7.8 in [34] uses the technique in the works [7] and [8] and the Fourier-Bros-Jagolnitzer transformation (73

A simpler proof. df Theorem 7.8 can-be obtained by combining Theorems 7.5-7.7. .

7.3. Hoiomorphic Continuation of CR-Functions &en on Cow& : CR- Manifolds. The theory of CR-functions has its origin in theoiems of Bochner and H. Lewy which give conditions’(respectively global and local) under which a CR-function defined on a real hypersurface of @” necessarily has a.holpmorphic continuation to some domain in C”.

In 1960, H. ,Lewy found the first example of a CR-manifold of higher codimension in C” for which an analogous situation prevails.

The fundamental results on holomorphic continuation of CR-functions are formulated using a natural generalizatiqn of the Levi.-form for arbi!rary CR-manifolds.

Let us denote by N,(M) the space normal to the CR-manifold (7.1) at the point ZE M. We shall assume further that the functioris {pi}, in the representation (7.1), are smooth‘of order C2(Q and so chosen that the vectorsgra$l pi(?) form an orthogonal basis for N,(M). The Leviform of the manifold M at the point z E M is the quadratic form L?(i) on T:(M) with values in N,(M) given by

where [ E T:(M).

88 G.M. Khenkin II. The Method of Integral Representations in Complex Analysis 89

We have the following geometric interpretation of the Levi form in terms of the Lie hracke$ of a complex tangent vector 6eld

where

is th& Cauchy-Riemann differential operator in the.direction of the complex tangent vector field [ with the property [(z) = [E T;(M). Here [[, 41 is the commutator of the differential operators [ and %.

A CR-manifold M in C” is said to be a standard manifold if it has the form

M = (z = (z’,.z”)~Q=~ x @n-k: p(z) = lm z’ - F(z”, 2”) = 0}, (7.9)

where F = (F,, . . . , Fk) is a Ck valued Hermitian form on Cn-k. If we identify the space N,,(M) with the space Rk in which the form F takes it,s

v&es, then Lf = F(z”, z”), Z”E QYk. For an arbitrary point 5 on a standard manifold (7.9), we have

T;(M) = (~4”: z’ - r’ E 2i(F(z”, [“) - F([“, z”))),

L?(Z)= jiI Fj(Z” - r, Z”-- r) grad pi(c).

‘The ‘importance of the role of standard CR-manifolds among all CR-manifolds is due to the fact that any CR-manifold is locally equivalent up to order two to a standard one. More precisely, if M is an arbitrary C2-smooth CR- manifold given by (7.1), then for any point z* E M there is a neighbourhood U and a biholomorphic mapping

cp:U+ncckxc”--k

such that cp(z*) = 0 and

’ cp(M n U) = ((z’, z”)E@~ x 6Yek: p = lm z’ - F(z”, 2”) + R(i) = 01, (7.10)

where F is a Ck-valued Hermitian form and the remainder term admits the estimate IR(z)l = o(lzj’).

A CR-mantfold M is called Levi-flat if L?(c) E 0 for each z E M. A real-analytic Levi-flat CR-manifold necessarily (s&e below) admits locally the following form

M = (( E R: pi = Re Fj(c) = 0, j = 1, . . . , k},

where ( Fj} are functions holomorphic in the domain R. For a standard CR-manifold one can precisely determine a domain to which

all CR-functions on M automatically continue holomoiphically (see [3]).

Theorem 7.9 (Naruki, 1970; R.A. Ajrapetyan, GM. Khenkin, 1984). Let M be any standard CR-manifold ofthe.Jbrm (7.9) and such that the convex cone V,

spanned by the vectors F(z”, z”), where z” E C”-lr, is solid in Rk. By D, we denote the domain g&n by

’ D, = {z E 6”: p = lm z’ - F(z”, z”) E V}.

Then, - a) each continuous (or generaliz,; !l CR-function on M has a holomorphic

continuation to the domain D,; b) ifmoreover the cone V is acute, and M0 is a relatively compact domain on M

of the form M, = (ZE M: Re m,,(z) <, 01, where QD, is some holomorphicfunction, then the maximal domain to which an arbitrary CR-function on M,, extends holomorphically has the form

D,, = (z E D,: Re @Jz) < 0). .

Assertion a) of Theorem 7.9. for continuous CR-functions was proved first in the rather difficult work of Naruki. The more precise assertion b) can be proved in a remarkably more simple fashion [3] with the help of integral formulas.

Namely, if f is a CR-function on M,, then a holomorphic function on DyO agreeing with f on M, can be given by the explicit formula

F = tn - 1Y (2nirtb + Fd, where

FM = s f(&o’ topgg + (1 - tO)g$ >

A fJJ(ih (i,f,f”k(iMg) x u x [O. I] 0

and iJ denotes the set of points t on the unit sphere Sk c (iw”)*, for which the scalar form (r, F) is positive definite,

w, z, t) = (PK, z, t), i-z); @o(L z) = (P,(i, z), i - z). Only recently [Z], [IS], [39] was there success in obtaining an analogue to

Theorem 7.9 for arbitrary CR-manifolds.

Theorem 7.10 (G.M. Khenkin, 1980; R.A. Ajrapetyan, G.M. Khenkin, 1981; Boggess. Polking, 1981). Let M be a CR-mantfold given by (7.1) and such that

for a point ZE M, th e convex hull VZ of the vectors L:(w), where WE T$(M), is a solid cone in N,(M). Tht-n, for any open subcon’e c c V, with the property ii r/p n ? VZ = (z}, there exists a neighbourhood R, of the point z such that each continuous (or generalized) CR-function f on M extends to afunction F holomorphic in the domain (M + k$)nR,.

90 ,,/ G.M. Kheikin

For the case V, F y;(M), Theorem 7.10 was first obtained in [39),&1d solve9 the Naruki problem (@69). /’

In the above form, Theoreni 7.10 is a rapid consequence of Theorem 7.6 on the edge-of-the-wedge for CR -functions and Theorem 7.16 on suspending analytic discs on CR-manifolds. An exp!icit formulation and proof %f Theorem 7.10 based on the approximation Theorem 7.4 was first given in [lS].

The method of integral representations ([2], [3], [39]) allows one to prove Theorem 7.10 just as we did for Theorem 7.9, with an explicit formula. In order to state this formula, it is necessary to introduce several definittins.

By the Levi form of the CR-mani$oajfZ A4 (7.1) at the point z E J&i; the direction , t, = c t’ grad p](z) or simply in the direction t =! (t’, . . . , t’), we shall mean the

scalai quadratic form

m4 = oz, Jww),

whdre pr = (t, p), w E T$$V). %.’ The manifold M is said to be q-concave (respectively weakly q-concaue) at the

point ZE M in the direction t E S’ if the form L;*(w) has at least q negative (respectively q non-positive) eigen-values on T$(M). We shall say that the manifold M is’ q-concave (respectively weakly q-concave) at the point z E M if it is q-concave (respectively) weakly q-concave) in each direction t E Sk.

Let us denote by S,, the closed seaof those points t on the sphere Sk for which the form E’(w), WE T$(M) has less ihan q negative eigenvalues. In particular, S, ,z is the set of points t E Sk for which the form (t, LF) iipositive semi-definite on T$(M).

Integral representations of CR-functions (and CR-forms [3]) on q-c+cave CR-manifolds make us of the following important assertion.

Proposition 7.11 ([3]). Let M-be a C2-smooth CR-manifold gioen by (7.1), t* EM, and U(S, ,*) a contractible neighbourhood of S, + Then there exists a neighbourhood f&, of the point z* and a constant A > 0 such that for ea,ch t ES’, there exists ta strongly regular barrier vqctor-function .P(& z, t) for the levelfunction d, = (t,p) + A(p:-+ . . . + p,‘) in the domain R,. Moreover, P is a Cl-smooth function oft with the property !!

(&P([,z,t) A dC)n-k-q”l = 0,

for each (c, z) of&, x R, and t E Sk\U(Sq, z*). ,I

Making use of the barrier function from Proposition 7.11, we can write down a function F satisfying the requiremen explicit formuia [4]: IF;

)pf Theorem 7.10. Indeed, we have the

F (n - l)! = FW”f + K3f)P

**mn. (7.11)

II. The Method of Integral Rcplmntations in Complex Analysis

. where : __ i

~ .:, .

U is a neighbourhood of S,,ti on SL; and P(J, z, t) is the barrier vector-function satisfying Propositon 7.11 for q = 1 and Q((c,,z, f)i~ (P([, z, t), c-z).

7.4. The Phenomena of Hartogs-Bochner ~PII& H. Dewy on l-Concave CR- Manifolds. Theorems 7.2, 7.3, 7;4, 7.10 and their generalizations to CR-,.,, forms (see [3]) lead to [50] a Hartogs-Bochn& type effect for CR-functioris od: ’ arbitrary l-concave CR-manifolds. ’

Tbxem 7.12 (G.M. Khenkin, 1984). Let ncc” be 4 pseudoconvex domain ,,’ and .p=(pI,. . . &IA . . .

,pc} a collection of real fdnctions of class C2(i2) such that + a,*#0 in $2 ,and each CR-manifold M, giuen by M,=.{z&::.

p(z)=&}, where E=(Q,. . . i&k), is .l-concave. &et Q o = (z E $2 : pa(z) < 0) b$;;$ relatively compact subdomain of h and let, E’ = (E:, . . . , $} be a ‘cbllection suck that thk manifold M a,,,A . . .

o= Mzo n Q, has a connected complement ‘in IU,,, cirid h a& A 8p. # 0 almost everywhere on ah4, Then, any &ntinuous CR;

finction f on aMo can be extended to a continuous CR-fun&ion F on M,.

We remark that under the conditions of Theorem 7.12, e manifold aM, is, in general, only an almost CR-manifold. Nevertheless, we &@ (by definition) that a function f on aM, is a &function if

I’ ‘.

. . for each smooth (n,n- k&2)-form cp. This definition i$ satisfied, for example, by the restrictiod to aM& of any CR-function defined in a neighbourhood of dM on Moo. !

In case the diameter manifo!d MO is sufficiently’small, a function F satisfying Theorem 7.12 given by the explicit formula

(7.12)

where z E MO and the fun&on a= (P, c-z) satisfies the requirement of Proposition 7.11. :

Jd .,

92 GM. Khenkin

From Theorem 7.10 or directly from formula 7.12 it follows that the CR- function F constructed on M0 extends further to a hoiomorphic function in ywle neighbourhood of M. in Q,.

-The resuit of Theorem 7.12 is probably valid far arbitrary weakly l-concave CR--folds if the function f in the formulation ofifue>heorem is taken to be a smooth CR-function. uzx;j -s-

The method of integral representations allows one to prove this conjecture for those CR-maaifolds M of type (7.1) for which there exists a weakly concave regular barrier function, i.e. a smooth vector-function P =(P(*), . . . , PC”)) of the variables 1: E SJ\M and z E M such that the following conditions are satisfied:

UKzhC - z>l 2 rkmx9. l~gw)l s YlPm - x ’ (8-&,z)~dC)“-’ = 0,

where (c, z) E (L?\M) x M; y > 0; x > 9. ‘x, .

‘Xi Aside from strictly 1-concav e manifolds, such b&&r -functions ‘can be

constructed also In two important particular cases: for standard CR-manifolds and for CR-manifolds admitting foiiiations into hoiomorphic curves.

T&ore& 7.12’ ([SO]). Let M be an arbitrary weakly l-concave standard CR- mi;nfild given by (7.9), where-dime TC(M)22, and let M, be a subdomain bf M with smooth boundary %d-~omtec.@ complement’ M\M,. Then, for any C”- smooth CR-function f on aM,, there &sts a Cm-smooth CR-function F on M0 agreeing with f on dM,.

-

We remark that by a Cm-smooth CR-function on dM, here, we haie in mind a function f of class C” on aM, which admits a continuation ,F to a neighbourhood of aM, with the property 1?“=0 on aM, along with ail its derivatives.

We shall regard that-a smooth CR-manifold M given by (7.1) admits a smooth fdiiation into hoiomorphic curves if there exists a smooth mapping of maximal rank II: M -+ ,?Y t jW2n-k-2 with the property: for each pe.9, the manifold n- l(p) i* a one-dimensional complex manifold.

Theorem 7.12” ([SOi). Let R be a pseudoconvex domain and p =‘(p,, . . . , pk} be a collection offinctions of class C2(Q) such that c?p, A . . . A 8~~ #O in Q and each CR-manifbld ME = (z E R : p(z) = E} admits a smooth folliation into holomorphic curves. Let k < n- 2, i.e. dime TC(M’) 2 2. Then, on each manifold M” the assertion of Theorem 7.12 holds for smooth, CR-functions f and F.

The classical theorem of Hartogs-Bochner 1.2 is of co&se a particular case of Theorems 7.12’, 7.12”.

We remark that the condition dime Tc(A4) 2 2 in Theorems 7.12’ and 7.12” may not, in general, be discarded. However from the Hartogs continuity principle (Theorem 0.1) it follows [ 131 that the assertion of Theorem 7.12” is

II. The Method of Integral Repmentatiaw ir! Complex Analysis s3

valid also for the case dime P(M) = 1 provided w suppose in’ additioti that dimRT(M j> 3; that the domain M&CM is such that the holomorphic leafs x 7 lb) n MO are -connected and simply connectad, and that the function f & ldxnorphic in a neighbourhood of dMo

Fro&-Theorem 7.12 follows a beautiful maximum principle for CR-f&tic&s obtained first by Sibony using the Mange-Amp&e equation.

+&orem 7.13 (Sibony, 1977). In order that an arbitrary smooth CR-function f on a CR-manifold M admit no local maxima on M, it is necessary and susjcient that the manifold M be weakly l-concave. ’

If a CR-manifold admits a smooth folliation into holomorphic curves, then on such a manifold, not only does the Hartogs-Bochner phenomenon prevail but also the equally acclaimed effect of H. Lewy (see [So]).

‘Ebeorem 7.14 (Hill, 1977; G.M. Khenkin, 1984). Suppose a CR-matiifold M given by (7.1) folliates smooChly into hoiomorbhic curves I-, depending on the parameter p E 9 c Iw 2n-kT2, k<n-2. L.et thedomain9be separated by a smooth hypersurface into subdomains P; and P- and set _

. if. M * =. u rr

PEP* : ;

.- Finally, let M0 be a compact subdomain of M whose bouiuiary is an almost CR- I!,, manifold and let r: = M0 n I-,,, p&+‘. Then any continuous CR-function f& on “i aM,, n M * extends to a continuous CR-functions F, on the CR-manifold M, A M * respectively.

The result of Theorem 7.14, under the hypothesis that the folliation {r,,, can be holomorphically straightened, was obtained by Hill and generalized significant works of H. Dewy. The validity of Theorem 7.14 in the general case is proyed in [50] and makes use of the following simple but very convenient property of CR-functions (see Vol. 9, article VI).

Proposition 7.15 (A.E. Tumanov, G.M. Khenkin, 1983). Let N be a smooth , CR-manifold in @” which folliates smoothly into holomorphic curves rp, p E B c 64” with smooth boundaries X,. Suppose moreover that aN contains a smoath’CR- ” manifold M with the property: each analytic curve r, intersects M transversally in a real arc yp C

T p and also

M=UYp PEP

Then any continuous function f on N which is holomorphic on each curve rp and is a CR-function on M satisfies also the tangential Cauchy-Riemann equations on N.

On account of ,Proposition 7.15, for the proof of Theorem 7.14 it is sufficient to show that for each PG.!?+ the function f+ extends holomorphically to an analytic disc ri. For this, since the analytic disc r: subtends, it is sufficient toi

94 GM. IC,hcnkin j,. : II. The Method of Integral Representation8 in Complex Analysis 95 ,.

C) the manifold R follfates as afamily of compiex one-dimensional analytic discs,- -.i with boundaties-on M and depending smoothly on (2n - k - 1) real parameters;

d) if M is a real~analytic manifold, then fi is also real analytic, and moreover, the manifold fi extends to a larger real analytic manifold i@, such that’ Mn&=-MnD;

e) if M is smooth of class C”, then the manifold Ii? is smooth of class C(J-21/3. ‘i

The method, of constructing (under the conditions of Theorem 7.16)~the , manifold @ by attaching to M a family of analytic discs with boundaries on M is

based on the following construction. In a neighbourhood V,. of a fixed point z* E M a Cl-smooth generic CR-

manifold M can be represented, by (7.10), in the form . it

M = {(z,w)~C’ x Q=“-k: x.= h(y,w)}, 1

t:..

where z= x+iy~ @’ and h is a Cl-smooth function on Wk x c”-’ with the properties h(O) = 0 and dh(0) = 0.

Let S’ denote the boundary of the unit disc D’ in @‘. If we are given mappings v : S’-*R’ and w : S’ +cadk, then we denote by H(o, w) the mapping from S1 to Rk given by

H(o, w)(e’+‘) = h(u(et’), w(e’@)).

.verify that the moment condition Y _- I . .

-~f,hhEO .’ , ;

:

holomorphic 3PP

is Wiled for each I. (.

l-form with &$omial coefiicients. ,Indeed, ‘supfise the disc ri has the form 1’ : ’ i ’

,’ r; = {z E M, : q,l(Z).~r ,, , : ., = (Pi-k- 1(Z) = a},!~ ’ r . . , where (~a,} are functions holomorphic in RPBy the Bochner-NIartinelli formula (1.2), we have .i t :~ ,.)

where

;. $nce the form h A c+, is a smooth &closed form of type (n-k, n-k - 1) on Me\i\r$ by. Stokes formula-we obtain, further, the equality , i -‘*

By the well-known theorem of Andreotti-Grauert-Hormander [44] which extends the approximation theorem of Oka-Weil to &closed forms, there exists in, the neighbourhood of no a sequence of &closed smooth (n - k,n- k- l)- lb%, o~,~ which approximate the form o+, uniformly on (8yo)n(8M+). From

,,%$iind from the previous equation’we have , .:: , :. :

,;i 3Lf+Ahr liin -P *-+oD (3Mo,!,3M ,f+ AhAww = OS + i.4. this proves the required moment condition. ,’

-, ‘-7.!5. Analytic Discs and the Holomorphic Hull of a CR-Manifold. Theorem 7.14 works well in conjunction with the following result on the local suspension of analytic discs on CR-manifolds with non-zero Levi form.

Theorem 7.16 (Hill, Taiani, 1978, see [43]). Let M be a C”-smooth CR- manifold given by (7.1), s 2 5. Suppose that for a fixed point z E M and for a fixed C E T$(M), we have that Ly(Q i t, #O. Then, there exists a neighbourhood D of the point z :nd a CR-manifold M cjfreal dimension greater by one with the properties:

a) aM=MnD, ,

b) T,(R) is equal to the linear hull of the space and the wctor tt;

By construction, any mapping of the form g(e’+‘) = U(v, w)(e’@)+ iu(e’@‘) carries the circle S’ to some closed contour on M. In order that the mapping g extend to a holomorphic mapping G of the disc B’ to C”, it is necessary and sufticient, as is well known, that the functions lY(o, w) and u be,the boundary values of conjugate harmonic functions, i.e. satisfy the Bishop equation (1965, see [14], [43])

u(e’*) = T(H(v, w))(e”) + y, ” ’

where T is the classical Hilbert transform and y i’s a vector in W”. !i Theorem 7.17 (Greenfield, Wells, 1968, see [82]. Hill, Taiani, 1978, see [43]).

For any smooth mapping w : S’-K”-lr with suficiently small Cl-norm and any 1

’

vector y with sufficiently small norm in Rk, there exists a unique continuous ’ mapping II*

v(w, y): s' --, R',

satisfying the Bishop equation. Moreover the mapping v(w, y) depends continuously on w and y and for any a< 1 /

llv(w, y)Il&l, + 0, if II wllcqsq 4, II,Y IlrP + 0. )

From Theorems 7.14 and 7.16 an important ‘corollary follows!

Theorem 7.18 (Hill, Taiani, 1978, 1981, see 1141). Under the hypotheses of Theorem 7.16, any continuous CR-function f on a CR-manifold M with non-zero I

Levi form extends to a continuous CR-function F on a CR-mdnifold @nD of dimension greater by one. I

I !

, i I I

96 GM. Khenkin

A significant strengthening of Theorem 7.18 was obtained not long ago by Boggess and Pitts [14] for the-case when a higher order Levi form of the CR- manifold M differs from zero:

From Theorem 7.18 follow, in turn, criteria for local holomorphic convexity of suficicntly smooth CR-manifolds which were obtained earlier (see [82]).

Theorem 7.19 (Greenfield, 1968; Wells, 1968). A C2-smooth CR-manifold M in C” is lor~ily ho/omorphicallJ~ con~x $und oidy if the Levi form Lr = 0 for all ZEM.

Stein (1937) established this result first for hypersurfaces in @“. From the Frobenius integrability theorem, it follows (Sommer, Nirenberg)

that any C2-smooth CR-manifold M, whose Levi form vanishes identically, folhates into complex manifolds of maximal possible complex dimension, CR dim M (see [58)).

$8: The &Cohomology of p-Convex and q-concave Manifolds and the Radon-Penrose Transfqfm

0

In 941-7 the method of integral representations has been demonstrated basically on problems from the theory of functions on domains in the space @“. Here we introduce examples of applications of integral formuli to problems from the cohomology theory of complex manifolds which go beyond.the limits of ;Iassical function theory.

X.1. The &Cohomology. Theorems of Andreotti and Grairt. T-, begin with we present a short survey of classical results from the theory of &cohomology. (see also the article of A.L. Onishchshik Vol. 10, I).

Let Q be a complex manifold, {&},i E J, an open locally finite cover of R and GL(N, Q the group of invertible complex N x N matrices. tet holomorphic functions gp be defined in the intersections fIjn&, taking their values in GUN, Q, and forming multiplicative cocycles, i.e.

~jk(z)'Cl&l(z)~~ljtz) = IT zERjn12~nnR,.

Here I denotes the identity matrix. i’

These compatibility conditions on the functions {gp) allow us to introduce an equivalence relation on the disjomt union

E = (J (nj x CN) by setting jeJ

(Z, 0) N (Z’, U’), (Z, 0) E (nj x CN), (~, U’) e n, ’ CN ,

if and only if z =z’ and u’=tjjku.


The complex-manifold E obtained from E by factoring by the above equivalence relation is called a holomtirphic uecfor bundle with base R and with fibre QZN. This definition is also closely related to the holomorphic mapping n: ,540 projecting a representative point (z, u) of E onto the point ?EQ. Here II- l(z)

CN. The functions (gjk} which determine this bundle E are called the Eansition functions of this bundle with respect to the trivializations lsZj x a=“}.

The bundle given by the same covering {a,} and the multiplicative cocycle of matrix functions Igil,‘) is called the dual bundle E* to the bundle E.

If D is a domain in 0, then a section of the bundle E ov,er D is a system of mappings, {hi} from the domains D nfij to CN such that hj=gjkh, on DnQjnRp.

The simplest non-trivial example of a holomorphic vector bundle is given by the universal line bundle @(-I) given by the natural projection n: C”+‘\

11 ‘0 +@IFp”, where Q=P” is n-dimensional complex projective space. The transition matrices gjr, for the bundle Cfi( - 1) have the form gjr = Zj/Zk, where z E U, n Ulr.

By O(I) we denote the line bundle over UP” given by the transition functions 1:*‘zj)‘. The sections of the bundle O(I) are the functions of the variable

,*. ., z”+’ which are homogeneous of degree 1. A form f of type (0, r) on D c 0 with values in the bundle E is a section over D

of the bundle E&A’*’ T*(R) c, where T*(R)c is the complexification of the cotangent bundle on a. In other words, a (O,r)-form f with values in E is a system (fj) of @‘-valued (0, r) forms on D n nj such that

fj = gjJ; on D n Qj’n f&. VW

In local coordinates on R, the formfj can be written:

where P= (Ply* * * %P,)Y d?’ = d.?” A . . . A’ dZpr.

Let us denote by Cg!JD,E) the space of forms of type (0,r) on D whose coefficients are sections of class C@) of the vector bundle. E over I). In view of (8.1) and the holomorphicity of the transition functions gs, for each form f~ @!,@I, E), the form afE C$!,+ 1 (0, E) is well-defined.

A form.fc C&(0, E) is called &closed if af=O and is called &exact tit=&, where u E Cg: !‘, (I), E).

Let #,(d, E) denote the quotient group which is the group of &closed form in Cg’,(D, E) factored by the group of J-exact forms. The group H$(D, E) is called the Dolheault (0, r)-cohomology group of the manifold D with coefficients in E.

Let H’(D, E) denote the r-dimensional &ch cohomology group Of-D with coefficients in the sheaf of germs of holomorphic sections of E. In particular H”(D, E) is the space of holomorphic sections of E over D. The following classical result holds.

98 GM. Khenkin

Theorem 8.1 (Dolbeault, 1953). For any natural number r and any s, there isa ‘canonical isomorphism between the groups H&(D, E) and H’(D, E).

We have already made essential use of a particu$ar case ofthis isomorphism in Y, the hroof of the equivalence of the assertion& Theorems 6.1 and 6.2.

Now we formulate .a fundamental result from the cohomology theory of complex manifolds (see [31], [33]).

Theorem 8.2 (Kodaira, 1953; Grauert, 1958). Let R be a strictly pseudo- --J-canvex complex mantfold with C2-smooth-exhaustion function p,. i.e.

‘1 -- - Q= u D,; a2ao

where D? = (z E fk p(z) < a} e R and p is strictly plurisubharmonic on $2\D,;; Let E be a holdmorphic vector bundle over SI. Then

a) for any r zz 1 and a z ao, we have

i dim; H’(Q, E) = dime H’(D,; E) < co;

b) $ D,, = 0, then for all r > 1, we have c

dimcH’(Q, E),=0;

c) &‘I is a compact mangold and Da, =. a--then the assertion a) holds for all r > 0; d): FQ is non-compact, then for each a.2 ao, we, have: - . . , .

,, 7’ dimcHe(R;E)=dimcH”(D,,‘E)=co. .-, ’ . _ , ‘* Assertionc) of Theorem 8.2, ,first obtained by Kodaira; can be seen as a far.

reaching generalization of the classical Liouwille Theorem. Assertions a), .b), and (d), first obtained by‘Grauert, areextensions of the theorems of Oka on the

: solvability of the problems of Cousin and. Levi in pseudoconvex Riemann . domains. Theorem 3.12 of Grauert is a simple consequence of assertion aj. From

.. Theorem 3.12, in turn, assertion d) of Theorem 8.2. follows quickly ’ We present a simple, example which illustrates the result of Kodaira. The space H”( @ P”, O(t)) is zero-dimensional for each m = 1, . , . , n- 1, The

space HO(Q:P”, O(1)) is zero-dimensional if 1 <O and consists of homogeneous h&morphic polynomials if i>O. The space H”(@P”, O(1)) is canonically isomorphic to the space [HO(Q=P”, @(-n-s-l))]*.

: The clarification of more precise conditions on a manifold f2, under which- the ass&ions of Theorem 8.2 hold for the cohomology H’(Q E) of fixed order r 2 0,

(. led to’ the important notions of p-convex and q-concave manifolds. A realjitnction p of class C2 on a domain D c @” is called p-convex (respect-

ively strongly p-convex) if its Levi form

II. The Method of Integral Representations in Complex Analy$s 99

has for each C E D no less than (p + 1) non-negative (respectively positive) eigenvalues.

The property of p-convexity of a function p is independent of the choice of holomorphic coordinates in the domain D and hence can be well-defined for functions on complex manifolds.

A relatively compact &main D + (respectively D _ ) of a complex manifold% called strongly p-convex (respectively strongly p-concave) if it can be represented in the form .

D, ={z&: p(z)<O} (8.2)

’

.

respectively D- = {z ES~Z p(z)> c}, where p(z) is a strongly p-convex function in a neighbourhood of aD *. In particular, a strictly pseudoconvex domain is a strongly (n - l)-convex domain.

The notion of strong p-convexity of domains’was introduced, by Rothstein in connection with the following effective generalization of a theorem of E. Levi.

Theorem 8.3 (Rothstein, 1955). Let the domains D, given by_(8.2) have smooth boundaries, where the domain D, is strongly p-convex and the domain D- is strongly p-concave at a point I ED, . Then, there exists a neighbourhood U, of the point c such that

a) in the domain U, n 6-, there exists a p-dimensional analytic submanifold intersecting D, r~ U, only in the point c;

b) any closed (n-p + l)-dimensional analytic subset of U, A D- extends to a closed analytic subset of U,.

For related results on the extension of analytic sets across pseudoconcave surfaces, see article III of E.M. Chirka.

A complex manifold R, (respectively a-) is called strongly p-convex (respectively strongly p-concave) if on Q, (respectively’&) there exists a C2-smooth real function p and a number a0 such that the domains given by

D$ =(zeQ,: +,p<a), +_a>ao, (8.3)

are relatively compact in “* and respectively strongly p-convex or strongly p-concave.

The following result significantly generalizes assertions a) and b) of Theorem 8.2 (see 6221; [44]).

Theorem 8.4 (Andreotti, Grauert, 1962). Let R, (respectively Q-) be a strongly p-convex (respectively stongly p-concave) manifold with exhaustion function given by (8.3). Let E be a holomorphic vector bundle over 0,. Then a’) for each r?n-p and.a>a,, we have

i* dimcH’(n+, E)=dimcH’(D:, E)<co,

a-) fir each r-zq and a< -ao, we have

dimcH’(Q-, E)=dimcH’(D;, E)<oo,

I

100 G.M. Khenkin

b) ifD,‘,=%, then for r>n-p, we have

dimcH’(R+, E) =(?.

We remark that Theorem 8.3 of Rothstein can be deduced from Theorem 8.4 if we make use of the equation of Poincar&Lelong and Theorem 3.1 of E. Levi.

A relatively compact domain D in a complex manifold 0 is called strictly q- concave if D is strongly q-concave and at the same time strongly (n-q)- convex.

The following result gives an interesting generalization of Grauert’s Theorems 3.12 and 8.2 (see [30]).

Theorem 8.5 (Andreotti, Norguet, 1966). Let D- be an arbitrary strictly 9- concave domain on a complex manifold R and let E be an arbitrary holomorphic vector bundle over $2. Then

dime Hq(D-, E) = 00.

Moreover, for any neighbokrhood U[.ofany point < E aD-, there exists an element ~EH~(D-, E) which d oes not extend.to an element of Hq(lJc u D-, E).

In the language of Dolbeault cohomology, the’ last assertion of Theorem 8.5 signifies the following. For-any point [IS~D-, there exists a &closed form fE C$,yJ(D’, E) such that there is no rp E CL:!- 1 (D-, E) such that f+ &Q extends to a &closed form in the domain U, u D-.

,

Theorems 8.4 and 8.5 show that on a strongly q-concave manifold RI, the role of hofomorphic functions is played by the elements of the space l$(ia-, E), For q>O the space Hq(R-, E) can have a very complicated strricture, even for strictly q-concave manifolds. For exami>le, Rossi (1965) constructed examples of two-dimensional strictly 1-coxicave manifolds R- for which the spuce H’(f2-, 0) are not separable. In the language of Dolbeault cohomology. this means that for such f2-, for all s, the set of all &exact forms in the class C$)l(Q-, 0) does not form a closed subspace of Cg’l(Q-, 0).

tie formulate a result giving a convenient sufficient condition in order that the space Hq(D-, E) be separable (see [223, [443).

Theorem 8.6 (Andreotti, Vesentini, 1965). Let D be a domain on a complex manifold R such that D =D’\D”, where 6” c D’, D’ c Q. the domain D’ is strongly (n -q)-convex, and the domain D” is strongly q-convex. Then, fQr any holomorphic vector bundle E over Q, the spcce Hq(D, E) is separable. Moreover, u j?orm $E C~~~(D, E) is &exact on D if and only if the “moment” conditik

p A cp=o J ’ ’

is satrsjied, for each &closed form cp E C!,TJ(D, E *) of Finite support in D.

II. The Method of Integral Representations m Complex Analysis 101

The conditi&ls df Theorems 8.5 and 8.6 are satisfied, for example, by domains in CP” of the form

D={z~a,lP”:(z,(~+- . . . +(z,+,~~-~z~+~I~- . . . -Iz,+,12>O} (8.4)

where(z,, . . . , z,+~) are homogeneous coordinates in 63 5’“. The dpmain (8.4) is a strictly q-concave domain, According to Theorems 8.4-8.6, for this domain, the space H’(D, U(1)) is finite dimensional for r fq and infinite dimensional but separable for r = q.

The result ofTheorems 8.2 and 8.4-8.6 were first obtained by the methods of coherent analytic sheafs (see the article Vol. 10, I).

Kohn, Andreotti and Vesentini, and HGrmander obtained these results anew with boundary L2-estimates using the method-of the a-problem of Neuman- Spencer. We formuIate here a fundamental consequence of these works (see WI, C443,C551)*

For a domain Don a complex manifold R and a vector bundle E over R, we denote by L$(D, E) the space of (0, r)-forms on D whose coefficients are sections of E of class LzVs(D). Let H$$(D, E) denote the quotient space of the space of &closed forms in Lz$(D, E) by the space of &exact such forms.

Theorem 8.7 (Kohn, 1963, 1975; Andreotti, Vesentini, 1965; Hiirmander, 1965). Suppose under the hypotheses of Theorems 8.4-8.6 that the domains D,’ , 0; and D have C x-smooth boundaries. Then, the assertions of these theorems remain valid if we replace the cohomology spaces without estimates H’(L2, E) by the correspuizding cohomology with L2-estimates H$:(D, E). Moreover, under the hypothek of Theorem 8.6, for any &exact form f 6 Lg;:(D, E), the equation & =f has a solu$idn with estimate

I f ,/’ i II~IP~D~ 5 rP)llf lb(o).

8.2. Integral Representations of Differential Forms and &Cohomology with 4Jniform Estimates. Integral formulas of Cauchy-Fantappik type have made it possible to give elementary proofs of the above results on the &cohomology of complex manifolds with precise uniform boundary estimates and to extend these results to domains with piecewise smooth boundaries, Thanks to these refine- ments, it has been possible to obtain criteria fdi.&lqbility as well as formulas for the solutions of the tangential Cauchy-Riemann e$atiQps,(see section 8.3), and a!so to write down the kernel and the image for the Radon-Penrose transform in precise form (see section 8.4). ’

In order to obtain these results, it turned out to be necessary to extend the fundamental formulas of Cauchy-FantappiC in $4 to the case of differential forms on p-convex and q-concave domains in @“. ‘For a smooth @“-valued vector function q =q(C, z, t) of the variables c, ZE C” and t E @ we-have the formula (see 1493 ): PI-1

el)= c w;m q=o

102 G.M. Khenkin _ ‘.

where w;(q) is a form of order q with respect to di and respectively of order \ (n-q- 1) with respect to dcand dt. Integral formulas for differential forms make

use of the forms {ob} and of the following important relation between them .

j d,w~(rl)+~ob(rl)+~w;-,(tt)=O. (8.5) An important rare in the sequel is played by the following generalization of

the Martinelli-Bochney integral representation 1.1 to the case of differential forms (see [4]. (403).

Theorem 8.8 (Koppelman, 1967). Let D be a bounded domain in C” with rectijable boundary and f a (0, r)-form in D such that both f and 8f are continuous on 6. Then,for any r=O, 1,. . . , n and zeD we have

(&) A W(c)].

The following generalization to differential forms of integral fol’mulas in Theorem 4.6 was obtained on the basis of Theorem 8.8 together with (8.5) and the Stokes formula.

Theorem 8.9 (Lieb, 1971; Range, Siu, 1973; R.A. Ajrapetyan, G.M. Khenkin, 1984). Let G be a domain in @” having piecewise-smooth boundary given by (4.16) and let { qJ ) a family of vector-functions having the property (4.17). Then for each z E G and each (0, r)-form f such that both f and 8f are continuous on e, the equality

holds, where f(z)= K’f- Tr”8f-8Tlf (8.6)

K’f= (- 1)’ $$;& K;f,

T;f=(- l)lJ’

(1 -+$+to ~J& 2, t)

<tlJ& 2, t), l-z> A w(c)

T’,f=i f(i) A w:-, A W(r).


The a&ertion of Theorem 8.9, for domains with smooth boundary, was first - obtained by Lieb (1971). For domains with piecewise-syooth boundary and under the supplementary condition 4.21, it was obtained m the work of Range and Sin (1973),Jnits general form, the result of Theorem-g.9 is contained in [3].

Applicatidns to the tangential Cauchy-Riemann equations (see section 8.3) require that Theorems 8.4-8.7 be generalized to p-convex and q-concave domains with piecewise-smooth boundaries. .

Let G be a domain on a complex manifolds f2. A q-dimensional complex manifold W(lJ passing through a point CE aG is called a local q-dimensional barrier to G at the poirit c if there exists a neighbourhood U’ of the point i such that iVfq(<)n U,nG={[}. 0

Taking off from Rothstein’s Theorem 8.3, we give the following definition. A domain G given by 4.16 on a complex manifold Q is called q-convex with

piecewise-smooth boundary if for each multi-index J c (1, . . . ,N} and each point [E rJ c dG, there exists a family of local q-dimensional barriers Mj([, t) to G at the point {, depending on the parameter t EAJ; such that the barriers M4J(C, t) are smooth of order Lip 1 in the variables [G r,, t EAT, and for

we have

A domain G is called q-concave with piecewise-smooth .boundary if the domain a\6 is q-convex at each point LEaG. ,.

The following assertion yields an iniportant class of p-convex (and corresponding!y.q-concave) domains with piecewise smooth boundaries.

Prd$sition 8.10’( [39],‘[3]). In order that a ~domhin G in Q be q-conuex with piece&se-smooth boundary in the- neighbourhood U of a point z* EX, it is $&cient that the domain U n G be representable in one of the two following ways

3 .,,, Gn U={ZEU: pl(z)<O,. . . ,p&)cO} (8.7)

where (p,) are function of class C”‘(U) such that Pj(z*)=O and for each t silk: IZ tj= 1, tj>O, the function (t, p) is strictly q-convex; or

.. .-

, GnU=,~,{zEU:pj(z)<O}, .

k where {pi}’ are fin&ions of class Ct2)(U) such that Pj(Z*)=O and for each . t t Rk: I: tj = 1, $2 0, the function (t, p) is strictly (q + k - l)-cbnvex.

f The c&truction of integral formuli for solving &equations on q-convex

; domains relies on the following elementary assertion, similar to Proposition 7.11.

~Proposition 8.11 ([SO], [39]). Let G be a q-convex domain with piecewise- smootb boundary given by (4.16) on R and {MQ1(5, t)} a family of local

,‘q-dimensional barriers at the point [E rJ c aG. Then, for each .I c { 1, . . . , N},

104 GM. Khenkin

and each [~l-‘~, tcA, and z~Uc, there exists a function oJ(c, z, t)= (pJ([, z, t), c-z) of class C” in ZE U, and class Lip 1 in (r, t)Er, x AJ such that:

a) MJ([, t) n UC= {Ze U,: @Jr, Z, t)=O}, b) @.&, z, t)= %,(L z, t), K, t) E I-, +,,

tLt)ErJxAJ; ’

c) @P&,z, t) A dz)“-q+l=O, ZEUS;

tr, tkb x AJ, ZE q.

Let I@,(& E) denote the space of (0, r)-cohomology of the closure d of a domain D c Sz with C.@‘(d) estimates, i.e. the quotient space of the &closed forms f of class C!$(& E) by the forms f= &, with u E Cg,:!‘,(& E).

Propositions 8.9-8.11 together with simple reductions to the classical theorem of Fredholm allow one to prove the following generalization -of Theorems 8.4-8.7 (see [3], t-491, [30], [39], [4O], [66]).

Theorem 8.12 (Grauert, 1981; GM. Khenkin, 1981). Let D be a domain on an n-dimensional complex manifold R of the form D = D’\ D”, where D’ in an (n -q)- convex domain with piecewise-smooth boundary and D” is a q-convkx domain with piecewise-smooth boundary. Then:

a) dim&q”!@‘, E)=dim&‘#‘, E)< 03, for r2q; b) dim&q”!(R\D”, E)=dimcH@!(R\D”, E)<cQ, for r<q; c) the spaces Hbq”!(D, E) and Hbq”!(D, E) are separable; d) ifmoreover the domain D’ locally has the form (8.7) and the domain D” has the

form (8.8) and dp,(z) A . . . A dp,(z)#O, ZE al>, then the assertions a), b), and c) are valid also for the cohomology spaces with untform estimates

Hg,‘,(6’, E), fl&Q\D”, E), cij-J’,(& E).

The assertions a) and b) for cohomology spaces without estimates HbTj(D’, E) and H$,yj(Q\B”, E) follows from the work of Grauert [30]. The remaining assertions of the theorem follow from the works [39], [3]. We remark that assertion d), for strictly pseudoconvex domains D’, was first obtained in the works of Kerzman (1971), Lieb (1971), Ovrelid (1971); for piecewise strictly pseudoconvex domains D’ in the works of P.L. Polyakov (1972) and Range, Siu (1973); for strictly (n -q)-convex domains D’ with smooth boundary in the work of Lieb, Fischer (1.974); for strictly q-convex domains Q\D” with smooth boundary in the works of S.A. Dautov (1972), 0vrelid (1976), G.M. Khenkin (1977). ,

be 8.3. The Cauchy-Riemann Equations on q-Concave CR-Manifolds. Let 121

a CR-submanifold of a complex manifold 52, i.e. a smooth, closed submanifold of ir locally representable in the form (7.1). Let E be a holomorphic vector bundle over R. By Cb”(,(M, E) we denote the space of differential forms of type (p, q) on M whose coefficients take their values in E and are smooth of class C(“. Here. two forms f and y in CF,‘,(M, E) are considered equal if and only if for each form cp E C!,Epi, ,,“- c-&Q, E*) of compact support, we have

jfv=@X . l8.Q


We denote by L’,-,“‘(M, E) the space of (p, q)-forms on M dual to the space Cp’, n-k-q@4, E+.

Wd define the-tangential Cauchy-Riemann operato? on forms in L’,T$I4, E). If u E C$!,(M, E), s2 1, then u can be extended to a smooth form U’FZ Cg!,(Sr, E) and we may set

Moreover, the condition for equality of forms on M shows that the given definition does not depend on the choice of th& extended form 6. In general, for forms u E L$,;$ 1 (M, E) and f E L$,;:‘(M, E), by definition,

&u=f (8.10)

will mean that for each form PE C$$-k-q(n, E*) of compact support we have

As a (local) necessary condition for the solvability on M of equation (8. lo), we. ;hust bve first of all aM f = 0 on M. Forms satisfying this condition will be called ’ CR-forms. In casefis a function, this definition reduces to the definition of a CR-

fu&tion given in 57. If the manifold M and the form f are both real analytic, then the condition

aM f=O is not only necessary but also sufficient for the local solvability of the equatib’n (8.10). Here, the size of the domains on M on which (8.10) has a solution depends not only on the manifold M but also on the real-analyticity

,

properties of the CR-form f (see [6]). If, however, the form f is not real-analytic on M, then the condition aM f= 0

in general is no longer sufficient for the local solvability of (8.10). This unexpected effect was first observed by H. Lewy in 1957 in the following example. Let M = {zeC2: Imz,- (zi I2 =O). For functions u on this hypersurface, we have

au au zMu=Fdz,+aZdF2=

1 2

Thus equation (8.10) in this case is equivalent to the following equation in Iw3

H. Lewy constructed a function f of class C (m) such that this equation is not

locally solvable even in the class of generalized functions u E L’- m). Treves [79] showed, however, that on any CR-manifold M of codimension k

there is an approximate solvability of (8.10) for a CR-form fE Cg”!n-lr(M).

Theorem 8.13 (Treves, 1981). Let M be an arbitrary Cs-smooth CR-mani$Ad

of codimension k on a complex manifold R. Then for each point z EM there exists a

106 GM. Khenkin

neighbourhood U, such that any CR-form f~ Cg”lnek(M), can be approximated on M n U, with arbitrary precision, in the C’“‘-topology, by au-exact forms of class Cbq”,!-,(U nM). *

This result is a natural generalization of the approximation theorem for CR- forms of Range, Siu (Math. Ann., 1974, 210, 105-122).

The clarification of conditions for actual (rather than approximate) local and global solvability of the tangential Cauchy-Riemann equations (8.10) has advanced rather far for the case when M is a hypersurface (Kohn, 1965, 1975; Andreotti, Hill, 1973; Sato, 1972; Folland, 1972; Folland, Stein, 1974; A.V. Romanov, 1975; Greiner, Kohn, Stein, 1975; G.M. Khenkin, 1975; Skoda, 1975).

We formulate here only one of the criteria for the global solvability of the equatio::.(8.10) on a compact hypersurface (see [49], [SS]).

Theorem 8.14 (Kohn, 1965,1975, s= a; A.V. Romanov, 1975; G.M. Khenkin 1975; Skoda, 1975, s < 00). Let M = {ZG 0: p(z) =0} be a compact P-smooth hypersurface on a holomorphically convex manifold R; let E be a holomorphic vector bundle over R. Suppose the Levi form of the function p has on T:(M), z E M, either max (q, n-q) eigenvalues of the same sign or min(q, n-q) pairs of corresponding values of opposite signs. Then, the equation aMu=f, @ere

fc C&(M, E) has a solution u E CS,~r?t(M, E), if and only if 0 !

If A cp=o

for each &closed form cp E C!,Fi- 1 -,(M, E*).

In case M is the boundary of a strictly convex domain in C”, the result of the Theorem 8.14 is a rapid consequence of the following integral representation for an arbitrary form fECg!,(M), q=O, 1, . . . ,n- 1 (see [49]):

where f=~~T,f+r,,,(a,f)+K,f-K:f, (8.11)

Tqf(+(-l)q (n WY

p*tc, d = P(C, z) =$(z).

11. The Method of Integral Repr-ntations in Complex Analysis 102

For manifolds of arbitrary codimension, rather general results on the solvability of theYtangential Cauchy-Riemann equations have been.obtamed for thf

, class of,q-concave (or q-convex) manifolds (see [31, [39]). _ .We &note by ,Hl;l,(JU, E) .the .q.uotient space of the %closed forms f in

‘C$!,(M; E) by the subspace,of forms $= aMu,. with u E C!$ i (M, E). k

Theorem 8.15 (PIJaiuki; 1972; G.M. Khenkin, 1981). Let R be an n-dimen-

‘- sional comp’rex mdnlfold, E’ a holomor~hicvector bundle over Sz, and M a strongly q-concave P-smooth submanifold of codimension k in a. Z%en, for each r, 1 I r ‘< q (respectiu& n - k -‘q < r <.n -k), for each, str,&igly, (ti - l)-concave (respecgvely strongly@ 1 l)-convex) domain D c Q and for ,each ‘CR-form f E C$!,(y.:n,D, E), the equation (8.10) has a solution u~Co,~-~ (s+1/2-r)(M n8, E), s>O, ifand:only#

:

for each &c-closed form cp E C!,Ti-k ‘, (&ET) with support in the domain D. More-

over, for each, r,t;<q’ or r>n.-k- q, the space H&(A4 n8, E) is finite

dimensionah i e For the case r > n - k - q and s = & the result of Theorem 8.15 was obtained

first by Naruki on the basis of the techniques of Kohn and Hiirmander. The general case is considered in [39].

The proof of Theorem g.15 ‘is based on representing a q-concave or q-convex CR-manifold as the intersection respectively of q-concave or a-convex domains with piecewise-smooth boundaries and applying to these domains Theorem 8.12 as well as the formulas yielding this theorem (see [393, [3], and [66]).

The requirement of q-concavity of the CR-manifold M in order that the

assertion of Theorem 8.15 hold is essential because of the following result which sheds light on the effect of H. Lewy (see [6] and [3]).

Theorem 8.16 (Andreotti, Nacinovich, Fredricks, 1981). Let M be a C”- smooth CR-manifold of codimension k in R. Zf the CR-manifald M is simultaneously strongly q-concave and strongly (n-k - q)-convex at some p&n’i?e M in some direction t E S’, then in each neighbourhood U of the point z, there exists a CR-folrm fc Cz ,(M) which is not aM-exact on M n U.

8.4. The Radon-Penrose Transform. A domain (or compact set), D in the

space CP” is called q-linearly concave if for each point ZE D there exists a q- dimensional’complex linear subspace A(z) c D containing the point z. and depending smoothly on z e D. A domain D c CP” is called q-linearly convex if the compact set fXP\D is q-linearly concave. ’ Examples of q-linearly concave (or (n - q - I)-linearly convex) domains are

/ given by domains in CP’” of the form (8.4). The notion of q-linearly convex domains was introduced by Rothstein (1955)

without the requirement that the subspace A(i) c @P\D depend smoothly on

108 GM. fChenkin

the point CE CP”\D. The latter requirement was added by us in the definition of q-linearly concave arid q-linearly convex domains because it figures in subsequent formulas. For q-linearly concave domains it turns out that one can construct a globaL htegral representation of (0, q)-forms in which only integral-residues along a subspace A(z) c D play a role. Such representations allow us to significantly clarify the theory of Radon-Penrose transforms in. connection with the integration of (0, q)-forms on q-&mensiona{ complex subspaces of D.

For a q-dimensional projective subspace A t D, we set 2 = n-l(A) c II?+ 1. Let a = (a,, . . . ( an-*} be an orthonormal (n - q)-rep&e in C”+ 1 &ich &ed the subspace 1 c C”+l, i.e.

A’ = ((EC”+? A,(C) = (Uj, 0 = 0, j = 1, . . . , n - q}.

Let G(q + 1, n + 1) denote the manifold of (q + l)-dimensional subspaces of the space 8” + I, For a q-linearly concave domain D c CP”, we denote by D* the domain in G(q + 1, n + 1) consisting of all 2 such that ~(2) is a q-dimensigonal projective subspace A lying in D, and we denote by F the fibration by (n - q)- rep&esa={u,,..., u,,-~} over D:. Let C(D:, F) denote the space of con-

.: tinuous functions 1(1 on F satisfying the condition .__ . L-- ‘*- _-_ --

.- -1-s W4 = (de! d- ’ tW Vge GL(n -.q, C).

-._ By d, A we der%++Yholomorphic form of maximal degree (q ‘+ 1) on A’ such that at the points of A we have

Jt.1: ,q -> .*a A dc”+’ = d<A’ A dsAl A . . . A d<A n-q’ For forms JE Cb”,‘,(D, O(l)), we set i= IC*J

Following [28] and [SZJ we define the-Radon-Penrose trunsformutioti

by the formula R: C&D, O( - q - 1)) -+ c(D:, F)

Rft4 = j .&, d,A, InS

(8.12)

where fez _CO, ,{D, 0 (2 q - I)) and S is the unit sphere in @“+ l. Let a,“X m = (m,, . . . , m,,-,) denote the mixed derivative

aI4

(adp . . .(aa:-,p-q f

in the direction of an arbitrary rep&e a’ = (a;, . . . , u;-~}EQ=“+’ dual to the rep&e (a,,..., Umpq), i.e. (Uj,U;) = 6,. We complete the rep&-e {a,, . . . , a,.-,} to an orthonormal basis of C”+l by adding vectors {b,, . . . , b4+ I >; set Clj = (Uj, di), & = (bk, d<) and expand the (0, q)-form 7 using these basis forms

II. ‘& Method of Integral Representations in Complex Analysis 109

We denote the coefficient of the monomial kj, A . . . A tijZ in this expansion by cf-jjto *j..Forare*rea = u(z) depending smoothly on the point z E D (i.e. if

u is a smdoth frame) and for r 5 q, we set r- IhI ml mm-, 9-r-l

A- h -.

det:(u, z,J) = det(Z, c a,, . . , u,,-~, d.6 &u,, . . . , kq, 4)

ifr<q,and

r--l4 ml ‘“l-r -L -

deti(u,z)=det(%a,, . . .,u"-~, dZ &u,,. . .rdrua-q)

if r = q, where dZ means that column d.2 is repeated r ~lrnl times.

We introduce the inverse transform G”: C(D:, F) --, CO, ,(D, O( - q - 1)) by the _ _ -formula

,’

G”$(z) = i d(m, q) det$z, z) x (ml=0

. . . XI

->+&~ .; . ($--~~q~)(ul, . . . , uneq). _. (8.13) .

In addition,. -we intEjdu~..&&%&ators ,I: mapping from C$!z’-” (&~l(~ q~~~l)j6i%&- l(D, O( - q - 1)) by the formulas

, ’ ,

~Ui,I, A . . . A ~~ipjr A

(f<(jl,...,j,)<n+l

where @*(C,Z) = (5,c - z), o&z) = (e, c - z), and b(m), c(m), d(m) are constants depending only on m-and n.

It is possible ‘[SZ] to obtain global integral formulas for forms on CP” by making use of the integral representation (8.11) for forms on the sphere S in (c”+ l. For (0, q) forms on q-linearly concave domains in CP”, the formula is giving by the following.

Theorem 8.17 (G.M. Khenkin, P.L. Polyakov [53]). Let D be a q-linearly

concave domain in UP’“. We&x a smooth mapping z H u(z), where z E D and u(z) is an orthonormul (nj- q)-reptre giving a q-dimensional subspuce A(z) c D. Then, for

(” ~vformf~Co,qt , D O( - q - l)), s 2 q - 1, we huoe an integral representhtion

110 . . ‘. G.M. Khenkin

given by :

In [52] and [53], modifications of formulas (8.I2) and (8.13) for’the Radon- Penrose transform and the inverse transform G for the case of (0, q)-forms with coefficients in any vector bundle Lo( - I), I> 0 are also constructed. By using these modifications, the assertion of Theorem 8.17 extends to any form.

f E Cb”!,(D, 0 ( - l)), 1 > 0. Theorem 8.17 applied to &closed forms shows, in particular, that the operator

.G” is the inverse of the Radon-Penrose transformation restricted to the &cohomology space H*(D, 0( - q - 1)). In form, the operator G”’ is similar to the inverse transformations of I.M. Gel’fand, S.G. Gindikin and M.I. Graev (1980) for the Radon transformation in real integral geometry.. A different inverse for the Radon-Penrose transformation on the a-cohomology. spaces was donstructed earlier in [28].

We denote by G”(q + 1. n + 1) the dense subdomain of G(q + 1, n + 1) consisting of all (q + l)-dimensional subspaces of C”’ ’ given by the equations

*+l .zbp = .z, a,8S * z,, fl = q + 2, . . . , n + 1.

The elements of the matrix {a,,r,} can be considered as coordinates in the domain G”(q + 1, n + 1). We set 0:‘” = 0: n.G’(q + 1, n + 1). The restriction

-, to 0:’ ’ of a function in C(D$, F ) can naturally be considered as a functiorrof the variables a, 8’. An immediate verification shows that the Radon-Penrose transform $ = Rf given by (8:12) satisfies a system of “wave” equations

a’* haa, - aassT

av o - aaas.-dap,, = ’

(8.15)

on D:‘“, where a,/3 = 1, ; . . , q + 1; a’,/? = q + 2, . . . , n + 1. As a corollary of Theorem 8.17 we obtain the following result.

Theorem 8.18 (Penrose, 1977; S.G. Gindikin, G.M. Khenkin, 1978). Let D ’ be a q-linearly concave domain in @IF”‘. The Radon-Penrose transformdtion (8.12) , establishes an isomorphism between the cohomology space H4(D, O(_-- q\-- 1)) and the space of holomorphic functions in C(D:, F) satisfying the system of equations (8.15) on 0:“.

In the case q = n - 1 the result of Theorem 8.18 is equivalent to the theorem of Fantappit-Martineau 4.5. ‘In this situation, the equations (8ja 5) disappear. For the case q = 1, n = 3, the result of Theorem 8.18 was initiated by Penrose (1969, 1977). In this case the system (8.15) consists in one “wave” equation. The general situation was considered first in [28]. 4

Theorem 8.17 gives us a natural way of looking at the Radon-Penrose. , transformation not only on &closed (O,q)-forms on D but also on arbitrary *’


(0, q)-forms f-e @,(D, 0( - q - 1) for which the first term I;+ i (8f) on the right side of (8.14) vanishes, The class of such forms depends essentially on the choice of the map&g z H u(z), z ED. Let us indicate the effectiveness of such a point of view on a “physical” example. In the following let D be a l-linearly concave domain,m’ C[Fp3 given by

D = {C&p’: - 1r1t2 - ILIZ + 11312 + 1412 < 01. (8.16)

I In this case the domain 0: c G(2,4) is a classical domain in G0(2,4) 2: C4 and consists’of (2 x 2)-matrices d = {akj} satisfying the condition I - da.&‘* 2 0. Let us fix the mapping z H a(z) given by a(z) = {d(z), - I}, where

zlz3 - z4z2 Z2Z3 + z4z1

mm’ lz112 + lz212

ilZ4 + Z3Z2 f2Z4 - Z3Zl

lz112 + lz212’ lz112 + lz212

1 0 7 I=. c > 0 1.

The family of 2-rep&es a(z) in C4 gives a fibration over the domain D by non- intersecting projective lines A(z) t D. We introduce on 0: the coordinates

w = (wi, . . . , w4) such that a 11 = Wl + IW2, a 12 = w3 - iw,

azl = w3 + iw,, az2 = - w1 + iw,,

where wj = xi + iy,. Consider, in C4, the real Euclidean subspace E = {w E c4: y,= . . . -y, = 0). We remark that the four-dimensional ball E n 0: is pre- dsely the image of the mapping ZH’W(&(Z)), z E D.

Let us denote by 9:. I(D, 0( - 2)) the space of (*O, l)-forms f on D such that the coefficients of the form f and of af are generalized sections of measure type for the vector bundle and [Jf]j = 0, j = 1,2.

Proposition 8.19 ([53]). Let fE 58’:. ,(D, O( - 2)). Then there exists a generalized function of measure-type cp = q(x), XE E n DT such that

af (z) = 44x(z))% (a(z), z) A E2(a(zh z), (8.17)

I where x,(z) = Re w/(z); and w,(z) are the components of the mapping z H w(&(z)).

Let us introduce for consideration two more spaces: &$, l(D,O (- 2)) is the quotient space of Ut, I(D, 8( - 2)) by the subspace of J-exact forms; Xz(D:) is the space of all generalized functions of measure-type in the domain 0: which satisfy the equations a- .

&I EnD: = O,

7 stw) + & pk, jfw)$& = Ov

9. 1

I’

(8.18)

112

where ~EDT, Pij(“‘) = 2&i) ly 1’ for h- # .j.

On the basis of Theorem 8. Pcnrosc’s ‘Thcorcm.

G.M. Khenkin

‘I2 - ‘1 for k = j; and P,,(W) = pi,(w) = zy, .yj/

17, one obtains an unexpected strengthening of

rvhrre II/ = Rf, .Y E E n UT.

Theorem 8.20 also allows us to give the following analytic reformulation of a well-known result of Atiyah (1981).

Let US fix a point x E E and a projective line A0 c @iFp3 corresponding to the point x0. Let f” be ~7 (0, I)-form from Yt,,(@P3,c( - 2)) such that

VIE Cei (CIFD3, C ( - 2)). Then, for the form /’ we have Rf” = $‘(w) = C(M’-x0, w-.1-O>-‘, i.e. r(f” is a holomorphic extension to 0: of the fundamental solution to the Laplacc equation on E n 0:.

We have touched upon, here, only a few aspects of the theory of Radon- Penrose Transformations connected with integral representations of differential forms. Marc complete information on the Radon-Penrose transformation and its applications is contained in the papers of Vol. 9, VII, Vol. 10, II (see also [::I, [Sl] and L64]). ,

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i t i

I ’

- -.

EM. Chirka

Translated from tbc Rlrr;tn by P.M. Gautk

Contents’ *

Introduction . . . . . . . . . . . . . . . . , , . . . . . . . . . . . . . : ‘. . . . . . . . . * . . . 118 . 41. L,ocal Structure of Analytic Sets. ...........................

1.1. Zeros of Holomorphic Functions. ........................ ::E

* 12. Analytic Sets. Regular Points. Dimension ............. ; ... 120 1.3. SetsofCodimension l................ :. ............ >. 120 . 1.4. Proper Projections. .................................... 121 1.5. Analytic CoGerings. .................................. 122 1.6. irreducible Components. .............................. 124 1.7. One-Dimensional Analytic Sets

$2.0 Tangent Cones, Multiplicity and Intersectidkkny. ..................................... 125 ,

126 2.1. The Tangent Cone ..

_.&$,JV~~y.Cones., .......................................................................

:. iZ&

2.3. Multiplicity of Holomorphic Mappings. ................... @ .2.4. Multiplicity of an AnalyticSet at a Point .. ., ..... 1 ........ j30, 2.5. Indices of intersection. Complementary &dimensions ... 1 ... 131 2.6. Indices of Intersection. General Case. ..................... 132 2.7. Algebraic Sets. ...................................... 133

53. Metric Properties of Analytic Sets .......................... ,135 3.1. Fundamental Form and Theorem of Wirtinger ............... 135 3.2. Integration on Analytic Sets. ........................... 136 3.3. Crofton Formulas,. .................................. 138 3.4. The Lelong Number. .................................. 139 3.5. Lower Estimates for the Volume of Analytic Sets ........... 140 3.6. Volumes of Tubes ............. .: .................... 141 3.7. Integrability of Chern Classes fi ......................... 142

118 .I - EM. Chirka

k; H&morphic Chains. ........... . ........................ 143 4.1.. Characterization of Holomorphic Chains. ................. 143 4.2. Formulas of Poincare-Lelong .......................... 144 4.3. Characteristic Function. Jensen Formulas ................. 145 4.4, The Second Cousin Problem ............................ 147 4.5. Growth Estimates for,Traces, on Line+; Subspaces..

45. Analytig Continua&n a%& Bbun&iy -Prop&r& ...................... 148

.. . ... .;.a 149 5.1. Removal of Metrically Thin Singularities. ................. 149 5.2. Removal of Pluripolar Singularities. ...................... 150 5.3. Symmetry Principle ....... ;.‘.:i .. .‘)‘. : ................... 151 5.4. Obstructions of Small CR-Dimension,. ................. . .. 152 5.5. Continuation Through Pseudoconcave Hypersurfaces. ....... 153 5.6. The Plateau Problem ‘forAnaly%ti ‘Sets .l‘; ................. 154 5.7. Boundaries of One~Dimet&nB Analytic Sets. ............. 155 . 5.8. A Perspective and Prospects. ........................... 155

References...........................,...........; ........ 156 . . t

r’ r : ,.. ,__

. 1 Introduction ’ J : * 7

.. T _- L

-The theory of complex analytic sets is part of the moderngeometric theory of functions of several complex variables. Traditionally, the- presentation of the foundations of the theory of analytic sets is introduced in the algebraic language of ideals in Noetherian rings as, for example, in the books of Herve [23] or Gunning-Rossi [19]. However, the modem methods of this theory, the piincipal directions and appli&tions,‘ are basically related to ‘geometry and anaiysis-(without regard to the traditional direction which is essentially related to algebraic geometry). Thus, 3’: the beginning of this survey, the geometric construction of the local theory of analytic sets is presented. Its foundations are worked out in detail in the book of Gunning and Bossi [ 191 via the not&of analytic cover which fogether with analytid theorems on the removal of singularities, leads to the minimum of algebraic apparatus necessary in order to get the theory started.

The study of smgularities of analytic sets at the present time forms one of the ’ most fascinating areas of analysis, geometry, and topology. However, :his direction has developed to such a degree that it has practically distinguished itself as an independent branch of analysis, bordering on the theory of singularities of differentiable mappings. Without going deeply into this theory, we briefly set forth the infinitesimal properties of analytic sets, but only a first approximation thereof (sufficiently’rich for applications). In this context, tangent cones, variousmultiplicities and intersection theory come to the fore.

III. Corn&x Analytic Sets 119

In thelast 10 years the direction connected with differential geometry and the theory of currents (generalized differential forms) has flourished intensively. Initiated in the work of Lelong, it was significantly pushed ahead by Griffiths and his school. This direction is reflected in @3-4 here, we briefly set forth the basis of integration on complex analytic sets as Well as some metric; fUnCtpnaL

analytic, and differential-geometric properties. The boundary properties of complex analytic sets are perhaps most connected

with the classical theory of functions. This direction has begun to develop very recently, but already it has to its credit a series of impressive results (theorems on the removal of singularities, the theorem of Harvey-Lawson on the structure of the boundary and others). There are vereteresting, connections here with the boundary properties of holomorphic mappmgsA.Tbe basic results are presented in 45 along with several problems ofthis prospective direction.

$1. I+ocal Structure of Aklytic Sets

1.1. Zeros of Holomorphic Functions. Let 0, denote functions holomorphic in a neighbourhood of a point a, and O(D)-functions holomorphic in a domain D. A holomorphic function ff 0 of one complex variable can be represented, in the neighbourhood of one of its zeros%,, in the form f(z) = (z - z,)kg(z), where k = ord,,fE &I is the order (multiplicity) ofthe zero aqd gEoZO, g(z,) # 0. From this it follows that the zeros off form a discrete set 2,. Conversely, if 2 = {z,} is an arbitrary sequence in a domain D 5 C, having no limit points in the interior of d, and ( kj) are arbitrary natural numbers, then, there exists a function SE O( D) for which the set of zeros coincides with {z,} and ord,,f= k,.

‘The local structure of the zeros of a holomorphic function of several complex variables yields the following most important result..

Weierstrass Preparation Theorem. Let jc be a function holomorphic in a neighbourhood of0 in @“J(O) = 0, butf(O’, z,) + 0 in a neighbourhood of0. Then f(z) = (T4, + c,(z’)z;- l + . . . + ck(z’)) *g(z), where c, and g are holomorphic in a neighbotirhood of0, Cj(0) = 0, and g(0) # 0. (Here z = (z’, z,)‘.) ,

Thus, in the vicinity of 0, the zeros off have the same structure as the zeros of the Weierstrass polynomial 2 + c,(z’)$- ’ + . . . . The proof of this theorem IS in all textbooks on multidimensional complex analysis (see, for example, [39], [19], [53]). The only algebraic theorem which is used in its proof is the well- known theorem on symmetric polynomials.

.

From Rado’s Theorem (see [39]) follows easily the existence of,a function A(z’) + 0 holomorphic in a neighbourhood U’ 30 in C” - i and a disc U, such that for A(z’) # 0, the number of geometrically different zeros off(z’, z,,) in the disc U, is the maximum possible (and one and the same), while for A(z’) = 0 they

120 E.M. Chirka

are sort of “glued together”. From the classical Rouchc Theorem it follows easily that the projection Z, n (U’ x U,,) -+ U’ onto U’\Z, is locally biholomorphic and this yields a finite sheeted covering space.

1.2. Andytic Sets. Regular Points. Dimension. Let R be a complex mani- foId.‘A sc~. A c R is called a (closed) analytic set in 51 if for each point a E Q, there is a neighbourhood U 3 a and functions fi , . : Ar$J=Zf,n...

. ,jN holomorphic in U such that n Z,-- n U. Thus, locally, A is the set of common zeros of a

finite collection of holomorphic functions. Such a set is necessarily closed in 0. If the above representation holds only for points a E ‘4, then we obtain the broader class of locally analytic sets.

Several simple properties fallow from the definition: 1). The intersection of a finite number ‘of analytic sets is an analytic set. The

union of a finite number of analytic sets is an analytic set. (In this, analytic sets advantageously differ from submanifolds.)

2) If p: X + Y is a holomorphic mapping of complex manifolds, then the preimage of an arbitrary analytic set A c Y is an analytic set in X.

3) If R is connected and an analytic subset n c R contains a non-empty open subset of R, then A = Q (* if A # Q then A is nowhere dense in 0).

4) On a Riemann surface (= connected one-dimensional complex manifold), each proper analytic subset is locally finite.

5) If R is connected and A c R is a proper analytic subset, then 0\ A is arcwise connected.

A point a. of an analytic set 2 is called a regular point if there exists a neighbourhood U~U in R such that A n U is a complex submanifold of U. The complex dimension of A n U is then called the dimension of A at the point a and it is denoted by dim, A. The set of all regular points of an analytic set A is denoted by reg A. Its complement, A\reg A, is denoted by sng A. The set sng A is called the set of singular points of the set A.

It can be shown by induction on the dimension of the manifold R that sng A is nowhere dense (and closed) in A. This allows us to define the dimension of A at any point of A:.

_ dim,A= lim sup dim, A.. Z-P.ZCregA

The dimension of A itself is the number dim A = max,,,,, dim, A and the codimension of A is equal to dimR\dim A. The set A is called purely p-dimensional if dim., A 5 p, for all z E A. Along with sng A, it is convenient to single out also the set

S(A) = (sng A) u (z E A: dim,A < dim A).

1.3. kets of ,&dimension 1. From the Weierstrass Preparation Theorem, it follows that the zeros of a function f f 0 holomorphic on a connected n-dimensional manifold R form a set Z, of codimension 1. Using the dis- criminant set Z, ($1.1) and induction on n, it is not hard to show that the

III. Complex Analytic Sets 121 .

Hausdorff dime&ion of Z, is equal to 2n - 2 while the Hausdorff dimension of the set sng Zik at most 2n - 4. Since each analytic set is locally contained in the set of zeros of a single holomorphic function, then the following metric estimate on the size-of the set of singular points is satisfied: if A is a proper analytic subset of R, then the Hausdorfl dimension of S(A) is at most 2n - 4. Since sets of Hausdorff dimension I 2n - 4 are removable for bounded holomorphic functions in domains in C-r, we obtain; from this and from the Weierstrass Preparation Theorem, the following theorem on the local structure of analytic sets of codimension 1. Let #E denote the number of points in a set E.

Theorem 1. ket A be an analytic (n - lj-dimensional subset of the pofytfise U’x U, in @n(U’~Cn-l, U, c C) without limit points on U’ x XI,. Then:

(1) A can be represented us the union of two analytic sets A,,- 1I u A’, where A(,-,,=(zcA:dim,A=n-l),A’xS(A),anddimA’<n-1;

(2) there exists a Weierstruss polynomiulf on U’ x C such that A,, I I) = i?, and gor each fixed Z’E V/Z,, the polynomial f has only simple roots (the tatter,,is equivalent to the assertion that

degf= max #A n {z’ = c} j. cetl’

Thus, a purely (n - 1)-dimensional analytic set is locally the set of zeros of a single holomorphic function. A function fE 0(Q) is called a. m@imal defining firnction for an analytic set A c R, if A = Z, and for each domain U c Q, any function g E O( U), which is equal to zero on A n U, is divisible by,fin 8( U). The Weierstrass polynomialf, mentioned in the theorem, has this property, while in general, the minimality condition on f is equivalent to the condition that df# 0 at each regular point of A. More precisely, the following holds. ‘.

Proposition I. A holomorphic function f f 0 on a connected marCfold. is. a minimal defining function for A = Z, if and only if the set (2 E A: (df), = 0) is nowhere dense in A. If f is minimalfor A, then zng A coincides with the analytic set ;f= df= 01.

1.4. Proper Projections. The codimension of analytic sets in @” is diminished by suitable projections into subspaces of smaller dimension. Anaiytic&, in general, is not preserved by projections (for example, the coordinate project[oy, of the set A: zi z2 = 1 in C2). Additionalconditions are required. The fcilowiqg~ a fundamental assertion on the lowering of codimension (“excluded variables),

Lemma 1. Let vi c CP, U” c Cq be open subsets and z (z’, z”j -+ z’&?; ’ Let A be un unulytic subset of U’ x U” without limit points on U’ x ?U”. Then, A’ = 7t( A) is an analytic subset of U’ and A n n- ‘(z’) is locally finite in U’. Moreover, :\

dim,. A’ = max dim, A. n s(i) = I’ ,a.

In particular,.dim A’: = dim A.

122 E.M. Chirka

A rather simple proof of this’important assertion can be found, for example, in the book of Mumford [28$ The condition on A implies that xi,4 is a proper kpping, i.e. the preimage of any compact set in U’ is compact in A. The proof of the existence ofsuch projections (locally) is obtained rather sjmply by induction on the codimehsion. The following holds:

Lemma 2. Let A be an analytic I.-+ in C”, 6~ A, and dim,, A 5~. Let Cl,.. . ,4, be un arbitrary jinite collect p-dimensional subspaces of @” and rrI the orthogonal. projection on L,. Then, there exists a unitary transformation 1 and arbitrarily small neighbourhoods U,3 0 such that each Hestriction n,ll(A)nU,-+L,nU~,j=l,,.., k, is a proper mapping. Moreovel- the set of such 1 is open a@ everywhere dense in the gr rp of all unitary transformations in @".

The following assertions are easily obtained as a corollary of these two lemmas. . . .~

1) An analytic set A c C” is compact if and only if it is finite (‘.e. #A < 00). 2) An analytic subset of a complex, manifold is ZL

it is locally finite. ) -dimensiona, if and only if

3) If A, are analytic sets and aE h A,, then codim, 1 1 Aj s icodim, A). . 1 1 1

15. Analytic Coverings. The lowering of codimension by proper projections and the theorem on the local structure of analytic sets of codimension 1 lead to the following representation for the local structure of analytic sets of arbitrary codimension.

Theorem 1. Let A be an analytic set in C”, dim, A = p, 0 < p <‘n, U = U’ x U” a neighbourhood of a, with nlA n U + U’ c Cp (‘1 Then, there exists an analytic subset o c U’ of dimensior8

?per projection. and a natural

number k such that (1) nlAn U\x-‘(a)+ U’\ (2) 11-l

u is a locally biholomorphic k-sheeted covering, (a) is nowhere dense in ACp) A U, where AipI = { z E A: dim, A = p >.

For such a structure it is convenient to introduce a particular name (see [19]).

Definition 1. Let A be a locally closed set on a complex manifold X and fi A + Y a continuous proper mapping into another complex manifold Y such

+ that the preimage of each point is finite. The triple ( A,f, Y) is called an analytic covering (over Y) if there exists an analytic subset CJ c Y (perhaps empty) of dimension < dim Y and a natural number k such that

~U)flA\\f-‘( 1 c is a locally biholomorphic k-sheeted covering (over Y\o), (2) f- i (a) is nowhere dense in A.

If we drop the hypothesis that # f - l(w) < co and replace the condition of . analyticity of u by that of local removability, then we obtain an object (A,f; Y) called a generalized analytic covering. (A set cr is said to be locally removable if it isclosed and for each domain U t Y, each function bounded and holomorphic

III. Complex Analytic sets 123 l\

i

, in U\a e ends hoiomorphically to U,; an example of such a set is any closed subset of-

3 usdorff dimension < 2dimc Y - 1.)

Theorem asserts that each purely p-dimensional analytic set is locally covering over a domain Cp. The converse is valid

covering.

of a domain U = U’ x U” c @” and generalized hnalytic covering. Then, A

of U and rrIA + U’ is an analytic i - \

A convenient analy ‘c tool for the study and the application of an&& 1 coverings is given by canonically defined functions representing mult sional analogues of the pblynomials TI(z - a,) on the plane (see Whitne They are defined in the hollowing way. Let sl,, . . , & be points of cm,- ‘not necessarily distinct, and set PJz, w) = (HI, z 2 a1 ) . . . ( w, z - ~8 ), where (a,b)=a,b,+... + a,,,b,. The con&ion P,,(z, w) s 0 for a fixed z E Cm, as is easily seen, is equivalent to the assertion that z belongs to the (unordered) collection u = (al, . . . , ~“1. Expanding P,(z, w) in powers of w,

Pa@, 4 = c Q,(z; a) WI, 14-k we obtain that this condition is equivalent to the system @,(z; a) = 0,111 = k, of

(“.‘,“T ‘) equations of degree I k for z. Thus, the set of solutions of this system is precisely the set a. The functions @, depend nicely on a:

@‘I (z; a) = Jk h’ w’,

where t,kIJ are,polynomials of k * m variables a’, of degree no greater than k. In the situation, when a = a(?) are the fibres A n R - ’ (z’) of the generalized analytic covering AI A + U’, z’ 4 o(as in Theorem 2), $Jtl(z’)) are bounded holomorphic functions on U’\a (since +Ja) is independent of the enumeration of the collection a). Since+iS removable, we obtain holomorphic functions

P

Q),(z’, z”) = ,‘$ k (P,,wIz”)J, 111 = k,

whert cpIJ(z’) is the holomorphic continuation to U’ of the function #,i(a(z’)). These a, are called the canonical defining functions of the (generalized) analytic

overing 1~1 A + U’. The set of common zeros of the 0, in U’ x @“-p coincides

F ith A (and from this, of course, follows the analyticity of A).

124 E.M. Chiika

The analytic set in A, defined by [he condition /

rank (mz’, 2”) ---<n-p,

az” / is precisely the set of critical points br z IA of the projection n 1 points as well as those regular points z, for which rank, nl

, i.e.

f

the singular

easily follows, for example, the analyticity of sng A for pure1 < p. From this,

p-dimensional A.

1.6. Irreducible Components. An analytic set A is calle irredkible if it is not possible to represent A in the form .4, u A,, where A, a d A, are also analytic sets, closed in A and distinct Cram A (in the opposit case, A is said to be reducible). Irreducible sets have a simple characterizati n in terms of regularity.

Proposition 1. i connected.

An analytic set A is irreducible if , d only ij” the set reg A is

,

An analytic set A is said to be irreducible at a point aE A if there exists a fundamental system .6f neighbourhoods Ujsa such that each A A U. is irreducible. We say that -A is locally irreducible if.it is irreducible at each of its

i points. The simplest example of a locally-irreducib!e but reducible analytic set is ,I a disconnected Complex submanifold. The set A: zf = z: + .z: in C2 (the image

of C by the mapping I -+ (A2 - 1, R(i12 at the point z = 0. The set A: zf I

- 1)) is globally irreducible but reducible zj.z: in C3 is irreducible globally and at 0 but

‘\ is reducible at each point of the form (0, 0, z3), z3 # 0. . An analytic set A’ c A is said to be an irreducible componenr of A, if A’ is

irreducible and maximal, i.e. is contained in no other irreducible analytic set A” e A.

Theorem 1. Let A be an ana!ytic subset of b complex manifald P. Then: (1) each irreducible component of A has the form 3, where S is a connected

component of reg ‘4;

(2) ifreg A = U Si is the decomposition into conrsccted components (J-finite or je.J

countpble, Si f Sj,Jx i f j), ther; A = u sj, and this is the decomposition of A

into irreducible components; ’ jcJ

(3) the .decomposition into irreducible components is focally finite, i.e., any compact set K c R meets at most finite1.v many si.

From this and from $!.5, we have the following properties: 1) Let A be an ;inaIytic subset of R of dimension p and hfk) = {ZE A:

dim, A = k). Then &, is a purely k-dimensional or empty analytic subsei of R

and A = fi&,.

2) For iny analytic set A, the sets sng A and S(A) are also analytic. Moreover, dim S(A) < dim A and dim, sng A < dim; A, for each z t: sng A.

111. Compirx Analytic Sets 125

a of an analytic set A, there exists a neighbourhood U 3a component of the set A n U is also irreducible at a.

are analytic subsets, A, is irreducible and dim A, n A,

c AZ. Moreover, AI belongs to some irreducible component of the set A2. ’

a From these prop rties and from the decomposition into irreducible components, we easily obtakn the following:

Theorem 2. Let {halJtl be an arbitrary family of analytic s&sets $ a

complex man$old R. Then, A = fl A, is also an analytic subset of 0. Moreocer,

for each compact set K c Q, tl;e:lf.is a finite, subset J c I such that A r\ K = I \

1.7. One-Dimensional Analytic Sets. The local structure of one-dimensional analytic sets is the simplest. If dim, A = 1, Usa, and Z( A n U -+ U, is ,a.n analytic covering over a disc U, c C, then the critical set 0 c U, of this covermg is zero-diniensional, and hence, either empty or consists of the single point a, = n(a), if U, is sufficiently small. Since the fundamental group of the punctured disc is Z, from this one easily obtains, the following.

Proposition 1. [fan antilytic set A-is one-dimensional and irreducible at a point a, lherl there exists a neighbourhood U 3 a and u holomorphic mapping of the disc A c UZ into U which covers A n U in a one-to-one manner.

In the situation described above, when A n U c @” is irreducible, a = II- l,(O) n A = 0, (T = { 0), and the number qf sheets of A n U + U, is k, such a mapping (local parametrization) can be written in the form z1 = ck, zj = 4pj(sc), j=2,..., II, where qj are functions holomorphic in the disc A c @. In other words it can be written in the form z, = cp,i(z:‘k),j = 2, . . . , n (representation di Puiseux). Since the indicated mapping A- + A n U is one-to-one, the inverse . . mapping 5 = t(z) is a function contirluous on A n U and holomorpntc m , 1 n 0’ ‘, {O 1. It is calied a loccrl normcrli5tlq puramcter on A in a neighbourhood of the point 0.

It is not possible, in principle, to parametrize in this way analytic sets of dimension greater than 1, because even in the neighbourhood of an isolated singular point, an anal;tk set which. is irreduc!bIe at the point need not be a topological manifotd. For example, the cone A: z: = z2z1 in a neighbourhood of 0 in C”: A\(O) is the image of C;\(O) by the two-sheeted coveSing

(%,, i;,)~(~:,If,i,,;l,),a~ld hence AnU\{O) is not simply connected for any

U 30 and hence cannot be homeomorphic to the punctured ball. The global structure of one-dimensional analytic sets is also comparatively

simple. Let sng A = { aj} and Uj3aj be pairwise disjoint neighbourhoods such that A n Uj cofisists of a finite number of irreducible components Sj, each of

126 .;rka

which admits a normalizing parameter 5j,: Sj, -B A c @. Let Vj c neighbourhoods of the

where u denotes the disjoint union (points from different SjV are considered different). The set A* is naturaliy endowed with the stricture of a complex manifold, since the usual charts on A\ u 6 c reg A are holomorphically compatible with each of the charts (Sj,, tj,). Here, the natural projection A* + A is obviously locally biholomqrphic over reg A. The manifold A*, which we have constructed, is. called a normalization of the one-dimensional set A. If A is irreducible, then A* is connected, i.e., is a Riemann surface. Every Riemann surface has a universal covering surface which, by Riemann’s theorem is either

‘the disc, the plane4 or the Riemann sphere. Hence, for example, each ‘bounded one-dimensional irreducible analytic set A in 8=” is the image of the unit disc A by some hdlomorphic mapping A covering.

+ Q=” which over reg A is a locally biholomorphic

Any analytic set of positive dimension is locally representable as a union of one-dimensional analytic sets containing the given point (this follows easily from the local representation in terms of an analytic covering). From this, we obtain,.for’examp.le, the following important property.

Theorem (Ma&mum principle). Let u be a p&subharmonic function in the neighbourhood of a connected analytic set A. Xf u(u) = sup, u at some point a E A, then u = u(a) on 4.;‘ i;t

Remark. The resClb of tt& section are basically well known, The proofs can be found In the booke [19], 1231, [53] and the author’s book [lo].

62. Tangent Cones, Multiplicity and Intersection Theory i:

2.1. The Tangent Cone. Let E be a,? arbitrary set in R”. A vector D E IWO, is called a tangent vector to the set E at the point a if there exist a sequence of points ajE E and numbers tj > 0 such that a/ e n and tl(aj - a) + v, as j -+ co. The collection of all such vectors u is denoted by C( E, a) and is called the tangent cone to the set Eat the point a. If E is a manifold of class C’, then C(E, a) clearly coincides with the usual tangent space T,E of *the manifold E.

Let us examine some properties of tangent cones for analytic sets in @” x W2”. If A is a purely one-dimensional analytic set in 4=“, irreducible at a point aE A, then, from the local paramatrization in’5 1.7, it follows easily that C( A, a) is a complex line. Hence, the tangent cone to an arbitrary one-dimensional analytic set at any of its points is a finite union of complex lines. The case of higher

j

III. Complex Analytic Sets 127

dimensidn is considerably more complicated (for example, the tangent cone at 0 for a set 4 given by pj(z) = 0, j = 1, . . . , k, where the pj are’ homogeneous

polynomials, simply coincides with A). As regards an analytic description, the

siriiplest casi is that of a hypersurface.

Propodtion 1. Let ff 0 be a function holomorphic in the neighbourhood qf 0 in Q=“, ui ’ q 2 1, the multiplicity of the zero off ut 0, i.e., in the expansion

f(z) = f-f;(z) in homogeneous polynomials, f, $0. Then, C(Z,, 0) = { z:,f,(z) 11

= 0). In the general case, a good descripti6n is obtained in ter&.$the canonical

defining functions.

Proposition 2. Let A be a purely p-dimensional analytic set in C”, OE A, and C(A,O)n{z, =. . . =z,,=O}=O. Let Ug U’x U”beaneighbourhoodof0 such that the projection n[ A n U 3 U’ c cP is an analyfic covering, and let #,, 111 = k, be can&ical deJining functions of this covering. Then C( A, 0) = (z: @F(z) = 0, III = k}, w h ere 07 is the initial homogeneous polynomial in the expansion of the function cPI near 0.

Corollary. The tangent cone to an analytic set at any of its points is a homogeneous algebraic set.

For many questions, the tangent cone successfully replaces the tangent space and this is connected, iost of all, to the following proximity property.

Proposition 3. Let A be a purely p-dimensional analytic set:in C” and 0~ A. Then,. there exist positive constants E, C, and r0 such that dlst(A,, C(A, 0),)

I Cr’ CC for each r <‘r,, where dist denotes the standard distance and E, = En{lzJ <r}.

This proprty follows from Proposition 2; indeed, from there it follows that, as E one may choose l/k, where k is the multiplicity of the covering in Proposition 2.

2.2. Whitney Cones. The tangent cone introduced above is not the only possible analogue of the tangent space in the nonregular case. Other natural

“generalizations of this notion, useful in the study of analytic sets, were investigated by Whitney [51], [52]. We introduce the definition of Whitney cones C,(A,a),i = 1,. . . , 6, via their constituent vectors. Let A be an analytic set in @” and aE A.

1) A vector v G a=” belongs to C, (A, a) if there exists a neighbourhood U 3 a in @” and holomorphic vector field v(z) in U (i.e. a holomorph;c mapping U + Q=“)

I such that v(a) iv and u(.z)ET,A. for each z~regAn U. 2) u E C2 (A, a) if for each E > 0, there exists a 6 > 0 such that if z E reg A and

lz - al < 6, then iv - ti’( < E for some V’E T,A.

128 E.M. Chirka

3) C,(A, a) = C(A, a), the tangent cone from $2.1. 4) UE C,(A, a) if there exists a sequence of point zje reg A and vectors

vie TzjA such that zi + a and d -+ v as j + CO. 5) v E C,( A, a) if there exist sequences of points zj, LG A and numbers lie @,

such that zi -+ a, d --+ LI, and E,j(Z’ - w’) -+ L‘ as ,j -+ 0~:. 6) t’ E C, (A, (I) if (df),( c) = 0 for each function ,1; holomorphic in the vicinity

of a and equal to zero on A (the set C,(A, n) is denoted by T( A, a) and is called the tangent space to A at the point a).

Proposition 1 (Whitney [Sl J 1. At rucll point of an analytic set we haue the inclusions C, c C, c C, c C, c C, c C,.

The cones C, . C,, and C, are clearly linear stibspaces of @“. The cone C,, as shown above, is an algebraic subset of @“. For C, and Cs, this property issues from the following.

Proposition 2 (Whitney. [Sl]). Let A he 011 anulytic suhsrt of a domain D~Cna~~dCi(Aj=i(~,~~):~~A,U~Ci(A,Z)),i=4,5.Tlie~~,C,(A)urear~al~~tic subsets of D x @” of dimension 2 dim A, where Cj ( A ) is the closure in D x @” of the tangent bundle [ (z, c): z E reg A, I’ E 7; .4 ).

As a’ corollary we also have that the cones Ci( A, z), i = 4,5, are “upper- scmicontinuous” with respect to z, i.e. if A 3 zj -+ a E A, then the limit set of the family of cones Ci(A, $), asj

The cone + -x, is contained in Ci( A. a).

C 6, the tangent space, rellccts the “imbedding dimension” of the analytic set. The following is easily proved.

Proposition 3. Let A be an rtnulptic set in @“, a~ A, und him T(A, a) = d. T\ten, there exisis a neighhotrrhood U 3u in @” und a complex d-dimensional suhmrrcifold I!! c U such that .4 n U c ICI. Ma:Cfi~lds qf ;es.ser dimension with this propert~~ do not exist.

The cones C, and C, rcflcc: important local properties of analytic sets (see, for cxamplc, 1493 and [53]). If, for example, dim C,(A, a) = dim, A EE p, then a is a rcSular point of A; ifqim Cj(A, a) = p and nEsng A, then dim,(sngA) = p - I;

CIC. (see L4.91). The cone C, .:!so “assists” in normalizing crrtain singula)rities of codimensiori I (Stutz [49]):

Pro@sitioa 4. Let A he a p;ir~/~ p-dimensional analyti’c set in @” irredltcible at i!~ point 0 E reg (sng A ) ~lnd let dim c, (A, 0) = p. Then, there exists a neigh- boarhood U 30 and a one-to-one holnmorphic mapping : = z(r) of the polydisc A”: I(,iI < 1, j = 1, . . . , p, onto the set A n U such thut the Set sng A’n U = :(bP n ( <I = 0) j and the restriction A”\{ {I = 0) --b reg A n (j is biholomorphic. !f the local coordinates are chosen such that sng A n u = u n ( z1 = zp+l =... = z, = O> and C,(A, 0)‘n {zl = . . . = zP = 0) = 0, then one such parametrization has the form


where r > 0 is constant, and k is the multiplicity of the cornering A over Cf, , . zp in th&%?ighbourhood of 0, and zj(<) = <: (pj(@, where ‘pj are holomorphic in the neighbourhood of 0 in Cp.

This parametrization can be seen as a multi-dimensional generalization of the Puiseux representation and as a first step towards the “normalization” of an arbitrary analytic set. Since the set (2~ A: djm C,( A, z) > p} has dimension up - 2 (see [49]), then after such “unwindings” and “unglueings” at reducible points, we obtain a set with singularities of codimension 2 2; it turns out that this can be done while preserving analyticity (see [14] and [53]).

2.3. Multiplicity of Holomorphic Mappings. Let .4 be a purely p-dimensional analytic set in a complex manifold and f : A -+ Y a holomorphic mapping into another complex manifold. Suppose a~ A is an isolated point of the fibre ,f- ‘(,f(u)) and that d im Y = p. Then, clearly, there kxist neighbourhoods U 34 and V3 f(u) such that .I’1 A A U + V is a proper mapping and f - 1 (,f(a)) n A n e’= tl. According to 91.5, fj A n U is an analytic covering over V. Its multiplicity (number of sheets) is independent of the choice of U and V having the given properties: it is called the multiplicit~~ qf the mapping f at the point a and is denoted by p,(,fi A) or simply p,(,f). For each )Z’E V\o, where Q is the critical set of the indicated covering, the number of preimages in A n U is precisely p(,( ,f). Each of these points are regular on A and in the neighbourhood of such a point, the mapping,[is locally biholomorphic. For MJ E C-J, the number of preimages on A n U is strictly less than p,( f )? Since G is nowhere dense in V,

p,,(f) = Wi~, (I ) # f-I(w) U.

This equation may be taken as the definition of the multiplicity, even if dim Y > p; in this case, the only requirement on lJ is thatf - ’ (f(a))n A n fl = a.

Each mapping reduces to the projection of its own graph, and hence, the local properties of multiplicity can be studied in the standard situation of the analytic covering, namely, projection nl A A U +*U’ in C”. ’

The following properties of multiplicit) follow immediately from the definition.

1) The additivity of multiplicity: If A = u: Aj, where each A, is purely p-

dimensional and dimA,nAj <p for i #j, thenr,(flA)=~P.(f/Aj).

2) Multiplicativity ‘of multiplicity: If A--L Y ’ -----+Z are holomorphic map-

pings with discrete fibres and f is proper,.then

dgGflA) = ~,(flA).~/,,,(gIf(Aj).

For the calculation of the multiplicity the following generalization of Roucht’s Theorem is very convenient (see, for example, Cl]).

130 E.M. Chirka

Theorem 1. Let D. be a domain with oompact closure in an n-dimensional complex manifold. Let f and g be holomorphic mappings D + @” continuous up to the boundary and such that If- g 1 < IfI on dD. Then,

The calculation of the multiplicity of a mapping at a point begins with the study of the first non-zero terms of the Taylor expansion and the use of Rouchi’s Theorem for the remaining terms. In this direction, the following theorem of A.K. Tsikh and A.P. Yuzhakov is very useful (see Cl]).

Theorem 2. Let f = (fi, . . . , f, ) be a holomorphic mapping in a neighbourhood of OE@” such that f(0) = 0 and f-‘(O) = 0. Let ff be the initial homogeneous polynomial for 4 at 0 and f* = (f:, . . . , f,*). If (f*)- ’ (0) = 0, then

PO(f) = ~ord,f; = 0 degf:.

In the opposite situation, p0 (f) > fiord, 4. I

In place of homogeneous polynomials here, one may take quasi-homogeneous ones (assigning to each variable its own weight). The proof easily reduces to the homogeneous case (see, for example, Cl]).

2.4. Multiplicity of an Analytic Set at a Point. Let A be a purely p-dimensional analytic set in @” and aEA. For each complex (n - p)-plane LsO such

. that a is an isolated point of A n(a + L), the multiplicity p,(~~I,4) of the orthogonal projection to the plane L’ is defined ($2.3). The minimum of these integers over all (n - p)r&nes L (i.e. elements of the Grassmannian G(n - p, n) is called the multiplicity of A at the point a and is denoted by pa(A).

At regular points, the multiplicity of an analytic set is clearly 1. The converse is also true: if p,(A) = 1, then aE reg A. This follows easily from the local representation of A as a covering with a finite number p,(A) of sheets (see 0 1.5).

The following property of biholomorphic invariance allows one to define the multiplicity of an analytic set on an arbitrary complex manifold (with the help of local coordinates).

Proposition 1. Let A be a purely p-dimensional anilytic subset of a domain D in @” and cp: D + G c @” a biholomorphic mapping. Then,

for each a E A.

The proof can be obtained, for example, from RouchC’s Theorem and the fact that rp is homotopic to the identity mapping ZI+Z.

III. Complex Atialytic Sets 131

The tangent cone plays an im‘portant role in the study of the multiplicity’ of analytic sets. ,*: .

Theorem 1.: Let Abe a purely p-dimensionalanalytic set in the neighbourhood ofOinC”,OEA,andLEG(n--‘p,n):Theequalitypo(a,lA)=~o(A)holdsYand only if the subspace L’is transversal to A at 0, i.e. L n C(A, O)=O.

I The proof is fairly simpie in case p = n - 1. Indeed, if f is a minimal defining

function of such”an analytic set ‘in the neighbourhuod of 0 (see §1.3), then p. ( R‘ I A) = ord, f 1 L, and the tangent. cone is precisely the set of zeros of the initial homogeneous polynomial ofthe function f (see 92.1). In the general case, the proof reduces to the hypersurface case by suitable projections in UZp+ r and is technically rather bulky(see (lo]). 1

The function p;(A), z B A, is clearly upper semi-continuous. ,As mentioned already, (i: pZ.( A) 2’2) = sng A is ananalytic-set. This is also true for other level sets of p,(A) (Whitney [53]):

Theorem 2. Let A be a purely p-dimensional’ analytic set in C” and A(“) = {z E A: p,(A) 2 m}. Then, A(“‘, m = 1, 2, . . . , are analytic subsets in A.

A’ simple proof is obtained ,witli the help of Theorem I. ’ ,‘I- ’

This theorem allows us to introduce the notion of multiplicity~along a subset. If A’ is an irreducible analytic set and A’ c A, then by Theorem 2, p,( A j = m for each z E A’ with the possible exception of an analytic set A” c A’ of lower dimension (on which u,(A) > m). This genc:L; value m is naturally called the multiplicity of A along A’.

,

2.5. Indices of Intersection. Complementary Codimensions. Let A,, . . . , Ak,,

be analytic subsets of a domain D c Q=” of pure dimensions pt, . . . , pk re- k.

spectively such that the sum of codimensions of the Aj’S is n and Q Aj ,is zero-

dimensional. Then the set A, x . . . x Ak = IIA. in CL” = C$ x . . . x @$ has dimension Cp, 7 (k - 1)n and the diagonal L\: I1 = . . . = zk has complementary dimension n. If a is an isolated point of nAj then (a>” = (a, . . . , a)E ckn is an isolated point of the set (nAj) n A, and consequently, the multiplicity of the projection K& 1 nAj to the space A ’ is defined at this point. This number is called the intersection index of the sets A,, . . . , &at t&point a and is denoted by

i.( AI, . . . , A,). Thus

ia(Al,. . . , 4) = /+a~ (~a I nAj)- .

Fora#nAj,weseti,(A,,..., A,) = 0. This definition, at first glance, seems

somewhat artificial, however, technicaIly, it is very useful for studying the multiplicity properties of projections (42.3);The geometric meaning of the index of intersection (justifying such a definition) is clarified by the following assertion whose proof follows easily from the definition.

132 EM. Chirka

P~opofiJion 1. Let Al,..., A, be analytic subsets of d)” c$’ pure cample- k

wntury codimensions and a an ‘itilated point of n Aj. Then there exists a 1

neighbourhood U 3a and a number E > 0 such that for almost all c = (Cl,. . . , c+)ECk” with Ic’/ < E, the nuder of points in the set ,n, (cj + Aj) A CJ is equal to i,( .4,, . . . , A, ). For such c, all points ‘of the set fl( ci + Aj) n U are regular on each cj + Aj and the tangent spaces to d + Aj at thee points intersect transversally. The exceptional values of c (for which these properties do not hold) form a proper analytic set in the domain { Ic’l -c e, j ,= 1, . . . , k}, and for such c, the number of point in n(cj + Aj)n CJ is’strictly less than &(A:, . . . , Ak).

The intersection index can also be defined for holomorphic chains, i.e. formal locally finite sums CmjAj, where mj are integers and Aj are iwucible pairwise distinct analytic subsets of some fixed domain D. A chain Tis called a p-chain if each Aj is purely p-dimensional and positive if each coefficient mj 20. The supp6rt of a chain T is the set

The intersection index of chains q = Cmjl A,, 1 I j .< k, with complementarv pure cQdimensions (of their supports) at a point a~ ;T / q 1 is &fined algebraically:

i,(Tl, . . . , &)‘= xmll, . . . mklr’ia(Allt9 . . . , &(,).

The numb$r

is called the complete index of intersection of the chains T,, . . domain D and is denoted by io( T,, . . . , q).

. , Tk in the

From the definitions and the multiplicity properties of projectibns, it follows easily that interse’ction index for analytic sets and holomorphic chains is invariant with respect to biholomorphic transformations; the complete intersection index is invariant under “small motions”, of the chains (analogue of Roucht Theorem); etc.

2.5. indices of Intersection. General Case. Let T,, . . . , T, be holomorphic chains in a domain D c @” with supports 1 q J of pure dimension pj respectively. We shall say that these chains intersect properly at a point ae D if dim, n I q I = (Zpj) - (k - I)n (the smallest possible). If this property is satis- fiedatallpointsaEnI~I,thenT,,..., Tk intersect properly in the domain D. For an affine plane L 3a of dimension n - dim, n I q / and transversal to A = n I Tj} (i.e. (L - a) n C(A, a) = 0), the number i,( T,, . . . , Tkr L) is defined in $2.5, and it is not difficult to show that it/is independent of the choice of such an L (see [I 13). This number is called the; index of intersection of the chains

III. Complex Analytic Sets I55

T,, . . . , & at the point a and is denoted by i,( T,, . . . , Tk). As above, this notion is bihdomorphically invariant and heace carries over to arbitrary complex manifolds. From the definition it follows easily that the number i,(Ti,, . . , T, ) is locally constant on reg A, and hence, for each irreducible component S (= A = fl) Tj 1, the number is constant on S n reg A. It is calkd the intersection index of T,, . . . , & on S and is denoted by i,(T,, . . . , Tk). If .4 = u S, is the decomposition into irreducible components, then (in the case ‘of a proper intersection of the Tj’s in D) we define a holomorphic chain

TI A . . . A Tk = Ci,&( T,, . . . , T,)-S,.

which is called the. intersection chain of the chains T,, . . . , Tk in D. The properties of the intersection chain (following from the definition and the properties of the index) are summarized in the following theorem (see [l 11).

Theorem .I. Let T, , . . . , Tk be holomorphic chains of pure dimension and

intersecting properly in D. Then the holomorphic chain TI A . . . A Tk is defined in D, and has dimension @dim / Tjl) - (k - l)n and support in n I q;.1. Its multiplicity on each ‘irreducible’component S of the set n 1 q 1 is is( TI, . . . , Tk ). .4t each point a E n TTj 1, the multiplicity of the chain TI A . . . A Tk is

CAT,, . . . I q). The operation ( TI, . . . , Tk)++ T, A . . . A Tk (under the condition of proper intersection) is multilinea, r over the ring Z, commututice, associative,

invariant with respect to biholomorphic mappings of the domain D, and continuous.

The multiplicity of the chain CmjAj at the point a is by definition; Cmj,u,( Aj). l

_ We shall say that T,, . . . , T, intersect transversally at a, if di& h C( 1 rj /, a) 1

= ($dimlT,i)(k - l)n. Th e importance of this concept is seen, for exam-

ple, by the following. I’

Proposition 1. Let .T,, . . . , Tk be positive holomorphic chuins of constant

dimension in a domain D.c @” and intersecting properly at a point aE n I Tjl. Then,

iAT,, . . . , T,)=dT, A . . . A Tk)2b(Tl)...&(Tk),

where equality holds if and only if these _ Phains intersect transversallj ut the point a.

In the final analysis, this follows from Theore? 1 of $2.4.

2.7. Algebraic Sets. An affine algebraic’set is the set of common zeros in C” of a finite system of polynomials. A projective algebraic set is the set of common zeros in P, of a finite system of homogeneous polynomials of homogeneous coordinates. The ‘closure in P, I 43” of an arbit@ry affine algebraic set is a projective algebraic set. This follows, for example, from the results of 4 1.5. The famous theorem of Chow (see, for example, [I93 or [28]) asserts that each

E.M:Chirka

analyti< subset of P, is a projective algebraic set. For an analytic subset of Q=“, there are many different criteria, in terms of the behaviour at infinity, in.orderfor it to be algebraic:Here.are several of them..

Proposition 1. A purely p-dimensional inelytic subset. A c @” is ‘algebraic if and only tfits closure in P, 1 @” does not,meet some complex (n - p - l)-plane in P, \ C”. .’

The proof is, for example, in [17].‘ In terms of’@“, this criterion can be formulated as follows:

Corollary. A is algebraic ifand only ifajler.an appropriate unitary change of coordinates, A is contained in the .union of some sphere (z( < R and some cone (~“1 < CIz’(, where z = (z’, z”), z’ E ~7’. ‘I

The following criterion of Rudin [36] is analogous in form to the theorem of Liouville. :, ~.

Proposition 2. A purely p-dimensianaj analytic subset A’ c C” is algebraic if and only if after a suitable unitti;‘,) change.of coordinates, A belongs to a domain lz”l -= C(1 + lz’+ l)“,f or some constants 16 and s(z = (z’,.z”), Z’E C?‘).

The proof is. easily obtained by establishing that the canonical defining functions of an analytic covering A;t UZp are polynomials.

If A is a purely p-dimension_al analytic subset of P, and R: C”.+ l\ (0) + P’ is the canonical mapping, then A = rr- r(A) w (0) is a purely (p’+ l)-dimensioial analytic subset of@” ’ i ~--- . Its multiplicity at 0, as is easily understood, is precisely deg A, the degree of the algebraic set A, i.e., the maximal number of points of proper intersection of A with complex (n - p)-planes in P,. Such an equivalence allows us easily to obtain various properties of the degree of an algebraic set.

From RouchC’s Theorem (§2.3), one easily obtains the well-known The0re.m of Bezout on the number of solutions of polynomial equations (see Cl], [28]) which states that if pl, . . . , pB are homogeneous polynomials in P, having only finitely many common zeros, then this number, taking multiplicities into account, is equal to the product of the degrees of the polynomials. The multiplicity pa(p) at a point a = Cue, . . . , a,] for which, say, a, # 0, is defined as the multiplicity of the mapping ~(1, zi, . . :, z,) at the point (allao,. . . , a,/~,) of C”. From 52.5 we obtain the following geometric analogue of this theorem:’

Proposition 3. Let T,, . . . ., Tk be positive holomorphic chains of complementary pure codimenstons in P, with algebraic supp&ts and zero-dimensional intersection. Then, the complete index of intersection of the chains Ti , , . . , Tk in P, is equal to the product of thetr degrees, where &gZmjAj:= Zmj&gAj.

This is a particular case of the following theorem which is a consequence j of $2.6.


Proposition 4. Let T,, . . . , T, be positive chains with algebraic supports and

proper intersection in P,. Then,

deg(T, A . . . A Tk) = IItdegTj.

Remarks. Tangent cones and Whitney cones are well-studied in the works of Whitney [Sl], [52], and [53]. For the multiplicity of mappings of complex manifolds, see, for example, Cl], [29], and [47]. F or the multiplicity of analytic

sets, see the book of Whitney [53]. The intersection theory for analytic sets is exposed in the paper of Draper [ll], containing proofs for most of the properties of intersections introduced above. Additional proofs of results in this section appear also in the author’s book [lo].

$3. Metric Properties of Analytic Sets

3.1. Fundamental Form and Theorem of Wirtinger. Let R be a Hermitian

complex manifold, i.e. in each fibre T,R of the tangent bundle to R, there 1s given a positive definite quadratic Hermitian form H5(v, w) of class C” with respect to c. In a local coordinate z, it is represented in the form ~hj,dzj @ dZ,, i.e. H&J, w) = Z h,vjtik ( T,R has the standard complex structure). The form S = Re H is symmetric (S(v, w) = S(w, v)), R-bilinear, and also positive defimte,

i.e. it is a Riemannian metric (usually denoted by ds2). The form o = - Im H is clearly skew-symmetric, i.e. it is a differential form on R of degree 2. It is called the fundtkentalform of the Hermitian manifold (0, H). In local coordinates, where H = Z hikdz, @ &, the form w is given by (i/2) C hi, dzj A d& (this easily follows from the definition). .

In the space 6” with the natural Hermitian structure H = $&j @ dZj, the

fundamental form is o = (i/2)I:dzj A d~j = (i/2)aalz12 = ( 1/4)ddclz1’, where d = d + 2 and & = i(a-- a). If the local coordinate system z on Q is orthonormal with respect to H at the point a, then (o), = (i/2)Z(dz, A dZj),, i.e. w is locally Euclidean in “infinitely small neighbourhoods of d’. -

The form Go = (l/4) ddcioglz12 in @“+l\(O> is invariant under dilatations.

Thus, there is a corresponding’form w,, in P, such that i3, is the pull-back of or, by the canonical mapping. In a coordinate neighbourhood Uj: Zj # 0 in IP,, we have the representation

F he form o,, is called the Fubini-Study form on BP,. To wo, these corresponds a .

nique Hermitian structure Ho on P, such that o. = - Im Ho, and the

136 . E.M. Chirka

Riemannian metric S, = Re H0 ( S0 ( U, W) = LZ~( U, iw)) is called the Fubini-Study metric on P,. In the sequel, both o0 and 61, will be denoted by c+,.

The Euclidean volume form in @” is

ndxj A dyj = nfdzij A dzj 1 = --JlP.

If L is a linear subspace of @” of dimension p, then, by a unitary transformation, L is transformed to Cp = @p, . . Tp c C”. Since o = ( 1/4)ddc(zlZ ’ 1s invariant by such transformations and the volume form on cp is (I/p!)wp(@P, then the Euclidean volume form on L is (l/p!)o’lL, From this, it follows easily that on any complex manifold M of dimension p, imbedded in @“, the form ( I/p!)oJ’I M is the volume form on M with respect to the metric induced by @“. Since the fundamental form of an arbitrary Hermitian manifold is Euclidean in appropriate local coordinates (at a given point), we have the following.

Theorem of Wirtinger. Let R he a complex Hermitian manifold with finda- mental fkn CO. Then for arry p-dimensional complex manifold M, irnbgi led in R the restriction ( l/p!)oPIM coincides with the votume form on M in di; induced metric. In particular,

vol,, M = +L’mp.

From finer linear properties (the Wirtinger inequality, see [13] and [20]) there follows the property of minimality: if M is an arbitrary oriented 2~1 dimensional manifold of class Cl, imbedded in R, then

fiwp _< vol,, M.

3.2. Integration on Analytic Sets. From Wirtinger’s Theorem, Lemma 2 of $1.4, and the analyticity of sng A, we obtain the following theorem of Lelong (see c251).

Theorem 1.. Each purely p-dimensional analytic subset A of a complex mani- ,fold R has, locally-jinite Hausdorfl2p-measure, i,e., X’,,( A n K) <‘co, for any compact set K c R.

In @“, we have the formula

-I-cop= 1 dV,, P! . #I=p

where dV, G fidxit A dyiY’ 1

is the volume on a coordinate p-plane in the variables z. . . Zi, (the sum is over ordered multi-indices I). Hence, Wirtinger’s Theorem can ‘be interpreted as follows: the volume of a p-dimensional analytic set A c @“(i.e., 3EC;,(A)) is equal to the sum of its projections on p-dimensional coordjnate planes counting multiplicities of these projections (the volume of A and of reg A coincide).

111. Complex ;Inalytic Sets 137

if ,+I is a. purely p-dimensional analytic subset of a complex manifold Q, then by Leiong’s Theorem, we can define a current [ (43 of integration on the manifold reg~, a current of measure type and dimension 2p:

cp E g2P(Q), where the symbol gs($l) denotes the space of infinitely differentiable forms of degree s and of compact support in R (concerning currents, see, for example, [13], [20]). In the sequel, we write

We introduce some characteristic properties of integrals on analytic sets. .

Proposition. A current of integration [A] on a purely p-dimensional analytic subset of a complex manifold R has bidimension (p, p), i.e.

p=o for anyform cp &*r(Q) of bidegree (r, si # (p, p). Such a m-rent is positioe. That is,

j’p20, A

for each positive q E 63 *r(O). Also, [A] is closed, i.e.,

ldv=O,

for each cp E 9 2p- ’ (Q).

A form cp is called positive if at each point it is represerrted as a linear combination, with non-negative coefficients, of forms

where each 1; is a linear function of the local coordinates; for example, the fundamental iorm and its powers.

The last property of [A] can be strengthened. We shaJ1 say that A is an analytic set with border bA, if A is purely p-dimensional, A lies in some other purely p-dimensional analytic set 2, bA = A\, A has locally finite (2~ - I)-

dimensional HausdortT measure, and there exists a closed set Z c A such that Xzo- ,(C) = 0 and A\I.I is a Cl-submanifold with border (bA)\C in Q\X.

Theorem 2 (Stokes formula). Let A be a purely p-dimensional analytic set with

border on a complex manifold R. Then, for any (2~ - I)-@rn ofclass C1 (Q) and of

138 E.M. Chirka

compact support, we have

pJ=A(Pv where the integral on the right is also to be understood as an improper integral along (bA)\Z.

In terms of currents, Stokes’ formula has the form ([,4], dq) = ( [bA], cp) or, more briefly, d[ A] = - [bA]. We remark that if A is a purely p-dimensionalanalytic subset of R and p is a real function of class C2P(Q), then, by Sard’s theorem (see [26]), aimost every set A border ‘bA , = An{p = t}.

, = A n {p < t> is an analytic set with

One of the frequently applied equivalent forms of Stokes’ theorem is the following complex analogue of Green’s formula.

Corollary. For any functions u and u of class C2 (Q) and any closed form cp of bidegree (p - 1, p - I), we have thefollowing equality

J,(udev - ud’u) A rp = J(udd% - vdd’u) A rp.

3.3, Crofton Formulas. The volume of an arbitrary purely n-dimensional analytic set A in complex projective space ,P” (with respect to the Fubini-Study metric) can be calculated in terms of the volumes of its traces along complex k-dimensional planes, k 2 n - p, via the well-known “Cr$fton formula”

.@&f) = c(p, k n)&nj~2,P+k-‘)(A nL)dL

where G(k, n) is the Grassmannian or all k-dimensional complex planes in P dL is the standard normalized volume form in @k, n), invariant with respect ;d unitary transformations, and c(p, k, n) = A” -‘(p + k - n)!/p! (see [3], [15], and. [38]). Essentially, this is Fubini’s theorem on the incidence manifold, more precisely, on the manifold { (z, L) E P, x G”( k, n): z E (reg ,4 ) n L. }.

If A is a purely p-dimensional analytic set in C”, O$ A, and the rank of the restriction of the canonical mapping rr: C”\,(O) -+ BP,,- r to reg A is almost everywhere equal to p, then from the formula in P,,‘we have

where o. * (1/4)dd’log Iz12 and G(k, n) is the Grassmannian of all k-dimensional subspaces of C” (L 30). The transition from “projective” volume to. Euclidean volume’is further realized with the help of the relation

co$JT,A= sin’(z, ‘T=A)

~“1 CA, zEregA, 142p

III. Complex Analytic Sets ’ ‘139 I ?i@

which is easily verified in appropriate local coordinates. $0; an arbi’rary f :

analytic set A c C”, we obtain, in this way,.a rather unwi&fy formula, which m case A,is a subset of the ball, simplifies in the following way as a consequence of

,&

Stokes’ formula (53.2). ‘. ’

Proposition 1. Let A’ be a purely p-dimensional anal@c &set of the ball IzI < R and O$A. Then

~~wP=R2“jw{. ‘ A

From this and from the formula introduced earlier with ejO, we obtain ‘the following theorem of Alexander [3]:

Proposition 2. Let A be a purely. p-dimensional analytic subset of the ball . IzI < R in C” and O$ A. Then, for any k 2 n - p, we have,the equality

&,(A) = c(p, k,n)R”“-k)o~!;“l,~ztb+i-“,(A ri‘L)dL.

3.4. The Lelong Number. Let A be a purely p-dimensional analytic set in 43” containing 0 and let 4, = A n { Iz( <, r}. From Stokes’ Theorem,, one easily ,

obtains

Since the form wP, is positive, it follows that the,function.$zg(A,)/(c(p)r2!) decreases as .r + 0, and hence, has a limit which we denote by n( A, 0) and which. is called the Lelong number of the set A at 0. Here c(p) = xp/p! is the volume of the unit ball in Cp. At an. arbitrary point ae A, ,the Lelo.ng number n(A,.a) is defined analogously. Thus,

#‘2p( A,) = n( A, O)c(p)rZp + r2p 1 c&/p!. A,

If A is. a homogeneous algebraic set, then o&A = 0. Since oP- !IbA, = wg- ’ \bA 1, we ,have

where A* is the corresponding algebraic su’bset of P, -.r. ‘The volume of such an A* in the Fubini-Study metric is equal to A P- * *deg A*. This follows from the

fact that A* can homotopically be deformed into a k-fold plane of dimension p - 1, where k = deg A*, and from the fact that the integral of OS- r is invariant under such a homotopy since w. is closed (for more details see, for example, [28]). Thus, for cones, the Lelong number n( A, 0) coincides with the degre of A* i e. the multiplicity pa(A). ‘From the homotopy of A. to the tangent cone (wAh* fhe covering multiplicity) and with the help of the dilation z Hz/r, we obtain analogously (see [ll] and [20]):

. . N 1. The Lelong number of a purely p-dimensional analytic xt A ‘c @” at an arbkrary point a E A agrees with the multiplici$y of A at this point c&, in purticidar, is an integer.

4 Thus, for analytic sets, this concept yields nothing new, however, it is more

lkx&le than the notion of multiplicity and easily extends to essentially more eral objects (fee, for example, [20]):

Mtioh 2 Let T be a positive d-closed current of bidknensinn (p, p) in II ~nDc~“~letn(T,a,r)=(nr2)-P(T,~~,,o~P),w~e~~,isthech~rac- teristicjiuraion of the ball /: - al < r. Then n( T, a, r) deereares monotoniCally as r --, 0 rrt each point a E D and there exists

lim n( T, u, r ) =: n(T, a). r-0

35. Lnrvermtes for the VokDe of mytic Sets. From the positivity of $k form w,, and from the previous section we easily obtain the following well- known estimate (Rutishauser, Lelong (1950), see [6] ).

.a hrqoeara &t A be a purely p-dimensional analytic subsezof the bail 1.~1 < r ia 4?, containing 0. Then, JI?~,,( A) 2 n( A, O)*c(pjrzP.

+autiful applications of this estimate to the question of removing sing- &r&s of analytic sets were given by Bishop [6]. It is also interesting to obtain analogous lower estimates for arbitrary convex domains. instead of c(p)r2p, it is necessary, of course, to take the minimal volume of traces on complex p-planes, Such estimates are valid for cubes in C2, for certain tubular domains etc. However, not every convex symmetric domain has this property. For example, in the domain D: 1x1 < 1, lyI < R in C3(z = x -t iy), the analytic set Dn {- = 4 + .$ } has Qvolume -C 20, while any trace of D on a complex 2-plane L ii has Qvolume 2 R-4-* > 20 for large R.

.

By Wirtingeis Theorem, .YY’,,(A) is the sum of the projections of A on respective coordinate planes counting multiplicities. Hence, the following estimate (Alexander-B. Taylor-llllman [4]) is significantly stronger than the above one.

Tkorem. Let A be a purely p-dimensional unolytic subset of the ball i”z ( < r in @” and OE .A. Then,

2’ .%f2,(lr,(/l)) 2 CpTzp, ‘ XI=!,

where cp > 0 is a constant depending only on p (for p = 1, we may take cl = n).

Here I denotes the sum over ordered multi-indices I, i.e. i, < . < i,, and xr is the orthogonal projection of C” onto the plane of the variables q,, . . . ,iip.

For Q = 1 iGk%mm rollows’ea6ily ikom t&: ak

for a bous&d hokxncrphic function f in the unit dkc A with f(0) = 0. One then n&ins the 01gt of arbitrary p by Fubini’s rr#ntm (with constants cp = T -4p+1cp-1 not best possible). !See [4].

Y, VtidTti LetMbapdimensicmalm+niroldinC.Theuaion ofJIbalbrB(Cr)nN,M(whenNdeeotesthe~mnmaSspafe)forailCEM is c&d a tubular tighbourhocd of radius r a.rou& M and is &not& by T,(W). .il tk allhe normal spots N+3< at various points of AI do not intasmt in some r,-lltighbourkxl of M( r, > O), SLar iix r jr,,, the tube z,(M) is naturally &&omorphk t0.Y x (ItI.< r),tEC-5 Thus, hwm the limnda for the tzbags of variables in an integrak it &Sows that the volume Of

r,(M)isapolynomialinroftheform

f CjT? R-P

From the local approximation by tangent spaces, it is clear that the prin- _

cipai -part of this Bolynomial (as r -O), just as for the plane, is equal to (vd U )- c(n - p)?@ - p). The remaining coelkients can be expressed, by the curvature of the manifold M in the following way.. I& rl, . . . , TP be holomorphic vector fklds on’M, forming a basis at each T, M (everything local). Let H = (h,), where (h,) = (rj, f ), be Hermitian scalar products in C”. Then 8 = (X)* H7 r is the canonical Euclidean connection matrix on M (here ?H = (ah,)), and Q, = &?, is the curvature matrix (of the tangent bundle) of the manifold M; its elements are differential forms of bidegree (1, i). The differential form c(M) = det (E + i(2x)-‘R,), where E is the unit.ip x p)- matrix, is called the total Chern class of the manifold M. Expanding c(M) according to degrees, we obtain c(M) = 1 + cr( M) -t . . . + cP( M), where c,(M) is called the q-th Chern class of the manifold M. A rather tedious calculation of the voiume of r,(M) according to the above scheme act H. Weyl (with essential use of the complex coordinate 5, t ) leads to the following formula of Griffiths [l;]:

lkom!ln 1.

for r Sr,.

From the definitions and by calculating in local coordinates, it follows that the forms ( - l)qc,,( M) are positive, and hence the signs in the Griffiths Formula are alternating.

142 EM. C@rJca

If M = reg A, where.,4 is apzan&tic set, then, in the vicinity of.a singularity, the normal bundle has a very bad structure (it is not holomorphic, unlike the tangent bundle), and the Griffiths .‘Formula .already no. longer has such a .geometric meaning as for manifolds. From the local representation in terms of analytic covers, it is, n,ot difficult to deduce that the volume:of an r-tube about

’ A( p vol t,(ieg A))Js, equal to. ‘.‘. I

X2d(A)*c(n -.pjr*@-P)(i 4 i(i)),

’ as r + 0. That is, asymptotically, the: singularity does ‘not influence the main tgrp of vol T,(A). ; ‘. !

: _ I I ,- :

_’ 3.7. Integrabilitj of Chem Classes. The Chem -forms cq ( M ) of a complex p- di.mensional manifold M (imbedded) in C” are -the preimages of the standard forms on the Grassmannian G(p;%n) under ,the. Gauss mapping yM: z + T, M, z E.;M. More precisely, over G(p, n) there is,a so-called tmiversal bund1e.U which is no other than the incidence manifold {(L, z): L. E G( p, n), z E L c C” }. With the help of the local bases in the fibres of this bundle we define, just as before, the ctiature matrix @, and the complete Chern class c = det (E + i(2~t)-~Q,) -gj’+.cl+ ... + cpI, Comparing this with the already defined classes c,(M), it is not difficult to’see that c&M) = (yu)*(c,J, q ‘= 0, . . . , p. The forms cq on

c T(p$) are” really universal; they depend. only on p, n, and q. Hence, the Y$i&igation of the forms c,(M) ‘reduces to the study of the corresponding properties of the Gauss mapping. We introduce some examples. ;

Lemma 1. . Let A be a purely. p-dimensional analytic subset of a domain D c C”, and A+ the.’ closure of the graph ((z, T,A,): zoreg A} of the Gauss mapping in D x ‘G(p, n), Then A* is a purely p-dimensional analytic subset of D i G(p, n). ., ,. ‘,

,I From this and from the theorem of Lelong ($3.2), clearly follows

. Proposition 1, ..Let A,be a purely,p-dimensional analytic set’in Q=“and let.@ A),

q - 0, . . . . , p, be the Chern forms of the manifold reg A. Then the forms . copwq A cq( A) are locally integrable on A i.e.

e :I. ’

s KnregA

COP-~,/\ c,(A) < co,

for any compact K c A.

Recall that the forms (- 1)4*c,(A) are positive on reg A. Using this fact, Stokes Theorem, and, the Lemma, we may more precisely estimate the local behaviour of such’integrals. .,

, : ‘.

I 1

/ III. Complex Analytic Sets 1, 143

I I i, . i Proposition 2. Let A be’a purely p-dimensional analy& set in C” and OE A.

Then, the forms CI$-~ A ‘cq( A) are locally integrabl? on Ai and the functions

(-II4 E’WP-‘hc(A) $(P-q) A 4 ’

q&o’ 9 * * * , p

,,‘4 ’ I I:

monotonically decrease as r -Q 0.

The limiting lues of these functio uninvestigated

f , Remarks. Thedesults of the first four sections are developed in detail by

Harvey in [20]. For’ lower estimates on volume, see Bishop [6] and [4]. Volumes of tubes and integrability ljroperties of Chern classes are studied& detail in the paper by Griffiths [15].

i !

! ‘@I. Iiolomorphic Chains i

:

4.1. Characterization of Holomorphic Chains. Formal holomorphic chains, which appear because of the necessity of considering various multiplicities in dealing with analytic sets (see $2), are,most easily treated in terms of currents. In the sequel a holomorphic p-chain in R will mean a current in n of the foti. Zmj[Aj], where Aj are the irreducible components of a purely p-dimensional analytic subset A = u A, of R, [A, J is the current of integration on A,, ang the multiplicity mi’is au integer (see 43.2). Of particular importance for applications are the positive holoniorphic chains (for ,which ml 2.0).

From the properties of integrals on analytic sets (see #f;2), it easily folIows that a holomorphic p-chain T is. a current of bidimensioi (p, p), is closed (dT = 0), and is of measure-type (i.e. extends to a functional on continuous 2p forms of compact support ‘in 0). If n is a domain inC”, then ‘at each point a&, each such chain T has a density which is an integer as in 13.4. The,density of a. current T of measure-type at a point a is defined as

.,.’ I lim ~~,,,,(T)/ldp)r2P), . :

,r+O

where B(a,r) = {lz -‘al s r} and .IbiKj(T) = s~p{lx~T(Cp)l: IIqII S 1) is th$ mass of the current T on the compact set K. For furthpr details, see ‘[lo] and‘ cw.

"..

It turns out, that this property fully characterizes holomorphic chains in the space of currents. The following theorem is due to Harvey and Schiffman [2(l),’ I c221.

144 EM. Chirka

Th&?orem 1. Let Tbe a~closed.current ofbidimension (p, p) and mea&e-type in a domain D oj C” whose density exists and is integral at .X 2,-almost-every point@ the support OJ T. Then T is a holomorphic p-chain in D.

The density condition can be replaced by the condition of local rectifiability of T as in the original foi-mulation of Harvey-Shiffman, however, rectifiability is perhaps a more complicated property. Each positive current is automatically of measure-type and moreover, for such currents, the Lelong number n(T, a) (see 43.4) is defined at each point aep. The following theorem is due to Bombieri -4 cx.. lraa rn2-i)

Theorem 2. Let T be a positive closed current of bidimensia (p, p) in a domain D in C”. Then, for any c > 0, the set (ZC D: n( T, z) 2 c) is an analytic subset ofD of dimension I p.

From this it follows, in particular, that each positive closed current of bidimension (p, p), whose Lelong number is a natural number on an everywhere dense subset of the support, is a holomorphic p-chain.

The description of holomorphic chains in terms df currents is useful for applications, in particular, for problems on the behaviour of analytic sets under limiting processes and the removal of their singularities (see $5). In this way, one easily proves the following well known theorem of: Bishop (See [6] and [20]).

Theorem 3. Let {A, > be a sequence of purely p-dimensional analytic subsets of a complex mamyold R having uniformly bounded volumes on each compact subset of Q, and suppose this sequence converges to a closed subset A c a. Then, A is alsoja purely p-dimensional analytic subset, of i2.

The convergence Aj + A here means that for each ZE A, there exists a sequence zje Aj, such that tj + z as j --, 00, and Aj n K is empty for any compact K c Q\ A and ail sufficiently large j. An analogous theorem is valid,‘bf course, also for holomorphic chains (see [20]).

4.2. Formuias of Poincar&Lelong. The zeros of a hglomorphic function f + 0 in a domain Q, counting multiplicities, define a positive holomorphic chain D, = Cnj[Zj], where Zj are-the irreducible components of the zero set Z,, and nJ is the multiplicity of the zeros ofj’ at points z E Zj n reg Z, (it is constant for all such z). This chain is called the divisor of the function f in Q. Analogously, one defines the divisor of a meromorphic function. in case f = h/g, where h, ggO(R), h, g f 0 (this is always local), the divisor of f is the difference D,- D,. If R is a domain in @ andfE @(Ct),f f 0, then log/J‘/ E L;,(R) and the generalized Laplacian Alog If 1 is equa! to cZnjSoj, where aj are the zeros ofj; nj are their multiplicities, and 6, is the delta-function at the point a (i.e. the current [a]), and c is an absolute constant. In the general case, log1 f 1 is also locally integrable in Q (for Poindart-Lelong holds.

meromorphic f f 0) and the following formula of

F 111. Complex Analytic Sets

* 145

Theorem 1. Let f be a meromorphic,;function on a complex manifold 0. Then

D, = $ioglf~

in the sense of currents on $1.

The assertion is local and its proof reduces to the one-dimensional case with the help of Fubini’s Theorem and the,representation as an analytic covering (see [I?] and [ZO]).

If A is a purely p-dimensional analytic subset of R and dim A n / D,f / < P, then the function log\ S\ is also locally integrable

on A (this follows from the

representation as a covering). Thus, on R, the current-

is defined, and in this more general situation, an analogous formula holds.

Theorem 21 Let f be a meromorphic function and T a holomorphic p-chain on

the complex manifold R such that dim1 TI n 10, I < p. Then,

h(T*loglfl)’ T A D, n

I in the sense of currents on Q.

The proof is technically rather difficult (see E17]); however, it fairly easily reduces to Theorem 1 and to the following non-trivial property on intersections: if T and T, are holomorphic chains with proper intersection in the domain D c’C’ and TIC is thi &-regularization of Tt, then ‘*

lim T,, A T2 = T, A T2 &-‘Q

{in the weak sense). Analogous formulas of PoincarbLelong are valid also fat holomorphtc.

mappings R + Ck and for more general objects (see [173). .>

4.5. . Characteristic. Function. Jensen. Formulas. if A .is ,a purely p-dimen-

signal analytic subset of the bait B: lzl < R 5 co in en-and A, = A n ( IzI < f }, then the function n(A, r) = volA,/(c(P)r * * ) is called the (unintegrated) counttng

function of the set A. For a holomorphic p-chain T = Cmj Aj in B, the counFing function is defined by additivity: n( T, c) = Xmjn(Aj, r),(if can also be negative). .

I The logarithmic, average r n(T, t)

N(T,r)=J-ddt 0 t

is called the characteristic (or counting) functibn of the chain T. For analytic sets.

E,M. Chirka

(and, positive chains) it represents the growth of the volume of A compared to the growth of the ipidimensional ball 1.~1 < r. Algebra: are WmpJetely characterized by the condition N(A, r) 5 c 10g(l + In the transcendental case, we define, as usual. the order nf A QE i .:

bCC; B3.Lli , ----- -- -- -”

k log N(A, r) .++Ci log r ’

Pc$tive cHains,of finite order can be ‘viewed as geometric analogues bf entire, ’ functions of finite order. :

For functionsf, mqpmorphic in a ball B, the characteristic fLnc+;An divisor is, as in the one-dimensional case, closelj connected to the log I J 1. The following “Jensen Foymula” holds: -

~l?+si!ion’I.

‘it!-

, Letf be meromorphic in the ball lzl < R-in @” and Og’lDll.

f7 = $@log,r, A 0;”

b the normalized volume form on the sphere 1 z 1 = r.

,I i In ptrticular, for holoinorphic functions f such that

..- ‘

iG j (lo&If&< 03, r-rRfz(-,

we have the estimate ZQD,, R) c co, which can be rewritten in the form

d (R - Izl)o”-’ < 03.

I For n - 1 and R - 1 this is the classical Blaschke condition Z( 1 - la .{) < Co for the zeros a, of the function 5 In CR this is also called the Blaschke cdndition. In

, ,more gene&l domains of the form 0: p < 0, where dp # 0 on 80, the growth of analytic sets near the boundary can be well estimated in te&s of the defining function p. With the help of the Green Formula.$j3.2), we obtain such an

; estimate, just as for the ball:

PIWMWII 2. “dp

Let D: p < 0 be a bounded domain in C”, where p E C*(D) and # 0 on BD. L.et f be a holomorphic function in the Nevanlinna class on D, i.e.

* O%‘lf Ma < 00,

where Dd .- D n {p < - b} and a, is the Euclidean volume form on aLI,. Then, the

111. Complex Analytic Sets 147

\

diois 4

of-f on D satisfies the Blaschke condition:

general situation arises in estimating the divisor off on an analytic the holomorphic chain A A D,. With the help of the averages

of the formulas of Stokes and Poincart-Lelong, one obtams

Let A be a purely p-dimensional analytic subset of the ball meromorphic in a neighbourhood of A such that

,

N(A A Di,r) = ,i flog\ f I)a, - -$j (log1 f I)wg - pdA)*loglf @)I, r ,

where

We remark that wt 1 A and bpJ bA, are positive forms (i.e. the coefficients of proportionality of the corresponding volume forms are non-negative), and this allows one also to obtain vati.lus estimates depending on the growth off I A.

4.4. The Second Cousin Problem. The equation of Poincati-Lelong

iff-ldZjlogI f) = / D can be considered as an equation with an unknown function f and a given right member, and then it becomes precisely the second Cousin prob!em of constructing a meromorphic function with a given divisor. The advantage of such a formulation is clear, particularly in connection with the great progress in solving &problems over the past 15-20 _ ears. Whereas the classical theorems (in the Oka-Cattan theory) yield only conditions for the existence of solutioils, with the-help of the &problem, one can obtain solutions. with estimates which depend on estimates for the right side. In this direction (in great part due to the efforts of the Lelong schoql), the study of divisors in C” and of entire functions of several complex variables has progressed significantly.(see $ C401, and CW.

The solution of so-called boundary &problems with estimates led to the. solution of the problem on the Blaschke condition in C”. G.M. Khenkin [24] and Skoda [45j proved the following. I

Theorem. Let D: p < 0 be a strictly pseudoconvex bounded domain in @” such thal H l (D, Iw) = 0, and let T be an arbitrary positive divisor (positive holomorphic (n - 1)-chain) in D, satisfying the Blaschke condition:

Then there exists a holomorphic function f in .D of Nevanlinna class such that T= D,.

; \

r, , I*0 E.M. Chirka

Finally, we remark that all characterization theorems for (see $4.1) were obtained by the method of solving the Poincar&-Le]ong e

with a given current on the right side (see [20], [22], and 11431). i

I 4.5. Growth Estimates for Traces on Linear Subspabes: 7’ e Formu!as (93.3) show tha! the growth of the volume of an analytib P

Crofton set ,4 in @”

admits two-sided estimates in terms of the growth oft-he volume of the trace of A on a “generic” linear subspace L 30 of fixed dimension, This problem has been more thoroughly investigated for traces on hyperplanes. From the Jensen Formula, we have the following.

Lemma 1. Let A be a purely p-dimensional unalytic subset ojtke hull 1 z; < R, p 2 1, and L”: (z, w) = 0. Then .

I(? w>l N(+d A L”, r) - N(A,i) = j log ___ W,., IZll~4

where A,,,, = A,\&Oe r. > 0, and oP is defined in $4.3.

Since this relation depends only on w/l WI, we may integrate with respect to a measure on BP.- t. In this-context, the potential

naturally arises. “Thin” sets for this problem are defined in terms of this potential. The following two-sided estimates are obtained in the work of Molzon, Shiffman, and Sibony [27].

Theorem 1. Let A be a purely p-dimensional analytic subset of the ball jz) < R in C”, p 2 1, and p.a probability measure bn lQ,- r whose potential is bounded: U,SC<CO. Then . *

2ce’4pcN(A, e-“r) I j N(A A L”, r)dp(w) I p,-,

for all k > 1 such that kr s R.

The upper estimate for N(A A L”, r) follows easily from Lemma 1. The finer lower estimate follows by averaging the following lemma of Gruman [ 181.

Lemma 2. Let 4 be a purely p-dimensional analytic subset of the ball lzl < R and’v be a plurisubharmonic function’in this ball of class C2 and such that log/z1


set E c P,- t is, by definition, of positive logarithmic capacity if measure p, concentrated on E, whose potential is

ounded. For example, all non-pluripolar compact sets and even some thinner sets are of positive capacity (for more details, see C27]). From ’ Theorem 1, , r&r example, one obtains the following consequences for purely p-dimensional analytic subsets A c @“, $ 2 1.

1) If the set A n L” is algebraic for each w lying in a compact set E c P.- 1 of positive logarithmic capacity, then A is’also an algebraic set.

2) If p is a probability measure on P,-, with bounded potential, then ord A A L” = ord A for p - a.e. w E P,- t (here ord A is the order of growth of A in C”, see 94.3).

For the traces on linear subspaces of codimension > 1, the picture is not yet completely clear. Logarithmic capacity is clearly not suitable, however, an, obvious substitute is not apparent. The following theorem was proved by Gruman [ 181.

Theorem 2. Let A be a purely p-dimensional analytic subset of the ball Iz[ < R in C”, p 2 1, and E a Bore/ set in G(q, n) ofpositiue dL-volume, p -t q 2 n. Then, there exist positive constants ct, c2 such that I

c,N(A,c,r) I JN(AnL,r)dL, rO<r<R. .E

Theorems on reciprocal estimates of the growth of analytic sets and of their traces on subspaces can be viewed as geometric analogues of theorems from the Nevanlinna theory of value distribution. In fact, such theorems arose from generalizing Nevanlinna theory to the multi-dimensional case.

I

Notes. For the results of the first four sections see the paper by Harvey in [ZO] and Griffiths-King [173. A relatively simple proof of the theorem of Khenkin-Skoda for the ball is in Kudin’s book [37). The estimates of $4.5 are proved in the paper by Molzon-Shiffman-Sibony [27].’

$5. Analytic Continuation and Boundary Properties

5.1. Removal of Metrically Tgin Singularities. The border,r$f a p-dimensional analytic set has (metric) dimension Jp - 1, and hence it is. intuitively understandable that a set of lesser dimension cannot be an authentic, obstruction

150 E.M. Chirka /’ /

for a purely p-dimensional analytic set. From the theorem on the analytic’ y of a, generalized analytic covering (9 1.5) one. easily obtains the following the rem of Shiffman [41]. i

Theorem 1. Let E be u ciosed subset of a complex manifold Q and let A be a / purely p-dimensional analytic subset of 0\E. If the Nausdor measure X’2P- 1(E) = 0, then the closure .I of the set A in R is a purely p- f imensional analytic subset oj’n.

ii The important particular case, when E is an analytic subset of Qlof dimension

less than p, constitutes the well-known theorem of RemmertStein (see, for example, [J 93).

Sets of positive (2p - I)-measure may serve as obstructions for A, however, there is here the following curious effect of “infectiousness” of the continuation (see [IO]).

Theorem 2. Let M be a connected (2p - l)-dimensional Cl-submanifold of a complex manifold Q and let A be a purely p-dimensional analytic subset of n\M. Suppose that in the neighbourhood of some point z” E M, the set 1 is analytic. Then A is an analytic subset of R. -

The proof is obtained in a rather standard fashion with the help-of analytic coverings and canonical defining functions. ’

5.2. Removal of Pluripolar Singularities. Obstructions of higher dimension can-also be removed for analytic sets under additional conditions on E and on the “holes” through which A continues. The following lemma was proved by Bishop [6].

Lemma 1. Let E be a closed totally pluripoiar subset of a bounded domain D = D’ x D” in C”, with D’ c Cp, and A a purely p-dimensional analytic subset of D\E having no limit _pojnis on D’ x aD”. Supposq that theie is u nonempty domain U’ in D’ such that A n(U’ x D”) isanalytic. Then the set An D is an analytic subset of D.

A set E c D is said to be totally pluripolar if there exists a plurisubharmonic function cp f - c;: in b such that E = {zED: q(z) = - co}-

From this one easily obtains the following theorem of Thullen on the “infectiousness” of continuation through analytic singularities.

Proposition 1. Let She an analytic p-dimensional subset ofa complex mantfold fi and A a purely p-dimensional analytic subset of Q\S. Suppose there is an open subset U of R which meets every irreducible component of S and such that the set 2 n U is analytic in U. Then A is an analytic subset of R.

The following theorem, essentially proved by Bishop [63, on the continuation through pluripolar singularities, also follows from the lemma without difficulty.


Theorem 1. Let E be a closed locally totally pluripolar subset of a complex ma~ifold-@arnd let A be a purely p-dimensional an .‘ytic subset of n\E. Then, tythe Hausdorfl measure Xzp(A n K) c co for eclch compact K c R or if 3Epz,(q n I$)? 0, &I ‘A is an analytic subset of n.i 1

In applications, the first condition has been particularly useful. Not long ago, El Mir [12] proved an analogous theorem for positive closed currents having locally fiaite mass in R An important special case is when E!is a proper analytic subset For example, if A is an analytic subset of C” such that N(A, r) 5 clog (1 + Y], so that the Fubini-Study volume of the set A in P, 3 @” is bounded, and if the obstruction is the analyticset E = la,\@“, then, by the theorem, Pi is an analytic and hence algebraic subset of P,. In this way, from Bishop’s theorem, we obiain detric criteiia, already mentioned in.$413, in order for an analytic set to be elgebraic.’

5.3. ‘Symmetry Principle. For the continuation of analytic sets through [W-analytic manifolds, a major role is played by a method analogous to the sjrmmetry principle for holomorphic functions. In this direction, one has, for

‘example, the following result (Alexander [2], Becker [S]).

Propositibn 1. Let D be a domain in @” and A a purely p-dimensional analytic subset of D\W. If p > 1 or p = 1 and A is symmetric with respect to W, then A n D is an analytic subset of D.

Any totally real n-dimensional R-analytic mani$old M in @” is locally biholomorphic to a domain in [w” c C”. Hence an analogous symmetry principle is valid also for such obstructions M. If M is given by a system of &analytic equations, then, an -anti;hoIomorphic symmetry tiith iespect to M can be written explicitly in a neighbourhood of M with the help of these equations. This symmetry @reserves analyticity of sets and for Iw” has the form ZH 5. If, for example, M = r, the distinguished boundary Izl 1 = . . . Jz,I = 1 of the unit polydisc U in C”, then the symmetry with respect to r is the anti-holomorphic mapping z i+ z* = (l/F,, . . . , l/Z,), and the following theorem of Shiffman [42] holds.

Proposition 2. Let A be a purely one-dimensional analytic subset of the unit po1ydisc.U in @” such that A c U v r, where r is the distinguished boundaiy of U, arrd let A* = (z*: z E A}. Then Au A* is an algebraic subset of Q=“.

If we don’t .require symmetry of the set A, then using the anti-holomorphic symmetry, we can obtain a c&tinuation to some neighbourhood of M. In this direction, we have ’

,’ * Theurem 1. Let M be a lkznalytic submanifold of a comp!ex manifold R, and let A be a purely p-dimensional analytic subset of Sl\M. ff the CR-dimension of M (i.e. the c-dimension ofthe mmplex tangent space to M) is.no greater than p - 1, then A extends analytically to some n.eighbourhood of the set A in Q.

152 E.M. Chirka

The proof reduces to continuation through totally real manifolds with the help ofappropriate traces and the following analogue of the Hartogs Lemma for analytic sets (Rothstein [34-J). , >

Lemma 1. Let A be a closed subset of a domain U = Cl’ x U” in Ck, with U’ c Cm and /et li” be a subdomain of U”. Suppose that. I) A n (U’ x V”) iS a purely p-dimensi&al unulytic subset of U’ x Y”; 2) A, = A n (2’ = cf is b purely +dimensional or empty analytic subset of c x U” for each c~ U’; and 3) each irreducible component of‘ the set A, meets c x V”. Then A is an analytic subset oj’ c:.

. The.lemma is also perhaps valid without supposing that P. is closed. Let us say a few words concerning the analytic continu’dtion of sets. If A and A

are purely p-dimensional locally analytic sets in R with ,4 r_ A”, and if each irreducible component of A’ has a p-dimensional intersection with A, then x is called an anulytic extension qf the set A. W,e say that A extends analytically to U if there exists an analytic subset 2 of V which is an analytic extension of the set A l~ 0’. The analytic continuation of sets has the following useful property of localness which i4 not difficult to prove.

Lemma 2. L!t E be a closed subset of a complex maniJold R and let A be a pprely p-dimensional analytic subset of Q\E. Suppose that for each point [g An E, there. is a neighbourhood =U,3[ in Q to which k extends analytically. Then A extends analytically to some neighbourhood of .i in Q.

5.4. Obstructions of Smail CR-Dimension. In the prese&e oi a rich complex i structure for a set E, an analytic subset of Q\E can approach E with trolled “wiggling” (as, for example, the graph of: w = e’jZ along the line z

uncon- = 0 in

C2), and in order to continue through E, it is necessary to. impose additional ccnditions (as, for example, in Bishop’s Theorem, $5.2). If, however, the complex structure of E is sufficiently poor, then, from all indications, E ccnnot be an obstruction for an analytic subset of O\,‘E of sufficiently high dimension. The following theorem was proved by the author [SJ. 9,

Theorem 1. Let M be p Cl-submanifold of a complex manifold R and let A be a puiely p-dimensional aoialytic subset of Q\ M, p > 1. Suppose the complex tangent spaLee TS(.WI has (com$~s) dimension < p - 1 for each [e An M, with the possible exception of a set of zero (2p - I)-measure. Then A is an analytic subset of R. :

We remark that the border bA of an analytic set A of dimension p (if it exists) has CR-dimension p - I. Thus, the CR-structure of M in the theorem is podrer than the possible CR-structure of a border. For positive closed currents of bidimension (p, p) in Q\M (for twice smooth M) an analogous’theorem wtis proved by EI Mir [12]. . .

III. Complex Analytic Sets 1.53 _

5.5. Crmtfnuatio~ Through Pseudoconcave Hypersurfaces. One of the central theorems on the continuation of hoiomorphic functions of scverai variables is the theorem on the continuation through a strongly pseudoconcave hyper- surfa& (see [39), [SO]). This remarkable property also carries over to analytic sets of dimension greater than 1 (see [33]).

Theorem i. Let D c G be domains in C” such that r = (ZD) n G is hypersurface of class C2 whose Leviform has at least q negatitle eigenaalues, and let A be a purely p-dimensional analytic subset of D, where p + q > n. Then, there exi!ts a neighbourhood U I> l- in G such that A extends analytic&’ to DU 0. In particular, if r is strictly pseudoconcave (i.e. q = n - l.), then each analytic subset

of D of pure dimension 2 2 extends analytically to a neighbourhood of D v r.

Here, the main particular case is when p = 2 and r is strictly pseudoconcave. The general case reduces to this one via the “Hartogs Lemma” of $5.3. We remark that Theorem 1 in $5.4, follows from this Theorem 1, in the case where M is of, class C*, for then, M cant. be represented as the intersection of “sufficiently pseudoconcave” hypersurfaces. Generally speaking, one-dimensional analytic sets do not continue through pseudoconcave hypersurfaces. It is easy to present appropriate examples.

From Theoreni 1 and $5.3, one can deduce the following geometric analogue of the removability of compact signuiarities for functions of several complex variables.

Theorem ‘2. Let D be a domain in @” (or on a Stein manifold a), K u compact subset ofD, und A II purely p-dimensional analytii subset of D\K, where p 2 2 and the closure in D of each irreducible component of the set A is non-compact (in Q). Then A extends analyticully to D. ,

One can proceed with a proof by contradiction. By Lemma 2 of 55.3, there exists a minimal compact E c K such that A extends analytically to D\E. Let <tzE be such that 151 = maxEIzI. Then, the hywsurface lzl = ItI is strictly/ pseudoconcave, as the boundary of D n (Izl > [<I;, and hence A extends through 5 by Theorem 1. However this contradicts the minimality of E.

5.4 The Plateau Problem for Analytic Sets. The border bf a p-dimensional analytic set (with border) is everywhere outside of a closed set 6f (2p - l)- measure zero a (2p - 1)-dimensional manifold of CR-dimension p - 1 (such an odd-dimensional manifold is said to be maximally complex). A remarkable theorem of Harvey and Lawson [21} asserts that, for.p 2 2, this infinitesimal condition is essentially a characterization. This is a geometric anaiogue of the fact that each CR-function on the boundary of a bounded domain in @“, having connected complement, extends hoiomorphicaily to the interior of the domain. This is the theorem of Severi, see [39].

A closed subset n/i of a complex manifold Q is called a maximal complex cycle if there exists on M a closed subset X,-of zero (2p - 1)-measure such that M\E is

154 E.M. Chirka

an oriented maximally complex (2p - 1 )-dimensional manifold of locally finite (2~ - 1):measure in R, p 2 2, and the current of integt’ation on A4 is closed in Q For example, we may take any maximally complex submanifold of R. Thd following theorem (Harvey, Lawson [21]) solves the problem of the existence of a complex-analytic “film” having a given compact boundary in C”.

1

2p Theorem 1. Let M be a compact maximally coniplex cycle of .dimension - 1 > 1 in C” (or on a Stein ma’nifold). Then there exists a unique holomorphic

p-chain T in @” \ &I having finite mass and compact support whose boundary is M . more precisely, dT = - [M]. M oreover, there is a set-E on M of zero (2p - 1):

measure such that for each point of M.\ E, in the Wghbourhood of which M is of class Ck, 1 5 k I co, there exists a neighbourhood in which 1 T( u M is a Cc- submanlfold (with or withouJ boundary).

For p = 1 the border of A sritisfies an infinite set of orthcigonality relations for holbmorphic (1, 0)-forms, For the correspondingly (already non-local) defined one-dimensional tiaximally complex cycles, the theorem on “soap films” is valid also for p = 1 (see [20-J), but this case was already studied long ago (see [48]). We introduce one of the possible generalizations of Theorem 1 in the spirit of Stolzenberg, 1481.

Theorem 2., Let K be a polynomially compact set in C” pnd M a maximally complex cycle qf dimension 2p + 1 > 1 in @“\K such that M v K is compact. Then;there exists an analytic purely p-dimensional subset A of C”\(w v K) such that A v M v K is compact and M t A. Moreover, outside of some closed’subset of M of (2p - I)-measure zero, 2 is a submanifold with border in. @” \ K having the same smoothness as M.

We remark that a polynomially convex compact set can be approximated by polynomial polyhedra which when lifted to a space of higher dimension become conve:: po!yhedra. The proof of the theorem of Harvey-Lawson in [20] carries .over to half-spaces almost word for word; and the gluing of films in various half- spaces is also feasible by this scheme. I

In the above describe& “complex Plateau problem”, one obtains a film in general with singularities (as in the classical case). For smoothness criteria for the film, scx the work of Yau [54]. Here one finds additional conditions in (not so e&y) terms of the cohomology of the tangential CauchyiRiemann complex, We introduce a result in this direction having a simple formulation and due to M.P. Gambaryan (see [lo]).’

Proposition 1. If a maximally complex manifold M lies on the boundary of a strictly pseudoconvex domain. D in C” and A is a purely p-dimensional analytic subset of C”\ M such that M = A\ A and A is compact, then sng A consists of jinitely many points and A\sng A is a submanifold with boundary in C”\sng A. Moreover, if 2p > n and’ there exists an (n - p)-subspace L c C” suck that L n T, M = 0, for each 5 E M, then A is a complex suhmanifold of C”\ M.

’

III. Contpkx Analytic Sets 155

57. Bumddes of ChwDhwoskd Andytic !3ets In the theorem of Harvey. and’ Lawson, there -is the assertion concerning the regularity of the boundary.behaviour of a p-dimensional analytic set A ,aiong a (2p - l)+m& sional ma&M M of class C”, k 2 1. Along manifolds of higher dimens)on, t!e bchaviour of A, is general, uncontrolled. However, it turns out that this is fundamefitally influenced by the CR-dimension of M, and if the latter isno more than p - 1 then one can expect smoothnessof A on the boundary, almost the same-as.in:the ‘theorem of Harvey-Lawson. The first difficult case is for p = 1 and .M a totally-real manifold.. The following theorem was proved by the author [O].

Tbeorem 11 Let D be a &Fin in Cc”, M a totally-real s&manifold Of d of class C?;k > 1, and A a purely one-dime&o& analytic subset of D\M. T@n, bA = 2 in M is locally rectijable. Moreover, on bA there is a closed (possibly

empty) sub&t E of length zero such that each point <E (bA)\b is e&r removable (il is a? a&al& subset ii the neighbourhood of <), or is a border point (A R U, is a manifold with border of smobtkness k - 0 in Utj, or a two-sided s&gufarity (A ti U, is the union of two bordered mangolds whose borders are smooth of order 7 1 and have tangent cones at tke point c which constitute a complex plane). Moreover, tke renwoable singularities and the border points fo? an open egery- where dense subset of bA.

A manifold S has smoothness of order- k E R + if SE CtL1, where Fk] is the integral part of k, and the tangent space to S, considered as a pomt of the corresponding Grassmannian, saiisfies a Hiilder condition with -exponent

k - [kj. Smoothness of order k - 0 means that SE d, if k is non-integral and

SE c” for each k’ < k;- if k is integral. The loss of smoothness from k to k - 0 actually takes place as is seen for example in the graphs of conformal mappings of plane domains. From the proof of the theorem, one obtains, for example, the following’ result on the regularity of suspended d&s (used in many multi- dimensi&al problems).

& 1. . . Let M c C” be a totally-real manifold of class Ck; k.2): and f:

b + C”*a kolomorpkic mapping of tke unit disc II c C suck that tke boundary dues off along an open arc y c ab lie in M. Then f E C

‘-O(Au y).

Results on the regularity of the boundary. of onedimensional analytic sets should apparently lead to analogous results for p-dimensional analytic sets in the neiihbourhood of manifolds of CR-dimension I p - 1 (with the help pf the “Hartogs Lemma” from $5.3). However, this is not so simple, and the question of the boundary behavioar in this situation remains open.

58 A Peqective arid progpecb. At the present time there are already many analogous theorems for holomorphic functions (mappings) on the one hand and for analytic sets on the other, which preseit the theory of analytic sets as a geometric variant of the thedj of functions of several complex variables

156 EM. Chirka

Adding a geometric viewpoint to analytic results and tbe reformAtion and ,poof of the analogous assertions for analydc sets undoubterdly enricbcs both directions of multi-dimensional :complex variabkzs even- more. ,One. should remark however on a characteristic partict&ritAof such a development. If one manages to find a correct (without obvious counterexamples) geometric formulation, then the subsequent proof is rat&r transparent (although perhaps teclinically very complicated as, for exampie, in the theorem of Harvey-Laws@. For this reason, at the present time, there an efforts in progress to formulate correct hypotheses by studying typical examples and the “inner geometry” of classical theorems of complex analysis. For example, concerning the removal of singularities, there is not yet an analogue to the theorem on the removal of singularities for holomorphic functions under the conditions of Hbkr an- tinuity, nor to the connection between the growth of a function along remova& singularities and a metric characterization of singularities etc. Analogues Qf the strong results of recent years 0~ the boundary proper& of holomorphic mappi’ngs of strictly pseudoconvex domains also have not yet been obtained. One of the few results of this program for analytic sets is the f&owing theorem of S.I. Pinchuk [31),

TIWUCVII 1. I-,

Let D he u domain in CZn and IL4 u &manifold of D of the form x r2, rvhue Fj ure strictlp psdoconwx ‘R-unulytic hypersu+ces in C”. Then

uuch purrlq, n-dimtwsionul subset A of D ‘: M extends analytically to some neigh: hourhood c!f M.

Here, the CR-dimension is completely sufficient to “block” A, however, this is not appareqt from the powerful Levi form of the manifold M. This phenomenon is, as yet, cort~plt%cly uninvestigated. Part of the hypotheses can certainly be weakened but which and how is so far unknown.

Notes. The results of this chapter are essentially found in the artiF’- mentioned in the text. An exception is the results &‘#5.7 and 2.8 which -._

proved in the author’s book [IO]. The theme of this chapter is perhaps the F”C~ current one in the tfieory of analytic sets. Among the works d rm*Pnt VPQ~I

..“UI I

draw attention to [J@J, [31]. [UJ: [12], [zI], ana [46].. -. .-- - - _ -_-.., ,-5 we

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2. ‘Ucxander, II.: Continuing I-dimensional analytic sets. Math. Ann. IYi, No. J, 143-144 (1971). Zbl. 2Il,lO2

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singularities I, , f complex analytic varieties. Duke Math. J. 45, NO., 3, 427-512 (197b). Zbl.

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409.53048 Griffiths. P., Harris, J.: Principles of algebraic geometry. Wiley. NW York 1978. Zbl. 408.14OOI &%iths, P.: King, J.: Nevanlinna theory and holomorphic mappings between algebraic rarie-

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26. Malgrange, B.: Ideals of differentiable functions. Oxford University Press. Oxford 1966. Zbl. 177,379

27. Molzon, R.E., Shiflman, B., Sibony. :d.: Avcrage growth estimates for hyperpiiinC &liOnS Of

entire analytic sets. ,vlath tnn. * ‘j ;. %o. I. 43-59 (1981). Zbl. 537.32009 28. Mumford, D.: Algebraic geometry. . . 1 Complex projective varieties. Springer-Verlag, Berlin

Heidelberg. New York 1976. Zbl. 356.1400.! 29. Palamodov, V.P.: The mulliplicily of a holomorphic transformation. Funk%. Anal. Prilozh. 1.

No. 3. 54-65 (1967); Engl. transl.: Funct. An,il. Appl. I. 11X-226 (19681. Zbl. 164.92

158 . EM. Chirka

30. Pinch&, S.I.: On holomorphic mappings 61 real analytic hypcrswfaces. Mat. Sb., Nov. Srr. 105, 574-593 (1978); Engl. transl.: Math. USSR, Sb: 34, No. 4,503-519 (1978). Z~I. 389.32008

‘31. Pinchuk. S.L: Boundary bctiviour of analytic sets and alkbroid mapping.s:hkl Akad, Nauk SSSR 268, 296-298 (1983); EngI. transl.: Sov. Math., Dokl. 27, NO.‘& 82-85 (1983). Zbl 577.32008

32. Ronkin, L.I.: Introduction to the theory of entire functions of several variables Nauka, M-w 1971; Engl. transl.: Am. Math. Sot., Providence, R.I. 1974. Zbl. 225.32001

. 33. Rothstein, W.: Zur ,Theorie der analytixhen Mannigfaltigkciten in RPmn von a komplexen Vednderlichcn. Math. Ann. 129, No. 1, 96-i38 (1955). Zbl. 64.g 133. NO. 3. 271-280 (1957). Zbl. 773289; 133, No. 5,400-&9 (1957). Zbl. 84,72

34. Rothstein, W.i Zur The&e der analytischen Mengen. Math. Ayn. 174. No. I,&32 (1967) ZbI. 172,378 ’

35. Rqthstcin. W:: Das Maximlimprinxip und die Singular&&en analytischer Mengen. Inyent. Math 6, No. 2,163-w (1968). Zbl. 164,382

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38. Santalo,,L.A.: Integral geometry in Hermitian spaces. Am. J. Math. 74, No. 242-34 (1952). Zbl. 46,161

39. Shabat, B.V.: Introduction to compkx analysis. Part II. Functions of Several variable; 3rd cd. (Russian). Nauka, Moscow 1985. Zbl. 578.32001

40. Shabat, B.V:. Distribution of values of hoIomor#hic mappings. Nauka, Moscow 1982 End. transl.: Amer. Math. Soc,‘Providence, R.I. 1985. Zbl. 537.32008

41. Shiffman, B.: OLI the,removaI ofsingularitics of analytic acts. Mich. Math. J. !S, No. 1,111~120 (1968). zbl. 165,405

42. ShiRtnan, B.: On thl cdntinuati?Q oiapalytic curves. Math. Ann. f84, No. 4,268-274 (1970). ZbI. 176,380

43. Siu, Yum-Tong Analyticity of sets associated ,to ‘&long n&bers and the ettension of do&d positive currents. Invent. Math. 27, No. I-2, 53-156 (1974). Zbl. 289.32003

44. Skoda, H.: Sous-en$embles analytiques d’ordre fini & i&i dans’c’. &ll. So& Math. Fr. IW, No. 4,353.4@3 (1972). Zbl. 246.32009

45. Skoda H.: Vakurs ai bord pour lea solutions di: l’ofiratcur 8. et aaracttition ha z&ox des fonc+s de la classc de NcvanIinna Bull. Sot. Math. Fr. I&, No. 3,. 225-299 (1976). zbl. 351.31007

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- -

IV. Holomorphic TMappings and the GeOmetry of Hypersurfaces

A.G. Vitushkin

Translated from the Russian by P.M. Gauthier

Contents

Introduction ................................................ 160

$1. The Normal Forin for Representmg a Hypersurface. ...........

,63

1 .l. The Linear Normal Form ........ : ............... ; ... 163

1.2. The Initial Data of a Normalization .................... 164

,‘C 1.3. Moser’s Theorem. ......... , ........................

1.4. The Classification of Hypersurfaces .....................

1.5. Proof of Moser’s Theorem. ............................

92. The Standard Normalization .. ........ ...

2.1. Definition of the Standard Normalization.

.... ............................

2.2. Approximation of a Normalization by ,a Linear-Fractignal Mapping ..........................................

2.3. Parametrization of Mappings .........................

fj3. Chains ............................................... 3.1. Definition of a Chain.

...............................

3.2. Chains’ on Quadrics. ................................

3.3. The Linear Normal Parameter ........................

-54. The Equation of a Chain ....... ..... ... ....... ...............

4.1. Straightening a Hypersurface Along an Analytic Curve. I 4.2. The Equation of a Chain in the Natural Parameter

........

4.3. Derivation of the. Equation ............................

-95. The Circular Normal Form ... _ ...........................

5.1. The Form of a Hypersurface in Circular Coordmates .......

5.2. The Initial Data for a Circul&r Normalization. ............

5.3. The Forti of a Subsiitulion Preserving a Chain ........ i . !

. 5.4. Continuation of’s Normalization Along a Chain. ..........

/ . . :-:- :* .J

.... % .

IUJ 165 166 172 172

172 172 173 174 174 175 176 176 176 177 178 178 179 179 180

160 A.G. V~tushkin

$6. Normal Parametrization of a Chain. . . . . . . . . . . . . . . . . . 180 6.1. The Circular Normai Parameter 1 . . . . . . . . . . . . . .

180 6.2. The Formula for Changing a Parameter. . . . . . . . . .

181 .6.3. The Initial Data of a Parametrization. . . . . . . lo- 6.4. Normal Parametrization and the Continuation _. ...“~~,“6. 6.5. The Equation for Passing to a Normal Parameter

97. The Non-Sphericity Characteristic of a Hvnersurfac’

. . . . . . . . . . . . IOL

nf Mannkng . 183 .“I . . . . . . . . . 183

,n ..- e.. . . . . . . . . 7.1. Estimate for the Radius of Convergence, and the Norms of the

184

Defining Series. . . . . . . 10” 7.2. The Non-Sphericitp Characteristic.

...........................

7.3. The Variation of the Characteristic Under a Mapping 185 : In

7.4. Chains on a Hypersurface Close to a Quadric. .... . . 1m

7.5. The Behaviour of a Chain Near Points of High . . . . . . . . . . . 187 Curvature. . . 189

.$S. Strictly Pseudoconvex Hypersurfades ....................... 1 nr 8.1. A Theorem on the Germ of a Mapping.

17b 101 1 ......

8.2. Properties of the Stability Group. ...................... 8.3. Compactness of the Group of Global Automorphisms

‘71 ‘n4 .... .

8.4. Continuation of a Germ of a Mapping Along a CompaLi IYL

.-t

Hypersurface . .. .. ...... ... .. ........ ..... .. 8.5. Classification of Covermgs.

....... 192 1 .................

8.6. Discussion of Examples of Non-Continuable M:~n~~~~:~ ’ ’ * ’ 8.7. Mapping of l-i:,,persurfaces with Indefinite Levi k ol,ll. 8.8. Proof of the Lemma in $8. I

...... IYJ

194 ........................... $9. Automorphisms of a Hypersurface .........................

9.1. Estimation of the Dimension of the Stability Group. 197 ‘I’-

9.2. Parantetrrration of the Group of Automorphismc ...... . IYI

9.3. Linearlration of the Group a.......... 197

SIX0 ........................... 9.4. Pmchf of the Kruzhilin-Loboda Theorem

IYO

................ ,1\<1 9.5.yroof of the Bcloshapka-Loboda 'rheorem

IYpi ....

4 10. Srnf ;oth l-iypersurfaces . . . . . . . . . . . I 99

7, ir 1o.t.

................................. In;ariarit Structure5 on a Smooth Hypersurfacc “” .........

10 2. Compactness of the Group of Global Automorrh;c,--- IO..?. Chains on Smo,>rh H~pcrsur!;lces 10.4. Properties of the Stahiiity Group.

..................... :I rn ....................

10.5. Lis?wrir;t!i:)n of l.ocal Atitomorphisms 209 ?,I\ ......

References ...... .... ..................... ........... L,!,

.......... 212

The principal topic of thi> p,~por i, non-dcpencratc tin the sense of Levi) hypersurfaces of complex manifolds and the automorphisms of such hvpersur- faces. The material on strictly pseudoconvex hypervurfaces is preen&d mns: comp!etely. We discuss in detwrl ;I form of \vriting the cqu:tti()ns of the hypcr-

IV. Holomorphic Mappings and the Geometry of Hypersurfaces 161

surface which allows one to carry out a classification of hypersurfaces. Certain biholomorphic invariants of hypersurfaces are considered. Especially, we consider in detail a biholomorphically invariant family of curves called chains. A lot of attention is given to constructing a continuation of a holomorphic mapping.

The posing of the problem under consideration and the first concrete results go back to Poincare [31]. His method of studying a hypersurface was to analyze its equations directly. In studying the classification of domains in C2, he formulated a series of concrete problems: the classification of real analytic hypersurfaces in terms of their defining equations, continuation of the germ of a mapping from one analytic kypersurfaqe to another preassigned hypersurface, and others. Poincart, showed that a germ of a biholomorphic mapping of the sphere into itself extends to the whole sphere and moreover is a linear-fractional transformation. As a consequence-the general form for an automorphism of the sphere was written out. Poincare remarked that a certain family of series, of a

‘special form having different coefficients of sufficiently high degree, yield pairwise non-equivalent hypersurfaces.

Segre [33], E. Cartan [lo], and later Tanaka [36] worked out different approaches to the construction of .a classification theory based on geometric methods. This topic acquired widespread popularity in the 70’s following the work of Alexander’ [l], and Chern and Moser [ll]. The paper of Alexander drew attention to itself by the clarity of its results. Therein, the above-mentioned result of PoincarC on the continuation of germs of mappings of. the sphere, till then forgotten, is obtained ane,w. The paper of Chern and Moser has many levels. This paper develops both analytic and geometric methodsand it gave rise to a lengthy cycle of works by other authors (Fefferman, Burns, Shnider, Diederich, Wells, Webster, Pinch’;lk; and others). The present chapter gives a survey of this theme over the past 10-15 years. In particular, we relate quite explicitly the results in this direction obtained in recent years in our.common seminar with MS. Mel’nikov at Moscow University (V.K. Bieloshapka, V.V. Ezhov, .S.M. Ivashkovich, N.G. Kruzhilin, A.V. Loboda, and others).

Let us dwell on some of the results of the topic under consideration. In the geometric theory of Chern, the surface is characterized in terms of a special fibration. The base of the fibration is the surface itself, while the fiber is the stability group of a quadric (a quadric is the set of zeros of a real polynomial of order two; the stability group of a surface is the group of automorphisms, defined in the neighbourhood of a fixed point of the surface, which- keep this point fixed). On this fiber bundle there is a finite set of differential forms which are invariant with respect to bihdlomorphic mappings of the base and which together uniquely determine the surface (see 9 10). The correspondence between

’ these families of forms and surfaces in geometry is called a classification of surfaces.

In the frame of analytic methods, a surface is characterized in terms of special equations which Moser. [ 11) calls the normal form of the surface (see § 1). In the general situation, one and the same surface has, generally. speaking, many different normal forms associated to it. The totality of all changes of coordinates,

162 A.G. Vitushkin

which bring the surface into normal form, forms a group which is isomorphic to the stability group of a quadric. This means that a stability group of an arbitrary surface is represented as a subgroup of the stability group of a quadric.

The class of surfaces with non-degenerate Levi form is divided into two types: in the first type we put quadrics and surfaces which are locally equivalent to them, and in the second we place all remaining surfaces, that is, surfaces not equivalent to quadrics. It turns out that the properties of surfaces of type I and. the properties of surfaces of type II have little in common. In fact many properties of these subclasses are different. Thus, for example, the stability group for the sphere is non-compact, while- for a strictly pseudoconvex surface of type II, it is compact [26]. On the other hand, from the point of view of local structure, almost everything is known about surfaces of type I. Consequently, interesting discoveries are, as a rule, in connection with surfaces of type II.

Among the results of a general nature which’follow from the theory of Moser, one should mention foremost.the theorem of V.K. Beloshapka [2] and A.V. ,Loboda [26]: if a surface is not equivalent to a quadric, its stability group can be embedded into the group of matrices preserving the Levi form of the surface (see 09). The dimension of the group of these matrices is not much smaller than the dimension of the stability group of a quadric. However, the group of these matrices is constructed essentially more simply than the group of a quadric. It is useful to compare.the latter with what was said earlier concerning automorphisms in relation to the work of Chern and Moser.

The class of strictly pseudoconvex surfaces is ,of particular interest. A fundamental result for this class of surfaces is the theorem on the germ of a mapping. If a surface passing through the origin is real analytic, strictly pseudoconvex, and non-spherical (locally non-equivalent to a sphere), then the germ of a biholomorphic mapping, from one such surface to another and fixing the origin, has a holomorphic extension to a neighbourhood common for all such mappings. Moieover, a guaranteed size of the neighbourhood as well as for a constant estimating the norm of the continuation are determined in terms of the parameters of analyticity of the surface and the degree of non- , sphericity (see §8).

The theorem on germs of mappings was the result of many years of/work in our seminar. At first we expected to obtain its proof by purely analytic methods, more precisely by describing and estimating the dependent parameters of the stability groups. The resulting theorem of V.K. Beloshapka and A.V. Loboda turned out to be no less interesting in itself. An essential consequence was also the proof of compactness of the stability group (141, .[37]). With regards to the continuation of mappings, we succeeded at first, in this direction, to obtain only a few special cases of the theorem under consideration ([4], [37]; for statements, see $7.1 and $8.3). The obstruction was that the estimate for the dependent parameters automatically contained some additional quantities. Further progress required not only a perfecting of the analytic methods but also a fundamental investigation of a family of curves called chains ([38], [13], [12], [23], [14]). For such chains, the so-called circular form for the equations of the

mult of this~,series of works .wa&T q.teqtl on t@ 9*ua~n of0 lhgping abng .-a compaq surface (iIS],- ,see §8.%, In [4OJ, we succe&d finally ha obtaining thc nccessaly eqig#q $y,tbe * sf-* gff@=wr m Lemma in 58.1) which’ allowed US to complete tke sob&n of the p&&n c+ gnnnsofmappings’ .; .

In dosing we point out one. more most +ro’mi&g resuk (N.G. Kruxb&, A.V. ~Mal.c In tk~ neigbbourbood .of %eack point of a strictly p+ndoeortye& ~-W=~~ ,s*,we .iqay ,c~oose, ~o@i&ates in wki& .& !a+ 4+* 4 morphism ii a lit&u tr&formati~‘(see~~9.3), T& kuestion of the exiatemz of

‘analogous coordinates on surfaces who& Levi form is indefinite remains open. ’ :

Tkis paper consists essential& of tke paper [40-J revised according to the requirements ,of tke present a+es and cont&ngq@mentary,, re&s on smooth s‘urfaces ($10) gkort pro& pf,tke fundamental reSulta w iqdqde$.:

.s’ ”

8 1. The Normal Form for Representing a Hypeitsurialce~’

In this section we’consider a special form of representing a surface that was introduced by MC&L It is in a certain sense the simplest for&; and,so it baa many applications.

,. ’ . -.

1.1; TLe Ihear ~~~BuII Farm. Let M’be a real-analytic hypersurface in ‘&I ndimensional complex ma&fold 3. ‘For every CE M we can~ehoose a jocaI

” coordinatesy~t~(z,,...,z,-,);w=ufiu,onXforwhicbCistbeorigin and M is def&d by F&z, &w, it)==0 where 61;,60) # a WC rotate M abottt tbe qrig@J)tltil tbemaxb~ ~4*tbae-mM.canke written iii the form I) = F,(z, 5, u)i’Wtren,~~@)~Qa~+I dF,(o)-0. Expanding F, as a &ri&

. in z, Z and isolating terms of tba f+rtn z,& we write the surface as

P = (2, 2) +F(z, z, ub (1.1.1)

wbere ( - -d=& (I‘, kZi& is a. Hem&n form and F’ is a real-analytic function -*

such that PF

F(O)=O, dF(O)=O, - I ,, . hJ% 0

-0 @k=l,2...,n-1) .

~fonn(t,z)iscalledtheLevifonnol1Watt~pointQWeshalIconsider bypmudaces for which the Levi form is non-degenerate at all pointa. .

If n = 1. tken the form (z, z) is not de&d, and does not #we into. t@ equation ol the su&aee (in this CBse, a -onal curve). Thtrdore,. by a change of coordinates any enrve can be reduced to tfm form o = 0. For A 2 2 no bibolomorpbic change of coordinates can remove (e, z)

\ lrom tbe equation of the surface. Tkix means that the form of the surface c&rot ,k Ggniftcantly simplified: I

4.G. Vitushkin

By making an appropriate change of variables we may assme that 1 n- I

(z,z> = ,& ZjSj” .-z- zjzj. ’ j=s+l

.We define the differential operator A by the formula

Moser [l l] showed that for each surface of the form (1.1.1) there is a biholomorphic map that sends this surface into one of the form

* u=(z,z)+ Fk,(?, 5; u), (1.12)

where Fk, is a polynomial of degree k in z and 1 in i with coefficients depending analytically on u, and Fz2, FS2, and %;, satisfy the conditions tr FZ2 = tr2 fS2 = tr3-F,, . The operator tr is a second order differential operator defined by the Levi form [l 1). If

n-1

& z> = i zkzk - 1 z& then trFk,&F,... k=t k=s+l kl

Regarding surfaces of the form (1.12) we shall say that they and their equations have linear normal form (along with the linear form we introduce below the so-called circular form). Any biholomorphic map sending a surface of the form (1.1.1) into one of the form (1.1.2), that is, into a surface written in normal form, and leaving the origin fixed, will be called a normalizing map, or siniply a normalization of the surface (or sometimes a reduction to normal form).

1.2. The Initial Data of a Normalization. We consider is an example the hyperquadiic’o = (z, z). Poincar&[31] and Tanaka [35] showed that any map defined and biholomorphic in a neighbourhood of the origin which sends this quadric into itself has the form

. . , z* = IU(z+uw)/(l-2i(z, a)-(r+i(u, a))~), .

w*~a~2w~{l-2i(z,u)-(r+i(u,u))w), , (1.2.1)

where cr = + 1, d > 0 and r are real number ((T can take the value - 1 only when the number of positive eigenvalues of (z, z) is equal to the number of negative ones), a is an (n - l)-dimensional vector, and U is an (n - 1) x (n - 1) matrix such that *

(Uz, Uz) =f Q(Z, z>. (1,2.2)

The converse is also true: if the set o = (II, a, 3., Q, I) satisfies (1.2.2) then the correspondingimap (1.2.1) sends the hyperquadric into itself. Thus, for ,hyper- quadrics there are as many normalizations as there are sets o = (U, u, 1,~, r) satisfying (1.2.2).

suppose that E r* 7 f (z, w), *IV* = g(z, w) aend! a surface of the form (l*lA) into one of the sam~.form, for, example, into a dfaceddincd in .no* ~~. ‘With every such map we associate a set o = (U, a, A, a, r) ddined by the systetil

IV. Holomorphii Mapping and the Geometry of Hyprautfaas MS

: :- iy- I = q, 4f

& ..’ I . . . I aw, =lZUa

. - .

&2.3)

ag aw, I = d2, Re a2g

aw2 = 2uA2t.

0

We shall call this set the set. of initial data of H. In particular if H is a normalization, the set is called the set of initial data of the normalization.

q 1.3. Moser’s Theorem. The main result in Moser’s paper [ 11) can be stated as follows.

. Theorem. For eucff~surface M of the form (l.l;l) and a?y set of initial data UJ,

there is a unique normalization H,(M) having initial data o. Zf M is defined in linear normal form u?p a = 0, then H, has the form .

z* = LUz/(l -r$ w* = a12w/(l -rw). L I$;;:

Here we must warn the reader who wishes to become acquainted w&l&~ ,,;‘I Moser’s work: in [ 1 I] the initial data are introduced in terms of the inyerae mgp ::.. of a normalization. It is more convenient for us to use a different definition; Let us explain this. A normalization is uniquely defined by the set of first derivatives ..a and a single parameter, which is calculated from the second derivatives. When . the initial data are defined in terms of- the inverse map, this parameter is expressed in terms of the second derjvatives.of the map and the coefficients ::’ a2,F~aziazs (see (1.1.1)) of the normalized surface (it is not good that the definition of the parameter depends on the coefficients of the surface). Moser

! , ~~

stated the theorem for surfaces that had been simplified beforehand (the holomorphic quadratic part of the series was removed by a substitution); in this case the formula turned out to be simple. In applications this is inconvenient because the preliminary processing hasto be carried through the whole statement. These difficulties do not arise if the initial data are defined as we do here.

Below we discuss several applications of this theorem to the classification of .’ hypersurfaces, to the estimation of the dimension of the automorphism group, and to the construction on hypersurfaces of a special family of curves called chains that is important in applications..

1.4. The Classification df Hypersurfdes. Let M and M* be two hype!; surfaces in C” containing the origin. We shall say that M and M* are equivalent if in a neighbourhood of the origin there is a biholomorphic change of variables sending M into M*. Clearly, if M and M* are equivalent, then a normal form of

. . dnmmamd case, this f” was proved by’+oincark [31-J. A detaikd discussion of . . m probiems:can .kfound in:Fefkrman’s.paper [l%YJ. . . .~ . “.

$ ,: . . . . ;. .I ‘” _’ .3 ‘.

I.!% Rad d lbae’s~Tleorem. For simplicity, we shall assume @at .the I4$v.i~~hasthe~~ . .i”

,.. +‘. ‘:, ,i ,;. ?

(z*z>=;&z~z,, L.. where&,= fl. (. 7, : -

IO this&e we have for the polynoniial F&z, 2, u), of degree k ‘m z and I in f that trFt, = (l/M)hF,. We list several properties of the operator 4 which w& .b smalalhthesesuel: J”i)Rthe ma&-U is such that (t.Jz;Uz)‘=o(i,z) (sikQ1.2), then from AP(z;&u)4l,itf&oWhat -’ :- :

i,

: A( vi, ~J(uj).= 0, fre3chfunction~ .

:,’ b) ‘if AF(z, 2, u)-O/then A’& z)F(z, 2, u)=Q .’ Q each real ~polynomial Fz2(z, 5,~) can be written

Paa A.<% E)(?AzZ)+N~~(Z, Z, u), where A(u) is a Hermitian matrix, that is 9b=-&md A2Nj2 =Q

. d): Erich real @ljn@mial F&, 5, u) +n be *t&p

,- c Fs~=<z,~)~(z,F(U)>+NJ~(S~,UX

where A2N32(z, Z, u)=O. .Let~6xahrpersurfaaMgiverqby(f.l.l)andacumy(t):z=p(l),w3dr),

l~ollM,~tisarealparameter,~paridqareanalyticfunctiaasJ\rcb that q(O) = 0 and q’(O) + 0. We shqw that there exists a biholomorphic mappin% in tae~bourbbodoftbcorigin*hichsends~totheliner=o,r,=o,sad~ M O$O a hypersurtace M(y) of the form

(13.1)

IV. Holomorphic Mappings and the Geometry of Hypesurfaces 167

where AF22-= $l and AyFJ3 = 0 (for the definition of Fk,, see 0 1.1). Moreover, impoging further conditions. on y, we obtain that A2Fj2 = 0.

.

.Lemma. There exists a biholomorphic mapping (in some neighbouthood’qfthe origin) wkibh maps ‘the curve y into the line z = 0, v = 0 and maps M mto .a hypersurface M(y) given by (1 S. 1). Any biholomorphic mapping of a hypersqface given by (1.5.1) onto another such hypersur$ace, and leaving the line z = 0,~ * 0 jxed;is ci linear fractional rransformation of the form

where u = _ , ’ + 1’ I > 0, r is a real number, and ’

(Uz, (h) = u(z, z;. ,/ We construct the desired mapping Q as the composition of Six biholomorphk

transformations cp = (P~Q~ . . . 'pl, In cdnstructing these transfokations, we shalt, at each step, denote the old variables by (z, w) atid the new variables by (z*, w*). We denote by Ni the hypersurface obtained from M by the transformation QfQi- 1 . . . &, and we write its equation as follows:

v = F”‘(z, 5, u) :’ F

Flfl(z, ;, u), i = 1,2, . . t , 6. ,I

1. : We de&e*the transformation Q; via its inverse

Z* +p(w*j, Q;': 2 = w = q(w*).

The transformation cpl, maps.the curve y to the line z* = 0, v* = 0, and the hypersurface M to Ml: v* 1: F”)(z*, if, us), where

Fylyz*, z*, 0) = ( P, z*), F(l)(O,O, ~*)=0;u*J,+,,~~~=t. ** .

2. By the transformation (p2: z* = z, w* = w +g(z, w), g(0, w)=O we map Ml: v = F”‘(z i u) to M,: v uniquely chosdn’such that

* = F(*)(z*, w*, u*), and we shoiv that g can be

F’,zd(z*, Z*, u*) and F$(z*, Z*, u*)=Q k= I, 2, . . . .

If in the equation for the hypersurfke Mz, we write z* and w* as functions of z and w, we obtain

. ( Ft2’ z, 2, u + ;(g(z, w) + @(z, = &(g(z. w) - #(i, @ii)) + F’“(z, 3, u; b

.

where w = u+ iF(‘)(z, 5, u). In this equation we set Z = 0. Then, from .the condition g(O, w) 70 it follows that the term #(Z, G) turns out to be zero. And

since F’*)(z 0 u) = , , P

%(z, Z, u), from the condition & Fi$(z, Z, u)=O, we

168 . G. Vitushkin

obtain that 1 . ! 1 D

0=-&z, u+if:l)(z,O, u))+F(“(z, 0, u). (1.52) /

Set 5 = u + iF”‘(z, Q, u). Since F($z, 0, a)JZZO = 0,. it follows from the implicit function theorem, ‘that u is a G(0, <)=O. On the other

fj~ F

ction hand,, from

of z .and 5: u = <+ G(z, <), where (1.5.2) we have 0=(1/2&(~,5)+

(l/i)(< - u), that is u = 5 +(f/&(z, r). Thus, g(z, w)= 2G(z, w) is the desired

:I* function. Dropping the asterisks, we can rewrite the equation of the hyper-* surface M, in the form

u = F’:l(z, i u)+ c F’,:‘(z, 2, u), k,IY 1

k+IL3

where F$)(z, 2, 0) 7 (z, z). 3. We define cp3 by its inverse transformation cp; r: z = c(w*)z*, w L ~9. We

“choose C(y*) in such a way that the form Ff3) r r , appearing in the equation for the hypersurface M, 5 q3(Mz) is independent of u and is equal to (z, z); namely, we choose C, satisfying the system

this equation and such that iu)z,-Ww; ;iom the equation for Mz, we obtain the equation for M,:

u = rzT(u)H@)C(u)5 + 8: 1 Fif’(z, i u) = (z, i) +

/I kk;’ i 3: ,I

c ’ Fi;‘(z, i, ir). k,lk 1 I k+lh 3

4. We chaos ‘” the mapping 8

d such that M, = (P4(MJ) is of the form:

. i ’ --*.q~l

u* s,//z*) + k & 2 F::‘(z*, i*, u*). .

To this end, we rewrite M3 in the form

u = (z, z> + 12; (d,(z, u) + &4k ~1) +.k 1c, ; F#‘(z, 5, u), :. ,- - ‘. where

d .+ I \

n-l

C CZi$dZ, u) + flAj(Z, U)) = f (F\T + F;:‘). i=l i -- j=l

IV. Holomorphjc Mappings and the Geometry of Hypersurfaces! 169

I I

We define (p4 as foflows: z w, wheref(z, w) is tl/e vector with coordinates (srAr(z, w), and Ei = +l(i = 1;2,. . . , n-l)

are the coefficients of the cing z* by z+f(z, 9) in the. form

(z. z), we have .

n-1 II-1

(Z*, Z*) = C EiZfZf = . i=l

& Ei(Zi + AiCZ, y))(F + Jitz9 w))=

II-1 ’

. . = (Z, Z)+ ill (ZiAi(Z, U)+ZiAi(Z, U))+ . . . .

That is, n-1

(Z,Z)+ 1 (ZiAi(Z, U)+Z*Ai(Z,U))E(Z*,:i*)+ : * " i=.l

11 1; -_

Since, for each i, the function Ai(z, w) is a sum of polynomials having degree at least 2 in z, then, expressing the variables (z, w! in terms of (z*, w*) in the equation of M,, we see that M4 has the desired form.

5. We write the change of variables cpj l:,z 4 V(w*)z*, w= w*, where ( V(u)z, V(u)z) = (z, 2). We choose V in such a wai that M, has the form. . ‘ f

u* = (z*, z*) + c F$‘(z*, ‘;t ;), k,lZ2

where AF$? = 0. Because of property c) of the operator A, the polynomial F22 can be written’in the form

I

FZ2(z, 2, ~)=(z,z)(‘zhi)+-;VZ2(z, 5, u).’ I

Since :

z = V(u* + iu*)z? = V(u*)z* + V(u*)(i J’ z*, z*) + . . .)z*, /

then making the substitution cp;’ in the equation for M4 and removing asterisks, we obtain for MS:

u.= (z, z) + (2, z)(i( Vz, Vz)-i( Vz, Vz))+ (z, z)(‘zrVAvZ) * -

+N,,( vz, vz, u)+ . . . .

Since ANz’z = 0, the equation AF$‘: = 0 is equivalent to the equation -

i( v’z, Vz)-i( Vz, v’z)+(‘z’VAJ-‘JVz)=O,

which, in turn, is equivalent to V(U) satisfying the equation V’(u)= . (1/2)JA(u) V(u). Let ‘us recall that J = J- ’ is the matrix of the form (z.z>. The

function .

V(u)=exp

/

k

170‘ A.G. Vitusbkin

satisfies this equa’tion and also

( Uu)z, V(u)z).= (z, z). Thus, we have cdnstructed the desired transformation vs.

6. For p6 we choose a’ transformation of the form z* 9: JQ’(w) WE, W* =

&@I), where ( uz, i/z > = &, z >, Q(O) = 0, Q(u) = Q(u), and Q’(0) > 0. The funo tion,Q will be chosen in such a way that for the hypersurface

Me: v* ^, (2*, z+j+ F

F$)(z’, z*, u’), k. 22

ihe conditions AF$q = 0, A3F&v = 0 are $atisfied. From the definition Of (ps we leave

-;~(U)-+~“+ . . . =t@‘v-;QttJv’+ . . . , .

. ,~.)(P’-ivQ”.-;“‘+ . . . . >3 “2<z,& I

= Q'a, 2) -f(P"'-yys(z,z)+. . . . From this, we obtain

&“1(z*, P, u*)=(Q!)~F$~(UZ, t?f, aQ(u))+ . . .

Ff$(z*, P, u*) - (Q’)s’2 Fi”1( Uz, UZ, oQ(u)) + . . . ’

F$!j&z*, f*, u’)=(Q’)~F$~(UZ, 05, (rQ(u))+ : . .

After a chcinge of variables, MS can be written in tfie form

oQ’u = ~Q’iz, z>+(Q’)~F~~(UZ, 02, oQ(~t))+@‘)~‘~Ff~(Uz, 02, oQ(u))+

+(Q’)’ p33(u~, t% a(Q(u))-d(2Q”-YQ”)‘/.~)a(z, z)‘+ . . . , tbtkt is, ’

J?f&,it u)=;F$q(U-5, O-‘5, Q-‘(U)),

From properties a) and d) of the operator A, we have AF f2 A3Ff3 = 0 is satisfkd if and only if Q satisfies the equation

= 0; The condition

Q’F(u)++“;3~;f) i 0, Q(i) = 0, &(O)>O,

where F is a rea\&tctik such that I % F33(z, f I+= F(u)(z, z>~ + N(s, 2, r),

with A’N = 0. T’be su b - tiMlq’(s(u))-2chplfBQ&equationtotbe~ equation S” = (3/2)F(rqS, and, hence, it is not w to see that tk ‘initial equation’ has a sdut& monotonic in u. Thus, q6 pad amuesktly p% m defined.

From & above calculstions, it tdlowr that any e wbkb mops the hypenurface ( I 5.1) into anotba aucb b y~rface, and &cb maps the line z=O,u=tiintoitseltisoftha6waa:z~1/~(w)Uz,~=a&o.~~thiscagQ satisfies the above difkntial equition in khieh F(u)=0 and berm has the form stkted in the theorem.

We now show that for a park&r choice d tbe eurvc 7, the condition A2 F,, = 0 will be satiafkd. From kxmubi (1.52) it is ekar that ifA2F32 = 0 for some choice of parameter on y, tben .*r a cb+ge of l&keter, this c0nditiod will still be satisfied. We show .tbat tbe’ca;#titiolf A2Fa2 = 0 is equivalent to y(t) satisfy’ing a certain &ke+ial equation d 0rder two. Tbe inverse mapping of (Ps’p4. * f cp,hastbeforin:‘ * ,

z = p(w’)+ T(w*)z* + . . .., w = q(w*)+ :. . .

For definiteness, we shall a&u& that the original parameter t is the v&e of t.be u-coordinate. In the equation for the original hypersurface M, if we iubstitute ’

the above expreksions for z and w, we obtain that the term Fit, in the equati~@~ for the hypersurface M,, has the form

F)‘Z= (z,Bp”)(z,2)2+K32.

where the matrix B as well as thi c0eflicients of the po!ynomial. analytically on p, jj, p’, and p’, and for smal1 1~1, tbe matrix B 3 Thus, the condition A2F$ = 0 is equivalent to y satisfying an ordianry dit ferential equation of order two. A geometric solution of this equation is *e defined by the dire&ion of a tangent vector at the origin.

-.

Thus, the transformation cp constructed with tbe.indicated choia Or tbe curve y is a normalization mapping. Here, the direction oi tbe curve 7, the rarl numbers Q’(0) and Q”(0) and the matrix U can he chosen arbitrarily PIpvidcd (Uz, uz> = a(z, z).‘It is not hard to convinceonesegthat &se pruusctararn be so chosen that the normalizing mapping p bas any preassigned set bl @itial data w.

We now show the uniqueness of a normaking mapping ha&g 3 given set 0f initial data Let cp and cp* he two -normalizations 0f A% having o aa set of initial .- data. .

Let us represent cp* in the form cp *=R(cp~Bythelemmaweproval~tbis section, R is a fractional linear transformation. Since cp and q* have tbe rsppc initial data, the set of initial data of the transformation R is (E, 0, 1.1. a), lred hence, R is the identity, i.e., (p* = cp. Tbe theorem is proved.

-4

172 A.G. Vitushkin

$2. The Standard kmnalization _

Here we present a standard procedure that associates with any hypersurface a normal form, which is in a certain sense the most natural one, and we discuss certain properties and applications of this form.

2.1. Definition of the Standard Normalization. For a hypersurface M the normalization H,(M) with the set of initial data e-= (E,O, 1, 1,O) (see $1.2) is’ called the standard normalization of M. We. write H; * in the form z = cp(z*, w*), w = $(z*, w*), It is easy to see.that cp and @ satisfy the relations

In ot‘her words, the tangent mappings of H, and Hi 1 are the same, and so are ’ the real parts of the distinguished second derivatives.

’ 2.2. Approximation of a Normalizatiofi by a Linear-Fraciional Mapping. Let M and M* be two hypersurfaces .defined in normal form, and H be a mapping sending M into M* with the set of initial data o. We considoer the composition HJR,), where R, is a linear-fractional mapping with the set of initial data 0 (see $1.2). and He is the standard normalization of the surface R, (M ). We write H, as

z* = ,” + f*(z, WJ; w* == M’ + g*(Z. IC).

Lemma. The mappiitg H,( R,,,) = H,,, and ,the corre.spondiny,firncti~n.s,f * and g*! are such that the order of smallness (with respect to (2. M: )) of/ * at the origirl is not less than third order, and qf g* not less than fourth order.

It follows from the lemma, in particular, that any normalizing mapping H of a hypersurface M can be represented as H = H,iR,.,(HCj), where H, (the first ma* ping) is the standard normalization of M, R,, is a lincat-fractional map with th(* set of init; i data o, and the last mapping in the compositijn is the standard nprmalization of R,( H,( M)).

It is proved in [iI] that f* and y* have second order of smallness at the origin. The result in the present paper was obtained by Kruzhilin [23]. The assertion c’an be restated thus: H, - R, is defined by functions with third order of smallness, and the fuilction corresponding to MT has fourth aider of smallness.

.2.3. Parametrization of Mappings. We first show that any mapping H: z* =f(z, w), w* = g(s, wJ sending a hypersurface M of the form (1.1.1) into another such hypedurface M can be represented as the composition of standard transformations and a’ linear-fractional transformation. Clearly, H can be written as H = Hi ‘( H*(H,)), where H, is a standard normalization anti H*

IV. Hotomorphic Mappings and the Geometry of Hypersurfaces

*‘> * sends M* G If$M) into I$* = H,(M). By the lemma in $2.2, fi. =I H,(Jb),

where ‘R,. is a libar-fractional transformation with the set of mrtia data w*. Thus H = (H; * that is, H is the composition of a ‘standard. ’ i reduction, a linear- ctional map, another standard reduction, and the inverse of the standard nornigli&ion. The representation of H in this form is unique, and so it uniQuely d&fines the set w . * The lelements of o* are called the

parameters of the inap, and o? the set of parameters of the map H. From the lemma in $2.2 and properties ofH, it can be de&ced that all the elements of o* apart frdm r* are the same as the corresponding elements of the set 6f initial data of H, that is, ;hat the fol owing system of equalities holds: I \

i a.i af ag iGo

= A*u*, dw = i*U*a*, aw = c~*A*~. 0 0

.

2

In general, the equality Re$ 0

= 20*IZ*2r* does not hold. Re$ is ex- 0

pressed in terms of elements of o* and ~- a2F (i,k=I,2,...,n-l),where? I ~ziazlrO _

is the function in the equation of M: v = (z, z) + F. It turl hypersurface of the form

; out that if .G is a

(2.3.1)

= 2a*I*‘r*. This

v = (z,z) + F,

.a22 where - aZ,az, o

=0 (i,k=l,2,.. , 829

n -- l), then Redw2 _. .

means, in particular, that if H is a normallzmg mappmg, then the set of its parameters completely coincides with the set of its initial data (sic 0 1.2). Hence the sets of parameters and of initial data are the same for any mapping sending a hypersuiface of the form (2.3.1) into the same hypersurface. The set of initial data (o for such a mapping and the set of initial data & of the inverse mapping are expressed in terms of one another in the same way as the anaiogous sets of a linear-fractional mapping

5 = u-1, (2.3.2)

$3. Chains

We introduce on a hypersurface a family of curves called chains. As we r;h;\ll see later, this- concepl works well in the study of holomorphic mapping:, of hypersurfaces. A family of chains was constructed in the two-dimensional i SC by E. Cartan [lo], and in the general &se by Chern and Moser [I !].

174 A.G. Vitushkin

. 3.1. Defirriti, of a Chin. A chin on a hypersurface ~$4 $s a maximatly continued curve which, after the hypersurface is reduced tb,normaJ form in a r&hbourhood of any point on the curve, by a certain norn$ization is defined Jocaily by .z = 0. 13 = 0.

CJearJy, a chain is an analytic curve, which at all point; is transversal to the. complex tangent space. Geometrically the construction of such curves is not simple. Fefferman [ 17l constructed an exampie of a hyp&urfa& with a nondegenerate Levi form, on which chains are constructed as spirals; a chain, whik keeping close to the complex tangent and slowly decreasing the diameter of the loops, approaches a point of the surface. It is still not clear whether a chain can approach a point while touching the complex tangeat space but without increasing its curvature.

We show that for any point <EM and any d&e&on transversal to the complex tangent space we can find a unique chain through this poiut and tangent to the @en direction. We choose a coordinate system with ; as origin andinwhichi\Bhastheformc=(z;z)+... (see(l.l.J)).Forahypersurfah in this form we can regard a change of yariables straightemng‘a chain as a normalization. Conversely, any normalization distinguishes a chain. A not&J- ixation I& with the set of initial data o = ( U, a, 4 cr, r) can be represented as the composition of three transformations: the linear transformation z* = : + aw,W,= w, the standard reduction 11, (see §2.f), and a certain transformation H,. The tangent map for H, has the form z* = 1 Uz, w* = aliZw . H sends' one n6rma.l form into another, and thewond ekment in its set of init% data (the vector a*) is zero. Consequently, by Moser’s theorem, H Jinear-fractional transformation leaving the line z = 0, t = 0 fixed. 6,” an” normalizations with the same second e,kment in the set of initial data (the vector a) send one and the same curve into& Jine z = 0, r = 0. This curve is tangent to tJxe direction ( - U, 1) at the origin; this means that in any direction there is no more than one chain. Such a chain exists, since the vector u in the construction of the normalization can be chosen arbitrarily.

Thus, the chain through a fixed point of a hypersurface is uniquely defined by its direction at this point. %or a hypersurface in the form c = (z, z* ) + . . . the direction of the chain through the origin can be characterized by the vector (4 1)1 where 4 = (a,, . . . , u. - t ). If a hypersurface M is defined by an equation F(z, 5, w, W) = 0, tkn at {EM we define the direction of the chain T through < by a vatof a; Jy4ng.m the Complex tangent space to the surface at this point. We d&e Q as tht projection onto the complex tangent space of the tangent vector to the chain at [, normalizd so that the length of its projeztion onto the normal to the complex tangent space is 1 (here we mean the normal lying in the t&gent space of the surface). We see that the vectors II and - II define one and the same chain.

3.2. chi on odirics. ‘On the hyperquadric r = ( 2, z) any chain through the origin can be obtained from the. line z = 0, 17 = 0 by an auto morphism of the quadric (see (1.2-l)). Hence, a chain on a quadric is a circle (or a

- ‘.

IV. Holomorphic Mappings and the Geometr) ol Hypersurfaces

straight linlj. It is the intersection of the quadric with a complex line. The chain in the direction (a, 1) has curvature x = 2(a, a>/[1 f iu, I2 liz. The angle x ]

between the direction of the chain and the complex tangent space is the same at all pointi of the chain, and is found from cot a = Ial.

In the sequel we shall need another type of hyperquadric, namely, hypersurfaces of the form 1 - lw12 = (z, z), This hyperquadric and the hyperquadric o = (z, z) considered earlier are obtained from one another by a linear- fractional transformation. The mapping sending IJ = {z, z) into 1 - Iw*12 = (P, z*) has the form z* = 2z/(i ‘+ w), w* = (i - w)/(i + w).

The general form of an automorphism of the hypersurface 1 - lw12 = (z, z > that leaves the point z = 0, w = 1 fixed is

Z' lUiz-4+-1)b w*-l- ai2(w -- 1)

=-- ----, ,

6 6

where 6 = 1 + (z, a) + (l/2)(1 - oJ2 - (ui u) -t- irfiw - 1). As before, Q = + 1. I > 0 and r are real numbers, a is an (n - l)-dimensional vector, and I/ is a

matrix satisfying the condition

(Uz, Uz) = a(z, z).

On the hypersurface 1 -- (~1’ = (z, z)’ the chain passing through the point = 0 w = 1 is again a circle (or a line). The curvature of the chain with the

krecbon a is given by x = (1 e (a, a>)/[1 + \c-I\~]‘!~, and the.aagle a between the chain and the complex tangent space is found from cot g =‘IG/, as before. This ‘condition defines a system of differential equations on the chain.

3.3. The Linear Normal Parameter. Suppose that a hypersurface is reduced

to linear normal form so that a segment of a chain y goes into the line z = 0, u = 0. In this case the values of the coordinate u define a certain parametrization on this segment of y. By considering different reductions of M to linear normal form, we obt%n on each chain a certain family of parameters. Any parameter of this family will be called a linear normal parameter. As we can easily set these parameters are expressed in terms of one another by linear-fractional transformations of the real line, of the form ti* = oI.~u/( 1 - nu) (see the theorem in $1.2).

For s # (n - 1)/2 the transformations preserve the orientation of the line. Recall that s is the number of positive eigenvalues of the Levi form. Thus, for s # (n - 1)/2 a certain orientation is distinguished on the chains. A first order differential form that iS invariant under biholomorphic transformations is delined in terms of normal families on hypersurfaces. If a hypersurface is given in linear normal form, th.en at the origin, it is written in the form kdu, where k is the norm of F22, that is, the square root of the sum of squares of the coefficients of the polynomial written in symmetrized form. This form characterizes the non- sphericity of a hypersurface: if it vanishes on a set of positive measure, then the hypersurface is spherical [42]. ’

176 A.G. Vitushkin

$4. The Equation of a Chain .

If a hypersurface M is defined by v = (z, z) + . . . then (2, U) can be regarded as a local system of coordinates ori M. In the coordinates (z, u), chains through the point z = 0 are the integral curves of a system of differential equations

a22 GFA z& ( >: with a realianalytic right hand side. Variants of this equation have been given by Moser [ 111, and Burns and Shnider [7]. There are also other interpretations of a family of chains (see [ll], [15] and [17]). FeffermanYconftruction [17] is interesting. He constru,cted a special bundle (with the surface as. base and a circle as a fibre) and a metr!c on this bundle that is invariant under biholomorphio transformations of the surface; projectio~ns of the light rays of this metric onto the surface are chains ([6] and [41]).

4.1. Straightening a’ Hypersurface Along an Analytic Curve. We explain how the equation of a chain arises. We fix on M an analytic curve y, which a; all’.& points .is transversal to the complex tangent space. We say that a mapping, straightens. 3’ if it setids points of this curve into the, lips. J = 0, ti?= 0. -By analyzing the,c&struction of a reduction to. riormal form, we can construct ‘a maps&g tharst&ighte& a giGen cu;ve.and sends the hypersurface into the form

c = (z, z) + 1. F(z, 5. u), k,1>2

where trF 22 = 0 and tr3 F33 = 0. It turns out that, iq general, the condition tr” FZ3 = 0 is ‘not fulfilled. We call the’constructed mapping a normalization strcri~~ht&uJ ;J. or a normulization of the surface along y. We emphasize that for a hypersurface of this f-u-m any mapping sending one hypcrsurface into another of the same form and leaving : he l.ine z = 0, t’ = 0 fixed is, as in the case of chains, a linear frdrtional trdnsf:;, matlon.

Clearly, a curve I5 a chain if and only if there is a normalization straightening it. after whrch the condition trZ F Zi dltftrcntial equations on the chain.

= 0 holds. This condition yields a system of

4.2. ‘I’he Equation of a Chain in the Natural Parameter. Let M c @” be a * h!lTcrsurface with a non-degenerate Levi form and defined by an equation

( ( ,; 1:) = 0 WC denote by (7 + z)~ the Levi form of M at thl point c. Let us state ‘Y(;:,’ ,)&ci~cly what we have in mind. We transfer the origin to [ and by a ;I,!j!:jp tr;mqformation reduce the hypersurface io the form u = (z, z)~ ,. i;- : -. J. II) (see (1.1.1)). This transformation is defined up to a unitary change of

:::r~~ibl::$ in the plane w = 0; this limitation is not essential, since we shall speak 1 .;!iy ;ri;.)llt the value of (z. z)( on one or another vector in the’complex tangent

IV. Holomorphic Mappings and the Geometry of Hypetsurfaces 177

. s;ace to-the surface at [, and this is independent of the choice of coordinates. We CPA

can regard (z, z)[ as the restriction of the form c ---= i,kd&%

d[, A dr;, to

the’cemplex tangent s!:rce of the hypersurface. ’ Let y(s) be a chain C,I’ a hypersurface M (s is the natuqal parameter), i 7 Y(S) a

point of the ch& a, a ;r in the compiex tangent sp?ce defining the dIrectIon very., of the chain at this .?oint, and (i, i), the Levi form qf M at. y(.s).

In certain situa:rons, when speaking of a hypersurface, It 1s necessary to characterize the parameters of analyticity of the functions defining the hyper- sllrface. To ihis end, we introduce the class M(& m) of hypersurfaces M satisfying the following conditions. The hypersurface M has the form (1 .l.l).

The corresponding function F(z, 5, u) is holomorphic in the -polydlsc (lz,l < 6(k =,1,2,. . . . ,‘n - l), 1.5k,1 < 6(k = i,2,. . . , n - l), (ul < b). Here thevariatiesz,,. . .,z,-~,Z~,. ..r&,-I, and u are considered as independent

complex variables, Also, I F(z, 5, u)l < m in this polydid.

Lemma. .Let MEM(& m), (see 97.1) and let y(.s) ;be a chain on M. Then in a small neighbourhood of the brigin y(s) sptisfes rhe equation

2(a,, as)s y”(s) = (1 + Ia,1 )

2 l,2 V(s) + rl(Y, Y’i

where ~(7, y’) is a vector such that \q(y, y’,)j < q*(6, m), and U* js a ,fiou’tion

satisfying the condition lim $(a, m) = Ofor every 6 > 0. The size of thr Ggh- m-+0

bourhood is ulso determined by (mi 6). If M is (I quadric, then q(s) s 0;and the curvature of the chain is 2(a,,p,)S/[ 1 + (aj2]“2 = const. I f M is given in linear normal form and ~(0) = 0, then ~(0) = 0.

Let the direction a, be such that jao( is sufficiently large and (a,, a,,) 5 In,j2. In this case, by using the equation it can be shown that in a large interval of variation of s the chain is close to a circle. Here we mean a circle that is a chain of the quadric v = (z, z)~ and is tangent to y(s) at the origin. The chain y(s) makes approximately Ia0 I-many loops near this circle. For a hypersurface’ with a

positive definite Levi form the cpndition (a,, a,,) y laOI h&!s; therefore, if the curvature of a chain on such a hypersurface is large at a point, then near the c&responding circle it makes several loops.

4.3. Derivation of the Equation. Let H be a normalization straightening a chain y. We fix the set of initial data for H as w = (E, a, 1, 1,O). ‘We recall that H = H (R (H,)) (see §2.2), where R, is a linear-fractional transformation. with the sa$e s:t of initial data. and H, is the standard normalization. By the lemma in 92.2, H and R,( I$) coincide up to terms of the third order of smallness at the origin. Therefore, the curves H-‘(0,~) and (R,(H,));‘(O,u)= H;‘(R;’

(0, u)) have the same curvature. Here H, - l is the mapping inverse to the standard

47% . A.G. Vitwhkin

normalization of M. We first write out the curvature x0 of the chain y+ L H,(y). The mapping R;’ sends points of the line z = 0, u = 0 into y*. R; l(u) has the form R;;‘(u)= &u/Cl -i(a,a)ul,‘whert<,isthevector(-a,,. . .,-a,-,,~) (see $1.2 a.nd the formula for the transition to an inverse mapping in $2.3). From the form of R,:’ we find that x0 = 2(a, a>/[1 + \u]!]“~. Since y = H-r(y*) and Hi ’ - E + H* (see the lemma in $7.4), we have y” = ikr + q, wheri q is a quantity whose modulus is bounded by the second derivatives of He (see ]ernma in $7.1), and so y(s) satisfies the above equation. ‘. _

$5. The Circular Normal Fork

The parametrization of chains introduced jn $3.3 is applicable to a study of local properties of hybrsurfaces. As is clear from the transition formula a function rcillizing a change of parameter has a singularity on a chain. Thus, b&h parilmeter and substitution are defined on only a part of the chain. This leads to an essential difhculty in studying chains in the large. By changing the form of rzpresenring a hypcrsurface we introduce a new parametrization, which is more convenient for a sludy of chains in the large. .

5.1. The Form of a Hypersurface in Cireufar Coordinates. We consider the space e’ with the coordinate functions z,, zs, . , . , z,- , , p, 0, where z z em.9 a-, are complex and p, 0 are real coordinates. We can-regard zt , . . . , z, -Ii, w = pe’o as Tocal complex coordinates, that is, En hasa natural complex structure, and so we can talk about holomor@lic transformations of C”, We emphasize. that the coordinates (2, w) in c” will always be understood as local coordinates. When speaking of one or another many:valued function defined in the coordinates (2,-w) we shall mean ‘some continuous branch of it. Whether it is specific or arbitrary will be clear from the context.

We consider a class of hypersurfaces.of the form . ‘1 - pz = (2, 2) + F(z, #, 0)s (51.1)

where F is a real-analytic function defined in a neighbourho,od.of z = 0,

a2F. . 8 = Oe, such that F(O,O, 0,) = 0 dF(O,O, 0,) = 0, and - az,ah io,eo, =

0.

A mapping,frorft c” ‘into @ sends any such hypersurface into one of the form

1 -p2= (z, z> + c @&,(Z, 2, 0). k.122

(5.1.2)

Here the G+, are polynomials of degree k in z and 1 in Z, with coefficients depending analytically on 0. It is assumed that their coefficients are defined on a

iv. Hoiomorphic Mappia&&htbc&tryofHypemrCua 1?9 .

,

. .,. in 2, i and 8’ i 6; &i& &&erg& & so~‘&hbourhood ‘of tb6 $mt i = 0, 5--O,@-eo= 0. The UXIIIS 02 jr &a, and.& satisfy the conditions

;. ., . . . .. ::;.-;:,, [email protected];3=Q trJ:fDjj=O ,, , .. -‘a

.(for the definition of the operator tr see $i.l): ” ’ ~

[ i ‘- , ; As regards hypekurfaces ofthis.f&m we say that they andtheir equations arc

&f&j in && j&p& form,’ &&& ‘the ~&d% &dt~ n~rtd CO- otdihates. The cm&po&ih$ mapp* &‘be a&d a normahzatio~~~ before: The circular normal form of a hypersurfike can be obtained from’the linear nor-ma1 foti. -I-he mapping Ro: . . . . . . _. ,:

,&.- .;aLi’-w ‘. -a = - ‘. *

,. (51.3) i+;iu’. .i + W :.

sends a hypersurfacc ,of the fo’rm ,y - .-‘L(z, Z) + .^.,-. s:(& $i.tfj intq’one of the

form 1 - /w*\2 = (z*,z*) + . . i (see (51.1)). The @verse mapping sends a

hyperjurfacc of the second type intoone of the fid. It can be shown that both dii and inverse mappings pksfxve’tiie normality of the form. that. 6, a hypcrsurface defined in linear form is taken by R. info one defined in circular normal form, and cmvetseIy. .

52. Tk In& Data for a circahr Normaliitioa. By analogy with the linear case ,we associate kith any normalization H: z + =f(z, IV), w+ = g(z, w), a

~ofinitial~~o=(L’,4,‘,u,r)dtfioed’bythesystem:

In contrast with the linear case. r is defined in terms of the imaginary part of the second derivative, rdther than the real part.

For any hypersurface and any set of initial data we can find a norrnaliiation with the given set of initial data, and this normaliition is-unique. This assertion

‘.-& easilv obtained from Moser’s theorem (see $9 1.2 and 1.3). The essential difkedce between linear and circular normal forms arises in the transmon formulae: if a hypenutface is defmed in circular normal form. then the normalization with a = 0 in the set of initial data w.tums out to be better in ? certain aen*. than in the case of linear normal forms.

.

5.3. ‘Ihe Form of a Substitution M a Chaii . @y map R that !caves

the line r=O,p= 1 fixed. and sends a hypk&fii& of the form (5.1.2). mto a

‘e A.G. Vitushkin * .

similar hywpurface, is a composition R = &,{ R( R, I)), where R is a linear- fractional tiansform’ation sending a hypersurface given in linear normal fo& inw..a similar hypersurface. and leaving the li’ne z = 0, u = 0 fixed (see the theorem in $1.3). ‘Therefore, in complex coordinates ?7 is written as

z* = ,,/@fijUz, w* = Q(w),

where U is such that ( Uz, Uz ) = a(~, z>, and Q(w) is- a linear-fractional ‘transformation sending [WI = 1 into itself.

It is clear from these formulae that a mapping sending one norma] form into another is holomorphic in a neighbourhdod of the whole line z = 0, p = 1. This property of circular coordinates gives a natural construction for the continuation of a mapping along a chain.

5.4. Continuation of a Normalization Along a Chain. Let yi and yZ be two arcs of a chain y on a hypersurface M that have a common point. Let H, and ‘HZ be two normalizations of A4 straightening y (that is, sending y into the line z = 0

. p = 1) defined in neighbourhoods of y1 and yz, respectively, and sendl*ng A4 intd a circular normal form. Then in some neighbourhood of the common point of these arcs we ha\e fZil = R”(H,), where R is a linear-fractional transformation (see 95.3). But since R does not have singularities on the line z = 0, p = 1, i( H2j is a continuation of H, from y1 to y2.

Theorem. Let M be’s hypersurface of a complex manifold that has a nondegenerate Levi form. Then every normalizing mapping that straightens y t M and sends ti ifto circular normat form can be continued indejinitety along y. Any two normalizations H, and H,. that straighten one and the same chain .are connected by a relation Hz = 8( HI), where R is a linear-fractional transformation (see Q;5.3) [38-j.

. Let us clarify the assertion of the theorem. We fix a chain y and a normalization H at some point XE y that straightens this chain. We denote by H the family of all norma’lizations at points of M that can be obtained by continui;g H holomorphically along-y. The family HY is called a normalization along the chain y.

96. Nornial Parametrization of a Chain

6.1. The Circular Normal Parameter. Let H: X + e be a normalization.of a hypcrsurface M along a chain y c M. The map defined by the corresponding family H, sends y into the interval (O;, 0,‘) (- co I 0, I 0; -< co) of the line z = 0, p = 1; The mapping H - ’ can be continued holomorphically to a neighbourhood of this interval, and so it associates with every point of this

. IV. Holomorphic Mapping and tkt Geometry of Hypcmurfaces 181

interval a poir&. # - 1 (0, 1,O) of y. We observe that if one.of ihe points 0,) 0; is a finite point d the Coordinate axis, then H-l cannot be automatically continued locally biholomorphicaliy to &y ncighbourhood of this point.

The continued mapping, which we denote by the same symbol H - ‘, sends the interval (S, , e,+ ) of the line z G 0, p = I onto the whole chain. The restriction ,

of H - ‘. to this inter&l can be regarded as a parametrization of y: .this mappiclg associates with each value of the parameter @a = Q from (0,) 0; ) the point H-‘(O,l,O) of the chain..

Such a’ parametrization is called normal, and the corresponding parameter a normal parameter.

If a chain is not,.closed and has nd multiple points, thin the interval of values of the parameter covers the chain y univalently; if the chain is closed, then the covering is infinitely-valued. We recall for comparison that under a parametrization defined by linear normal form (see $3.3), the interval of values of the parameter covers, in general, only .part of the chain.

As an example we consider normal parametrizations of chains of the quadric 1 - lwl2 = (z, z) passing tlirough the point z = 0, w = 1. Setting 1 w 1 = p we see that the hypersurface is defined in *circular normaI form. The branch of the mapping p = 1 WI, 0 = Arg w, z+ = z sends, ‘he circle yO: z = 0, e = 1 into the line z* = 0, p = 1. Therefore, y0 is a chain, and every branch of the argument of w is a normal parameter on it, and so varie’s from - 00 to + a. An arbitrary normal parameter on this circle can be obtained from those indicated above by a linear-fractional substitution sending the circle onto itself. Under a single circuit of y0 the normai parameter changes by + 27~.

An arbitrary chain y on the quadric in question can be obtained from ya by a suitable automorphism (see $3.2), and so under a single circuit of y each normal parameter of it changes by 211. Similarly, under a single circuit of a chain on the quadric v = (z, z) the normal parameter changes by 27~.

6.2. The Formula for Cbanging a Parameter. Let P denote the projection cf

6” into CR that associateswith the set (z, p, 0) the point (z, w = pe”). If M c C” is a hypersurface of the form (51.1) then P(M) cari be written ‘as 1 - 1~1’

I= iz.z> + c @k,l( z, Z, O), where 0 is a b&h of Arg w on 1wl = 1. The

interval (0’:‘6*‘) on the line z = 0, p k 1 is a chain on M. The ‘projection P

winds this interval ont6 the circle z = 0, [WI = 1. The’ multiplicity of the

covering, that is, the integer part p+l-p-(

2n: 1 , is independent of the choice -._

6f the normal parameter. In otherLwords, the n&&r of complete circuits m.ade by the point P(0, 1,O) as 0 varies in (o-, O+) is an invariant of the cham.

Lemma. If 0 and &* are two normal parameters on one and the same chain on a hypersurface M, then they are related by eie* = q(e”), where q(w) is a ’

. linear-fractional transformation sending jw/ = 1 into itself: If 0(x”) - 0(x’)

;. ..,’ -,: .. ,’ . : 63, The Ioitial Data of a ‘Parametrization- Now; having introduced, the

circular normal form, we xum again. to .hypersurfaces of the form .u F (5z.j + . parameter

. . . . When considering a hypersutfe ot thistype and a notmal on thirchains on this. hypersurface;we have to speak of a circular :

normalization of a hypersurface of this. form. In,this case the nor&l&g .map fez* Ff(Z, w), w* = ,g(z, w) of M can be re@esented by two compositions of. the form H = H&R,) and H = R,( Hd); .where & is a transformation sending a linear normal form into a circular one (see ga.lk,‘H ,̂ is a ~circtiku normalization of R,(M) with some set of initial dataab’=(U,,,s, A, a; r), and ‘H, is a linear normalization of M with a certain set CD+; In de6ning the initial data we saw to it -I that we can-now say that o = w*. We.shall call tbe set w the set of initial data of the normalization H. The elements of this set and the derivatives of the functions f and B are connected by the relations

;)r z, I = - 2iAU, af

Go=. I

-2im4

47 Kv, I

= 2id2, 828 Irn= I

= - &12r. 0

Let H be a circular normalization of the hypemurface M that straightens a chain y. This mapping defines a normal, parameter 8 on p. If M is given as ’ u=(z,z)+... or1 -~w12=(z,z)+ . . or the point z = 0, w =.l,’ respectively, thG

and y passes through the origin asetofinitialdatatorHis

determined. The parameter 8 is uniquely determined by the triple 0, set. We call this triple the set o/initial data of the paramet&ation. A, r of this

On any analytic curve y lying on the hypersurface A# and transversal to the complex tangent space we can define in the same wriy as on chains a fhmily of normal param&zations (linear and circular). A normalization of M that

. straightens the curve y (see 66.1) sends M. into a hypersurface of the form

* *= (z,z) +J/i,( z, Z, u) The transformation R, sends the resulting hypef-

surface into one of the form

1 -p2=: <z.z> + c e&gee), kIT2


The compositi&‘H of these maps can be continued indefinitely along y and thus defines a normal parameter 0 on y, We note that in contrast to the normalization of chains, when straightening arbitrary curves we cannot assert that a normalization with a given set of initial data is unique. But if we specify what curve is straightened by a given map, then the set of initial data o = ( U,a, 1,e, r) uniquely determines the map. Thus the triple (a, A, r) uniquely defines a parametrization of the curve just as in the case of chains.

6.4, Normal Parametrization and the Continuation of Mappings. Let there

be defined a map H of a hypersurface M into a hypersurface M* that sends XEM into a point X*EM *. Suppose a chain y c M passes through x and let y* = H(y). Let cp and cp * be normalizations sending M and M* into linear normal form and straightening y and y*, respectively.; We write H -as H = ‘p* - r ( R (cp)). Here, R is a linear-fractional transformatron leavmg the line z = 0, u = 0 fixed (see the theorem in $1.3). If cp and cp* are defined on large portions of y and y , * and R on a large part of the line z =‘O, o = 0, then the map H, which is defined, in general, in a small neighbourhood of x,‘can be continued to a large segment of y. However as is clear from the formulae for’ the transformation of linear normal forms (see §3.9), u = l/r is a singular point. Por large r the mapping R is holomorphiconly in a small neighbourhood of the origin, and this turns out to be an obstacle to its continuation. If cp and q* are circular normalizations, then the correspcnding R is holomorphic on ihe entire line z = 0, p = 1. Circular normalizations can be continued indefinitely along ,

chains. Thus, if, for example, the interval (Q* -, O* + ) of values of a normal parameter of a chain y is the whole line, then the map H can be continued to the whole chain y. A chain on the sphere in C” is a circle. In this case (O*-, O*+ ) is the whole line, and thus a locally defined map of an arbitrary hypersurface into a sphere can be continued to the whole hypersurface.

It is known that a locally defined map of one hypersurface into another cannot, in general, be continued (see [9] and [3]). In these examples a chain y, along which a mapping H cannot be continued, and y* = H(y) are constructed such that the variation of a normal parameter on y is infinite, and on y* finite.

63. The Equation for Passing to a Normal Parameter. We consider a family

ofhypersurfacesM,: 1 - lw12 = (z,z) +a(z,z)3,wherea~R.Thecirclez=0, IwI = 1 is a chain on each hypersurface of the family. However, as Ezhov [12] has shown, its normal parametrizations are different for different a, and 01= l/24 is found to be critical in the sense that for e = l/24 the variation of a normal parameter on the chain is 2x, while for a < l/24 it is infinite, and for a > l/24 it is strictly less than 2n.

In general it turns out ([12], [23], and [14]) that an estimation, of the variation of a normal parameter on a chain requires an analysis of aJ3. For

A.G. Vitushkin - 184

hypersurfaces of the form

1 - lw12 = (2, z> + 1 Q)kl(Z, 6 t), r; k.122 I

where-t = arg w, tr@,, = 0, and tr* @23 = 0, the transition function 0 = s(t) from a parameter t to a normal parameter 0 satisfies the equation J

a,+Q(g’)* + 2g”‘g’ - 3(g”)* + (g’)4 - (g’)* = 0,

where a = (:) ([12] and 1141). An analysis of this equation ,and of the whole construction of the reduction

enables us, in a number of cases, to give an estimate of the length of the interval of variation of a normal parameter. The hypersurfaces most studied are those with a positive definite Levi form. If a chain on such a hypersurface makes at some point a small angle with the complex tangent space, then it turns out that the interval of variation of a normal parameter on this chain is large. This will be discussed in more detail in $8.2. In particular, if the angle of inclination of the chain to the complex tangent space decreases to zero as a point moves along a chain, then the variation of any normal parameter on this chain is infinite (see [23]). The need for estimates of this sort arose in connection with problems of continuation of holomorphic maps ([38] and [14]). The estimation of the, term ?a3 involves complicated calculations. A property of chains necessary for the continuation of mappings will be stated differently (see gQ7.4, and 8.2). This will ,-..L,- __- I-. -..- 13 I-I- . . . . . CUUUIC us LO avoia iaoonous calculattons.

$7. The Non-Sphericity Characteristic of a Hypersurface

A connected hypersurface is called spherical if in a neighbourhood of each of its points it is equivalent to a quadric, that is, for a suitable choice of coordinates it can be written as v = (z, z). Otherwise a hypersurface is said to be non- spherical (when speaking of sphericity or non-sphericity it will always be assumed that the hypersurface is connected).

From the point of view of the local structure practically everything is known about spherical hypersurfaces: any normal form of such a hypersurface has the form o = (z, z), and the automorphism group of a quadric as been known from Poinca&s time (see $1.2). Therefore, as arule, our future discussion will concern non-spherical hypersurfaces. It turns out that in a number of problems it is essential to know not only whether a hypersurface is spherical or not, but to have the possibility of characterizing the “magnitude” of its non-sphericity. For this purpose we introduce a special numerical characteristic.

7.1. Estimate for the Radius of Convergence, and the Nor& of tile Defining Series. In order to write estimates for the radius of convergence and for the


.

norms, we introduce appropriate parameters for the hypersurface and the normalization mapping. We shall describe the analyticity of the hypersurface M as before ,by a pair of positive numbers 6 and m, writing ME M(6, m) (see section 4.2).

We characterize the set of initial data o = (U, a, 1, C, r) of a normalizing mapping by a number v, writing [WI it’ by which we mean the system of inequalities {l/u III U 11 Iv, Ial I 0, l/u Ii.50, Irl lo}.

Lemma. Let M i M(6, m) and JwI I v. Then a-normalizing mapping H, of a hypersurface M is representable as KU = R, + H(z, w), where R, is a linear- fractional transformation with initial data OJ (see tj 1.2) and H(z;w) is a mapping holomorphic in the polydisc.

(lzkl < S*(k = 1,2,. . . , n - l), lwj < S*} .

and satisfying IH(z, w)i s m*(lzli + [WI*) in this polydisc, where 6* > 0 and m* = m*(d, m, v) is such that lim m*(&m, v) = 0 for every 6 > 0, and 6*, m*

. m-0

depend only on 6, m, v, and the matrix of (z, z). The hypersurface obtained as the result of the normahzation belongs to M@‘, m’) (8 and m’ are also defined by m and v).

The assertion of the lemma is valid for linear and circular normalizations of hypersurfaces given in the form v = (z, z) + . . . or in the form 1 - lwl* = (z, z) + . 1 . . The proof is obtained from the fact that every normalizing mapping is a composition of standard normalizations, of their inverses, and a suitable linear-fractional transformation. The standard reduction is constructed quite concretely, and so the proof of the lemma reduces to purely technical estimates. This thankless task is carried out in [4]. From these estimates it is not difficult to obtain, in particular, the following result.

Theorem. Let M, M* E M(6, m) and let H(c) be a biholomorphic mapping .

defined in some neighbourhood of the origin that sends points of M into points of M* and is such that laH/tXlo cm, and la2H/a[*10 < m,. *Then H extends holomorphically to the ball Ill < b*, and in this ball I.H(c)l < m*;where 6* and m* depend only on 6, m, .nb, and the norm of the matrix of (z, z) (143, [39]).

i

7.2. The Non-Sphericity Characteristic. Let M be given by an equation A((; r) = 0. Fix [EM. By a unitary transformation we map [ into the origin and M into a hypersurface of the .form y = (z, z) + F (see (1.1.1)). This transformation is determined to within a unitary change of variables in the plane w = 0. Let U be such a unitary change of variables. Under U, the hypersurface v = (z, z) + F goes into a hypersurface of the same form, and we write it as v = (z, z>,, +.F,. Then we carry out the standard normalization of this hypersurface and write the result as v = (z,z)c + F,,,.

We associate with a point i of M a number Na,,,([, M), which we shall call the non-sphericity characteristic of M at <. N,., is defined at c provided the

186 A.G. Vitushkin %

&. corresponding hypersurface u = (z, z) + FU,c belongs to M(6, m) for all U. 1 S.i this case we set

where the maximum is taken over all (z, Z, u) satisfyihg 1~1 5 S/2 (k = 1, j 2, . . . n - l), l&l 5 b/2@ = 1,2, . * . , as independent complex variables).

n - l), I u 1 5 S/Z (here z, 2, IJ are regarded

For all points of the quadric u = (z, z) the non-sphericity characteristic is zero.. If the characteristic is zero at a point CE.M, then the standard normal- izationsends a neighbourhood of this point into a quadric, and therefore every non-sphericity characteristic is zero at all points of a neighbourhood of [. The lemma in 97.1 ensures that this neighbourhod is rather large. By progressively changing the point and repeating the argument, we find that the characteristic is zero at all points of the hypersurface M.

Let M be defined by A((, c’, = 0, and let MO be the part of M lying in the ball lzl* + iwl* 5 1. We shall assume that M,, is compact and connected, and the matrix bf its Levi form belongs to a compact set of non-singular matrices, which wk as&me is fixed. Fix two pairs of positive numbers 6, ti’and S*, m*. It is assumed that N,,, is defined at some point Co of the hypersurface M, and that N f,,,* is defined at all points of M,,.

Lemma. If Na,&, Mu) = N,, > 0, then Np,,,*(C, Mu) > Nt at each point CE M,,, where N,* > 0 is a finction of 6, m, 6*, m*, and N,,,

The family of hyperkfaces of the type M, wtisfyia the conditions of the lemma is compact. Therefore, assuming that the assertion of the lemma is not trut, we can find a hypersurface of the type M, such that NJ,&, M,) 2 IV, at some point [ of this hypersurface and N s,,&, MO) vanishes at some other point r which, as was mentioned above, is impossible;

We state a problem which arises in connectibn with the.definition of the non- iphericity characteristic. It would be r$ce to ha,ve 51 definition af a measure of non-sphericity that is not linked to (6, m). It is natural to try to define non- sphericity as ihe.lower bound of the deviations of a given hypersurface from spherical hypersurfaces. The first question that arises in this conn&tion can be stated thus, for exaniple: we fix a non-spherical hypersurfw and some compact. part of it. Can this compact part be approxim+d with any accuracy (in the metric of deviations) by a spherical hypersurface?

7.3. The Vpriatioo of the Cbaraeteristic Under a Mapping. sends a non-spherical hyperstiace M u

Suppose that H

into A: u = (z, z} + F(z, i; u) of the form (1.U)

= <z, z) + &z, Z, u) with H defined by z -+ p, w + p* i, where ~1 E R.

Then F(z, 5, 14) = -$ F(pz, p5, p*u). If M is given in normal form, then the

expansion of F begins with terms of degree at least 4. Hence IPI s pm for small p

IV. Holomorphrc Mappmgs anQ tne ueomerry 01 nypersunaus

and SO N,,(O, fi)‘*c*o a~ p sphe&ity"of~~~ hypps&aa

3 0. Thus, a large “dikilion? def{%eS _ ttik non’- 1, . . .fi ,I -. _’ : r . ,I . 2’ ./ i ’ ‘: ;,‘., z .‘;-

‘We consider +tti ex&hples of maps ,for. which it tums,out- that ‘the variafiondf the characterlstidaf a%@ersurfa& can be estimated. Let ## be a m+@ltg oE,M into .$ sukh %&at the, first,.‘ati &and aider ;p&tkl ‘derivatives ,of Hzancf’-its inve+ ai the tigilr are b&mded by a cons& m;. W& &y that such,mappings bklotig to the class W@, m, mg).: Bar fixed S and lit the, class of .hypersurfaccs M(&.hl)‘is; com@ct. It turns out that $Z(6, m, m,,) for fi&&mO~is also compact @see the,theorem in #Xl).-It fullows‘from%he coinpactness.of these klasse_s that if N&,(0, M) > lil > 0, then for ii? = H(M) (HE H@, m, mu)) Nd&(O; M.) ih % least N * > 0; where N* is 8’ functi6n of (6, m, ni,). iJext suppose that-M ,and F are hypersurftices with a pbsitive definite I&i form; and that R sends M into M and has the form z* = AUz/(l - rw), w* = A*w/(l - rw) (see $1.2.for the definition of U and A), where Iz < v and Irl;< u. The group of matrices U preserving a positive definite Levi form ib compact, and so the first and second derivatives of R for fixed u and a fixed Levi form can be bounded from above. If we bound d from below by assuming, fofoi kxa?ple, that l/A < ti; then a’ siniilar bound kolds also for the derivatives of R - l:. In this c&e, as is clear from tlie above exiunple; the’ variaiion of. the non-sphericity character&tic is bounded from both above &d below. It was shown at the beginning of this s&tiont$ai under a 1-e dilation, the oon-sphericity characteristic deereas&; Thus, without in$osing any lower bound on A, it can be said that Nd,m(O, 5) cannot be much less than N,,(d, M). Indeed, if Iz is small and. the mapping R significatitly decreays the non-sphericity characteristic, then the composition of

:I R and the dilation- z* = iz, w* =-w tionsiderably decr&ses the non- ’

: 12

I. sphe&ity. The latter is impossible, because this composition. is a linear- fractional transformation of the same forr& for .&hich L is equal to 1. The following assertion is easily dkriied from what has been said.

Lemma. Let M and‘ fi b4 strictly pseudoconuex bypersurjizces of the class M(& m) fir which ‘he cbaracterWc Ns,E is defied ,and N,,(O, M) > v > 0. Let H be a mapping sending M into M that is defined by the’ composition H = Hi(R(H,)), b w ere’ HI, H2 E H&m, u) and R is a linear-fa@onal trans-. formation of the&m e* = A Uz/(l 7 rw), w* = A*w/(l - PW), where A c u and Irl < u. Then N&(0, Ma) > N*, where N* > 0 is a function of& m, v, N arid the norm of tbe matrix. of the Leui form.

Similarly we cari odserv~ the variation of the non-spheridty characterigtic under mappings of the form H; ‘(R(H,)), where H, and H2 are’mappings frarrri _. . . rl c” into c”, and A is a linear-fractional transformation sending a arCUIar nOlW$ form into a circular norm form, and leaving the circle z = 0, Iwl = 1 fixed.: ‘r.z;‘,

7.4. Chains on a Hypersurface Close to a Quad+ ; #= j

Let r(O) bc a ch&ti pn $. hyper&aceM:u=(z,z)+ . . . that passes through the origin, and let y&8): ‘.

he a chain on the quad& u = (z, z> that passes through the origin and 4 tangent to y(0) there. Suppose for definiteness that y(O) = ye(O) = 0. We assums that the paramerrizations of r(0) and y,(0) are compatible, that is, they have the same initial dati (see W.3). The function ~~(0) is defined for all values of 0, and @) is defined on some interval (0-, 0’) (see $6.1). We assume that M E M(6, .m] is given in linear normal form, that NJ,,,, (0, M) for. this hypersurface is defined and is small and that the projection of the circle ye onto the plane w = 0 helongs to the polydisc Izkl ZG S/2 (k = 1, . . . , n - 1); The latter implies that for small 6 the modulus of the vector u defining the directionof y, and y (see n-4,. Ic. *... 93.11 is sumctenrw large.

Let H and R be normalizing mappings of M and the quadric v = (z, z) that define a parameter 0 on y and yo. We shall assume that the set of initial data of HandRis

(6 -a, 1, 1,O).

tenrmr. For ea+large p and saall E > 0 we canjnd an N > 0 such that if N&O, M) c N, then H - 1 is defined and is holomorphic in a 6*-neighbourhood of the segment z = 0, p = 1, -pIO~p,andIH-’ - R- ‘I < E everywhere in this neighbourhood, where 6* > 0 is a function of S, m, p, and lal.

The substancebf the lemma is that if the hypersurface M is nearly spherical then on a large part of the domain of 0 the point y(0) is close to thd corresponding point of a circle.

. . ProoJ The mapping RL ’ can be regarded as a composition R- ’ = R, 1 (R,), where R. is a linear-fractional transformation sending the quadric u = (z, z) into the quadric 1 - (~1’ = (z, z) (see $43.2 and 5.1), and R, is a linear- fractional transformation with initial data o = (E, -a, 1, 1,O) that sends 1 - 1~)s = (z, z) into itself. Since the paiametrizations of y and y. are compatible, H-’ = H,(R,‘(R,)), where H, is the.standard normalization (see 92.1). If the characteristic N&O, M).is small, then N,,,,(O, M) is also small for M = H(M) (see Q7.1, and 7.3) (6’ and m’ are definzd .bwhE triple 1 al, S, m). Hence Nf,,,,#,(O, M*).is small for M* = R;‘(R,(M)) @“and m” are again cefined by the triple [al, 8, m’) (see §7.3).. Thus, by the lemma in47.1 the standard normal- ‘ization H,, which sends M* into M, is close to the identity in a large neighbourhood of the origin. From what has been said we obtain that H.- ’ is close to R - ’ in a b,-neighb ourhood of (0, 1,O). Here So is defined by the triple ([al, 6, m), and the deviation (H-’ - R- ‘1 is defined by the same triple and N, JO, M). When the latter decreases in the indicated neighbourhood, IH- ’ - R’ ’ 1 tends

uniformly to zero. In particular, on the segment (0, 1, - t 6, 5 0 s 4 6,) !- H - 1

d2 & ” 4 -

and 7 H- ’ are close to _ff_ R -i and - R - ‘, respectively. Therefore, by the 4 . 4 dC2

. \ ‘* \

IV. Holomorphic Mappings and the Gametry of Hypcrsud~#s 189 ,.*

theorem in $7.1, H-l is holomorphic in a b*-neighbourhood oftbis segment and “,:

~H-‘~~~.~WemayassumethatF<bo.At(O,1,)6L)H-‘andR-Larc~~! in the metric of P, and so, by the theorem in 57.1, H- l yxp, holomorphically I_ I

to a S*-neighbourhood of this point. Since I$RJandI$RIareboundedon

the whole line z = 0, p = 1, by passing sucksivel:~ along the chain of points

f 0, 1, f ZP and using the proximity of H-t-and R-’ obtained at earlier \ stages, we continue H -Ltoaneighbourhoodofthesegment(O,l, -p5;0Sp) of the line z = ,O, p ,= 1. The. size of this neighbourhood and the constant that bounds. I H - i 1 are determined by ([al, 6, m) and N&O, M). The required accuracy E for fixed Ial, S, m, and p/6@ is ensured by.choosiag the characteristic N&O, M) sufficiently small.

7.5. The Ikaviour of aChain Near Points of High Curva$ue. We show that if a chain on a strictly pseudoconvex hype&face has large curvature at some point, then on a large interval pf variation of a normal parameter it is clok to a circle, and on each loop (a single circuit along this circle) the variation of the normal parameter is nearly 2x.

We recall that the curvature of a chain y at a point x E M: u = (z, z) k i . . is approximately equal to 2(a, a)/[1 + la12 1”’ (see.$4.2), where c is the vector in the direction of y at x (see 43.1). Therefore, for strictly pseudoconvex hypersurfaces the curvature x of y at x E M is equivalent to la 1 for large 1 a I. The angle a between the chain and the complex tangent space is defined by cot a = Ial. Hence, for chains on a strictly pseudoconvex hypersurface, to say that the curvature is large.or the angle is small is one and the same.

1

Let y(0) be a chain on M: D = (z, z) + . . . that passes through the origin, and io(0) a chain on the quadric u = (z, z) that passes through the origin and is tangent to y(0) there. Let a be the vector in the direction of y and y. at their point of contact, y(O) = ~~(0) ,= 0, and let H and R be normalizations of yand ye with initial data w = (E, -Q, 1, 1,O) that define a parameter 0 on them. We shall denote by u(0) the vector in the direction of y at y(0).

Lemma. Let M be a strictly pseudoconvex hypersurface from the class M(6, m) given in linear normal form. Then for each p > 0 and E > 0 we c&find an a, > 0 depending only on p, E, 6, m, aJ the norm of the matrix of (z, z), such that when Ia) = la(O)) is greater than ao, the segment [ - p, p] t (0-, O+ ), and at all points C-p, p] we have

IY(0) - ro(O)l < + la(O) - al < slal,

where& an absolute constant. - : .: , : .‘..__ ,/ L ,. ‘.‘., ‘- _ : This assertion can be reformulated as, follows. When.a normal parameter

varies by 2x, the corresponding segment &Pa chain is geometrically close ,to -a circle; the’ an& of inclination .of the chain to the complex tangent spade is almost constant on this segment; the derivatives of a norinahzing mapping on this segment and the derivatives .of its inverse’ cannot be much larger than laj2 +)al-‘. ” .._- r -.

Proof. We introduce the substitution z + p, w + pz w, and denote the ‘, .re@ting’images of M? Y* and y. ~y.$f*,,~*~.a~,yX, respectively: We choose cc ,,,,su@;that the radius of‘yg is- l.‘Let a:, be a vector in the direction of ii*. The

curvature of the circle: yz is given by 2,(a*, a*.)/[1 + Ia*12]1,‘T (see’j3.2). Thus ,@l;t k*12]1’2 = 2(a*, a*); but a* = c(h and so 1 + p21a12 .k 4p4(a,a)2. Since

.“the form (z, z) is positive definite, p k ‘lj lil and Ia*1 - 1: The above substitution sends M: v _i 7 (z, z) + F into $*:‘v = (2;;) i’Fi;

where. F* -= . $ e@z+.@, p2u). Since ME M(Q, m) and the expansion of F con-

&ins: no terms- of order less than 4, Py taking p3 out of the parentheses in

F(pz, $, p2u) and writing ;F(p, ~5, ~‘a) as F we obtain IF*i K pIPI. If b is /, small, then M* 6 M(3, P).. The value 3 is chosen so that y$ lies strictly within a large polydisc. For p small, M* is close to a quad&, and so bythe lemma in 97.4, HTi and R- 1 are close in a 6*-neighbourhood:of the segment .[z = 0, p = 1, -‘$::6 0 5 pJ Thus, for any point y(8) (0 E [ - p, p)), H and R are close in a large neighbourhood of y(Q). Consequentiy, the first and second derivatives of

to the corresponding derivatives of R and R-l, and the absolute constanta-

By making the inverse substitution it is not difficult to obtain the inequalities - in the lemma.

. ‘

.’ . $8. Strictly Pseudownvkx Hypexsurfaces

In this section we shall prove a theorem that brings out an interesting property of biholomorphic maps that send a real-analytic strictly pseudoconvex non-spherical hypersurface in C” into another such hypersurface. We obtain as corollaries several known results about this class of hypersurfaces.

IV. Holomorphic Mappings and the Geometry of Hypersurfaces 1q1

8.1. A Tbeorem on the Germ of a Mapping. We formulate our main result in terms of normal forms; it is simpler to talk about the non-sphericity of a hypersurface and properties of maps in these terms.

Let M: v = (z, z) + F(z, Z, u) be a hypersurface of class M(S, m) (see $7.1) defined in linear normal form. We charadterize its non-sphericity by N(M) = maxIF(z, i; u)], where z, 5, u are regarded as independent complex

variables,‘~~d ‘the maximum is take over a polydisc of radius J/2: 1~~1 z$-&?, k=l,2,... , i - 1, jZ,l 5 6/2, k = 1,2, . . . , n - 1, lul 5 S/2.

We fix positive 8, m, and N, and denote by M+ the class of all strictly pseudoconvex non-spherical hypersurfaces ME M(6, m) of the form 0 = (r,z)+ . . . defined in linear normal form, for which ‘N(M) 2 N. If M and M* belong to the class M+, then every biholomorphic map sending M into M* can be’regarded as a normalization of M with a certain set of i&al &v 0 = (U, a, 1, u, r).

We characterize the propertiespf this map by a number v, writing as before -. IwlS v (seeg7.1). ’ ’

Lemma. For each positive 6, m, and N we can find a v such .that if a’. normalization H, of M E M + sends it into M*E M +, then 1~01 5 v.

The droof will be given at the end of this section. We remark that the mentiai part of the lemma is the estimate on the parameters a, J, and r. Anestimate on the norm of the matrix U follows directly from the compactness of the group of unitary matrices. For indefinite forms the analogous group.of matrices is clearly non-compact, and hence the assertion of the lemma cannot be generalixed to the case of hypersurfaces with indefinite Levi form. In this situation, we might expect an estimate on the parameter 4 S-and r,‘which depends on the matrix CJ; In particular, for automorphistirs; it is known that the parameters n, A and r can h estimated in terms of the norm of the matrix U and some parameters which determine the hypersurface [4]. . ”

_ We now state a fundamental consequence of the lemma.. * :t : Theorem. Z’ a bihohmaorphic, map H fixing ihe origin sends Me M + into

M* E M + , then H can be continued holomorphically to a 6*-neighbourhood cfthe origin, and I Hj < m+ in this neighbourhood, where 6* and m* are junctions of b, nr, a& .Jq : -. , I. .-.a :.

The assertion follows easily from the above lemma and the lemma in 97.1 We now consider several corollaries of this theorem. ,

8.2. Pr6perties of tbe Stability Croup ‘. .,_

Corollary 1. Thegroup of automorphisms of a hypersutjace ME M + that leave the origin fixed is compact 137).

Thisfollows immediately from the theorem. We.$rnark that the group of ai quad&, and therefore of qny spherical surfaca, is non-compact. For. hyper- surlhces with indefinite Levi form, this group is, in general, non-compact even

192 A.G. Vitushkin

when the hypersurface is non-spherical. For example, the ‘stability grqup of the hypersurface u = (z, z) + (z, .z)~ is isomorphic to the group of matrices pre-’ serving a given form (z, z), and so is non-compact if (z, z) is indefinite. “’

Corollary 2. If ME M + , then by using a biholomorphic Change of variables leaving the origiy fixed, we can go ocer to new coordinates in which. all the automorphisms of the hypersurface that leave the origin fixed are linear transformations.

This follows from the compactness 0; the group in question (see Corollary 1) and Bochner’s theorem [YJ: close to any point of a real-analytic hypersurface we can find a coordinatk system in which all the transformations of any compact subgroup of the stability group of this hypersurface with centre at the fixed point are linear. In the next section a stronger assertion will be proved regarding the stability group of hypersurfaces from the class M + .

- 83. Compactness of the Group of Global Automorphisms

Corollary 3. Let M and M* be non-spherical compact real-analytic strictly pseudoconvex hypersurfaces of some complex manifolds. Then each biholomorphic map of M into M* can be continued holomorphicall!* to a neighbourjlood R of M, which is common f&r all these maps. Here the family qf all such maps is equicon- rim&s on R [373. . .

I&corollary is obtained from the.theorem on germs and thk lemma in $7.2. It means, in particular, that the group of+global automorphisms,of a hypersurface with the listed properties is compact. From this it follows in turn that if a doniain ‘bounded by such a hypersurface is defined in a complex manifold then its automorphism group is compact ([47] and [32]) for in this situatioi the automorphisms are extended holomoiphically to the boundary of the domain (see [36] and [16] or [30] and [27]). We note that the reverse implication from the compactness of groups and the symmetry principle to Corollary 3 is * impossible, since here we obtain more, namely, the uniformity of theestimates of the size of the domain of holomorphy.

8.4. Continuation of a Germ of a Mapping Along a Compact Hypersurface

Corollary 4. Suppose &I and M* satisfy the conditions of Corollary 3, and let H be a germ of a biholomorphic mapping from M into M*, that is, a biholomorphic mapping defined in a ,neighbourhood of some x E M and sending pointS of M into points of M*. Then H can be continued holomorphically along any path lyihg on M .and starting at x.

This assertion for the case of hypersurfaces from C”, that is hypersurfaces given by a Gngle chart, was pr+Gd by Pinchuk [33], and in the general case by Ezhov, Kruzhilin, and the author ([I31 and [ 141). The assertidn, just as Corollary 3, follows from the theorem on germs and the lemma in $7.2. In view

IV. Holomorphic. Mappings and the Geometry of Hypersurfaces 193

of the unifoanity of the estimates, the map can be continued to any point of the first hypersurface by a finite chain of expansions of the defining series.

8.5. Classifk~tio~ of Coverings

Corollary 5. Let M and M* satisfy the conditions of Corollary 3. Then if M and M* are bdiy equivalent (see §i.4), their universal coverings are globally equivalent (the complex structure of the coverings is obtained by lifting the structure of the original hypersurfaces).

This is a reformulation of Corollary 4. As simple exa&ples show the global equiiaience of M and MT ,does not, in general, follow from their’local equivalence; and so we have to speak of the.equivalence of their coverings. Of course, if M and M* are simply-connrzcted, then their globai equivalence follows frdm their local equivalence. The latter is also trie for spherical simply-connected hypersurfaces (this follows easily from PoincarC’s theorem stated in the preamble).

8.6. .Discussion of Examples of Non-Continuable Mapphgs

Corollary 6. If M satisfies the conditions of Corollary 3, then its unieisal covering A? is a surface which can be said to bk maximal in the following sense: if,Q hypersurface ‘%I is connected, and there is a locally biholomorphic map .on W sending h? into Sn, then H(G) = %I. ’ I

If the hypersurface is spherical, then its universal,covering d&s not,,in ‘general, have to be a maximai hypersurface. similarly, if the Levi form of a hypersurface is indefinite, then its covering does not have to be a maximal bypersurface. These hypersurfaces are used in the known’ examples of non-~continuable local,.maps.

We recall these examples. Burns and Shnide! [73 conceived a hypersurface (in Cz).that is reai-gnalytic, co,mpa& strictly pseudocon<ex, spherical,“but not simply-connected, and a no&continuable local’ map, from a. sphere into this hype&face. Beloshapka [3] constructed a non-continuable local map of a non-spherical hypersurface with an indefiuite Levi form into another such hypersurface: .

We recal! that the Question of why local maps are not always continuable yas discussed’ also i” $6.4.

8.7. Mapplqg of Hypkurfaces with Indefinite Levi Form. The continu-

ability of holombrphio .maps of strictly pseudoconvex hypersurfaces is, accoun- ted for by the compactness of the group of local automorphisms. Maps sending a hybrsprface with indefinite, Qvi form into another such hypersurface also

have the continuation property. But the rkason here is different. When the &i form, is indefinite, then in differen.t.dire@ions ..the hypersurface is convex. .pn diffeient sides. For‘any side tie ian.construct 6 one.-caiameter family of anaiyiic discs whose boundaries lie on’ the hypersuiface; under a \;ariation of the

194 A.G. Vitushkin

parameter the boundaries of the discscontract to a point,‘and the discs fill out 3 part of a neighbourhood of this point that lies on one side of the hype&face. ’ Therefore, every function holomorphic on a hypersurface (and consequently a map) can be continued to all the discs of this family.“There is an ‘extensive, Literature devoted to hulls of holomorphy of hypersurfaces, and the removal of singularities of holomordhic rnae (see, for example, [19], [34], [44], [21], [20], II453

We state one of these results. Sufipose that in U.P there are defined two ‘compact hypersurfaces with P-smoothness, and that their Levi forms ,are indefinite at all points. Then any locally bihoiomorphic map of one hvpersurface &to the other can be c&tin& holomorphically to the whole of @P, and so it turns out ‘to be a linear map (see c2i j).

,As regards applications of the technique of normal forms irk this context; so far we may speak only about possibilities. Although the group of local automorphisms of hypersurfaces with indefinite Levi form is non-compact, perhaps we might expect certain elements of the,set of initial data of automorphisms of a non-spherical hypersurface <to form in the aggregate over the whole group a compact set. We shall discuss some of these results in the next section. : ’ ‘. e- ‘:&8. Proof of the Lemma id gal. We fix M, MS, and H which maps M into

&*. Let y(0) and ~~(0) be chains on M and on the quadri: u = (z z) with the same vector b defining their direction at the origi~id (see $3.1) We gssume that the parameterizations of these chains are compatible (see 97.4) and ‘that y(O)‘= y,,(O) = 0. Let b* be a vector defining the direction of y*. We show that !b+l cannot be much greater than Ibl:Haturally, we can assume that lb*1 > lb1 (it IS unnecessary to force an open aoor). Furthermore we assume that Ibi is a large number. Then by the lemma in Q7.5 the interval’[O, 2x3 belongs to thedomain of ,$finition of y(0) and.,..y*(S);. the corresponding segments f = y(to,2z1 and Y = y*[to,2nl are close to the circles y. and y$; both circles and these segments of chains lie inside a polydisc of radius S/2 with centre at the origin, and I lb(@)1 - I4 I < 4hl for 63 e CO, 24.

We define p by lb*1 = (l/p)jbj, and make the substitution z --) pz, w + p2w. From M*, y*, $, and b*(8) we obtain similar sets M**, y**, T**, and b**(8). By the Lemma in 67.5 when lb1 is large, T** lies in a polydisc of radius S/2 (since lb*+1 = lbl), f** is close to s*, and lb**(O)1 is close to Ib**(O)l at all points of F**. By taking lb1 sufficiently large and so obtaining all we need for what follows,

:we also assume that lb1 s bo, ‘where b, is a function of S and m. We denote by N, and H2 normalizations of M and M** with initial data

w, =(E, -b, 1, 1,O)ando 2 = (E, -b**, 1.1, a), respectively, that send M and MT* into circular normal form. These normalizations straighten the chains y and y**. Weshall aas#tne that the parameter 0 on the chains y(0) and y*(0) is defined by’ these nor&izations. We consider a map H** that is the composition of a dilation followi 6y H.

sending’M into M**

IV. Holomorphic Mappings and the Geometry of Hypersurfaces

The idea.df the proof that lb*1 cannot be much bigger than lb1 consists in showing that g is- small for large lb*I; and so the hypersurface M** is close to spherical. On the other hand, by the lemma in $6.2, f** = H**(Y), and thus it is possible to find an x1 E 7 at which H ** has derivatives not large in modulus, and at the same time sends x1 into ?I,~~ -**. But by the lemma in 97.3, this is impossible since M has large non-sphericity at x1 and M** is%lmost spherical at

x2. We note that if N(M) > N, then Nd,,,, (0, M) ? N (N(M) is introduced’here

only to avoid binding the assertion to an awkward definition). We have

lb**( =“ptb*( = Jb(. If p is small, then M** is more spherical than M*. More precisely, the non-sphericity- characteristic N,,, of the hypersurface M** is defined ‘and Nb,,,(O, M**) < N*; where N* >O is such that N* -+ 0.a” P 40. .

We re$resent the map$ng H** as H** = H;‘(R(H,)), where R %’ H2(H **

, (H ; ! )). The mapping R sends a normal form into .a normal form and preserves the orientation of the line z = 0, p =. . 1 Thus, when R is written in complex

coordinates, it turns out to be a linear-fractional transformation (see $5.2). It sends the disc : = 0, /M’/ 5 I into itself. For any linear-fractional tran_sformation R sending the disc z = 0, jwJ 9 1 into itself there is a &,$[O, 2n] such that

i$O.i.o,,i<I and ~~~O,l,Ool(a~~(a.~,~o~~.

The minimum point of the derivative of the map satisfies these conditions. Thus, at (0. I 0,) the mapping R has the initial data (II@,), 0, 1, A(@,), r(Oo)), and so knowing the general form of R (see $5.2) it is not difficult to see from the inequalities

that each of A, jr1 does not exceed 1 in this case. We explain this: earlier, the initial data of a normalization were defined only at the origin (see $5.2). Here we are speaking of initial data at (0, 1, Og), having in mind the quantities obtained in the image and the inverse image of corresponding points after transfer to the origin.

The lemma in 97.5 gives an upper bound for the first and second derivatives of the mapping H 1 at points of 7, and of H; ’ at points of the segmyts z = 0, p = 1, 0 5 O. I 2~. Similarly, the derivatives of H2 on 7’ and of H2 on the segment

z=o,p= , 1 0 5 O. 5 27t can be estimated from above. Thus, havmg chosen N ;, ,,,, we can use the lemma in 97.3.

We choose s’ and m’ so that Nd’,,,’ is defined for all points of M and M** lying inside a polydisc of radius S/2, and in particular ‘for xi ~y(0c))~ M and x~E~**(@~)E M **. The pair 6’. m’ is defined by the pair 6, m. By the lemma in $7.2, N,. Jxt, M) can be estimated from below if 8, m, and N are fixed .. beforehand. Hence, by the lemma in $7.3, N8,,,,+(~2r M*) has an estimate from-

196 A.G. Vitushkin L

below. Therefore, by the lemma in $7.2, p = /6//1b*1 cannot be arbitrarily small. more precisely, it is bounded from&low for any positive 6, m, and N regard]esi of the choice of M and M*. From this it follows that for any vector b with a sufficiently large modulus, for example lb/ = b,, we have lb*/ < cb where c is a function of s, m, and .N. Similarly, for If: 1 we obtain thatif $;I = ko, then (bl<cb,. .

Let o = (V, a, I, 6, r) be the set of initial data of the mapping H in question. We estimate Ial and i,. From the definition of o (see $2.1) and the vectors b and b* we have b* = oVi.- ‘(b + a). By substituting in this an arbitrary vector b such that lb/ = b, we obtain that JgVI-.‘(b+a)l 5 cb,. Similarly if (b*( ‘G b

then (aAV-lb* -al 5 cb,. Substituting in the second inequality aiector b* su% that AU-lb* has the same direction as -a, we obtain Ial s cb,. Similarly using the first inequality we obtain l/El < c. Hence, for any M and M* and H iending M into M* we have obtained upper estimates for Ial,& l/3,. An estimate for the norm of the matiix V follows from the compactness of the group of unitary matrices.

on We now estimate the parameter r. The map H defines a change of parameter

a segment of the chain y’* in exactly the same way as in the above construction H** defined a reparametrization of a segment of y’**. Again we use the formula for a change of parameter. We represent H as H*.- ‘(R*(H )), where H* is a normalization of M* straightening y*, and R* = H*(H(Hl-l)) is a linear-fractional transformation sending a normal form for H (M)‘into the

‘normal form for H*(M*); We define the initial data o* and ‘w by w* = (E, - b*, 1, 1, O), and as before, wl = (E, - b,. 1, 1,O). As before we &all assume that lb/ is large, for example, lb1 = b,. B’y the lemma in 57.5 the first and second derivatives of H, and H*- 1

132nl I can be estimated from above. Thus I is bounded

above by r* = Is),, 1 o)/, b, and 1. Therefore, if r is large, then so is I*. But R*

(in complex coordinatksj is a linear-fractional transformat’ion sending the circle z = 0, w = 1 onto its+elf. Thus if r* is large, then we can’find a O,E[O, 2x1 such

that El,,. ‘. e”ll( . . 1s large. By the lemma in $7.5 the derivatives of H* at all

pair+ Tf F ahz’;he derivatives of H;’ at a!1 points such that z = 0, p = 1,

0 i 8s 2a can be estimated from above. Hence - /,“,I HI is large. This i; I”5 lY(wd I impossible. We explain why. Close to xl we choose coordinates with x1 as origin

and such ihat the t&axis’ is ‘perpendicular to the complex tangent space* we construct the standard normalization of M at xl. Similarly, close to x2 = Zfix ) we choose linear ndrmal coordinates for M*. Then we can find 6’ m’ and .& depending only on 6, m, &d N, and such that N,,,,.(O Mj >‘N’ and N,,,,,,.(O, M*) > N’. By rewriting H in.the new codrdinates, we tin assume that ldl I

r.1 I zo H remains laige. But we already know that the parameteirh and ,I for such

IV. Holomornhic Mappings and the Geometry of Hypcrsurfaces

- maps are bounded above, and so

Id .-- I I (4 o

H cannot be large. Hence we also have

an estimate on r.

99. Automorphisms of a Hypersurface

Can we associate with a hypersurfacea coordinate system in which all the automorphisms of the hypersurface are linear-fractional transformations?

9.1. Estimation of the Dimension-of the Stability Group. Let M be. a hyper-

surface with a non-degenerate Levi form. We fix a point [E M and estimate the dimensicn of the stability group (the group of automorphisms of M defined in a neighbourhooa of [ and leaving this point fixed). Let us introduce a coordinate

.

system with origin at [ in which M has normal form. Then every automorphis? can be regarded as a normalizafibn of the hypersurface, A normalization ,!s uniquely determined by g set of irfitial data 02 - (U, a, i, o , r) and every such set gives an automqrphism of a quadric (see $1.2). Hence, the set of !!I normahzation can be regarded as a group, which is isomorphic to ;he stablhty group of the quadric. Therefore, the stability group of M is isomorphic to a subgroup of the stability group of a quadric.

‘Let us now calculate the dimension of the stability group of a quadric; The dimension of the group in t’he parameters 0, A r is 2, in (I it is 2(n - l), and in thl ,

matrix. V it is (n - l)2. Thus the dimension of the stability group of a. qu’adric is n2 + 1 ‘and consequently, for an arbitrary hypersurface with.a non-degenerate

’ Levi form ‘the dimension does not exceed’ n2 + 1. This bound, as ye shall see later, is exact only for spherical hypersurfaces, fhat is, those equivalent to a

1 quadric.

9.2. Parametiization of the Croup of Automorphisms. A spherical hypersurface is written in normal coordinates in the same way as a quadric. Using such a representation, for tiny set w satisfying (1.2.2) we can find an auto-

morphism of the hypersurface with the given set of initial data. A different situation arises when M is non-spherical. In this case the initial

data (V, a, A, u, r) dotresponding to automorphisms are no longer independent. Beloshapka’and Loboda ([2] and [26]) have proved the following result.

Theorem. Let M be a real-analytic non-spherical hypersucface having a non- degenerate Levi form and given in normal ftirm. Then for any automorphism the initial data a, I, ts, r are uniquely defined by the matrix V. In ,other words, the stability group of$ is isomorphk to a subgroup of the group of matrices preserving __, the Levi form of this hypersurface. . . . .

198 A.G. Vifushkin

This assertion can be reformulated in geometrical terms. A local automor. phism of a non-spherical hypersurface is completely defined by the restriction ot the tangent map (at a fixed point) to the complex tangent.

It follows from this theorem that the dimension of the group of lccal automorphisms of a non-spherical hypersurface does not exceed (n - 1)2.

The very simple example of a hypersurface t’ = (2, z) + (z, z)~ shows that the highest possible dimension is actually realized. ,

We emphasize that although for large n the number of initial data of automorphisms that turn out to be dependent is relatively small, these parameters have significantly complicated the study of automorphism groups.

I 9.3. Linearization of the Group. Consider the stability group of a

pseudoconvex hypersurface. -Kruzhilin and Lob&s [25] have shown th,. stability group of such a hypersurface is linearizable in the following sense.

. ..”

Theorem. spherical,

If a real-analytic hypersurface is strictly pseudoconvex and non- then in a neighhourhood of any of its points we can choose normal,

coordinates in which every automorphism leaving this point Jixed is written as

z* = uz, w* = w,

where V is a matrix preservirtg the Levi form.

The initial data(V, a, E., r).of automorbhisms of a quadric are not connected with each other by any relations, and so the stability group of a spherical hypersurface is non-linearizable. The possibility of linearization in the non- spherical case is suggested by a theorem of Bochner [S]: for every compact subgroup of the stability group of a real-analytic hypersurface we can find a linearizing coordinate system. However, a direct application of this theorem is not acceptable, .since in the case in question we need a system of normal coordinates.

Let us recall the question stated at the beginning of this section. For hypersurfaces with a positive definite Levi form the answer turns out to be positive. If a hypersurface is spherical, then all normal coordinates have the required property. For a non-spherical hypersurface a suitable coordinate system is assured by thetheorem of Kruzhilin and Loboda. The question of the linearization of the stability group of a hypersurface with indefinite Levi form remains open. The theorem of $9.2 gives one possible approach to this problem (see also the parameter estimates in 8.1).

From this point of view the works of Burns, Shnider and Wells E93, and Webster [43] are also interesting. In these works, a linearization of the system of coordinates is constructed under certain restrictions on the type of hypersurface.

9.4. Proof of the Kruzhilirdohoda Theorem. Suppose M contains the origin and iA given in normal form. We denote by Q, the stability group of the hypersurface at the origin. First of all, we show that there exists a chain y in A4?

IL. Holomorphic Mappings and the Geometry of Hypersurfaces 199

r(O) ,= 0, wh&&‘p&ts are fixed for each transformation cp E @. Denote by p the vector tangent to M at the point (0,O) and having coordinates z = 0, w = 1. We set p = (dql,)p, where dq&, is the differential of the automorphism cp at (0,O).

By%orollary 1 of Theorem 8.1, @ is compact and therefore there is a finite positive mea&e on Cp, invariant with respect to the group action (the Haar measure). Let us denote by p this measure and by h the average’of pq with respect to this measure ” .. _

h’= jep.&.

Each automorphism cp can be considered as a normalization with a certain set of initial data.4 = (U, a, A, u, r). Since M is strictly pseudoconvex, B = I. Conse- quently, for each cp, the projection of pr’ on ‘the u axis is numerically equal to 2al*,.hence positive. Hence, the vector h also has a positive projection on the u axis and therefore, it’clearly is transversal to the complex tangent space to M.

Since h is obtained by averaging p,,,, for each $E@ we have

(d&h = (d$lo)cjo p,dp = j (dJllO)podp = ;(d+V/o)pdlr =

=jp~~~d~=~p,dll=x. cp

By Bochner’s Theorem [21], there is a system of coordinates (z*, w*), w* = u* + iu*, in which each automorphism rp E 0 is linear. We shall suppose that h has coordinates (0,l) in the coordinates z * iu* and M is given by the equation ,

u*. = G(z*, F, u*), where G(0) = 0, dGlo = 0. In this system of coordinates, each cp E@ is a linear transformation which

leaves the hyperplane w* = 0 invariant as well as the point z* = 0; w* = 1. It follows that the transformation cp leaves the coordinate w* invariant.

Consider a chain y(c) on M such that y(0) = 0 and (dy/dt)(O) = h. Since automorphisms of M send chains to chains and y is the only chain passing through the origin in the direction of the vector h, q(y) = y for each cp E @. And since the projection of y on the u-axis is one-to-one for t small, q(y(t)) = y(t) and the points of y remains fixed under the aptomorphism cp.

Let us now bring the equation of M into a linear normal form such that the chain y is carried to the line z = 0, u = 0. Each local automorphism cp fixes y and

’ hence is a linear-fractional transformation of a completely concrete type (see j Theorem 1.3). Since all points of y are fixed, A = 1 and r = 0, and so cp is a

unitary transformation of the variable z. This completes the proof of the

theorem.

! 9.5. Proof of tbe &losbapka-L.&da Theorem. We shall write the equation

c the hypersurface in the form

u= <?,z)+“$tF”(zm

200 A.G. Vitushkin

where F, are homogeneous polynomials of degree v, i.e. such that

F&z, t.T, c%)-= t’F,(z, f u). ,

Let M and M* be hypersurfaces given in normal form

v=(z,z)+F(z,5,u), v= (z,Z)+F*(Z,~,u)

respectively, and let h = (f; g) be a biholomorphic mapping of M to M* in a . . . . .~

neignbourhood of the-origin. In this case we have the identity .

(-Ims+<f;f>+F*(l;~Res))l,,.+i(l,r)+iFrr,i.U) =O. (9.51)

This is an analytic equation with the property that if the variables are bound by the equation for M, then the tralues are bound by the equation for M*.

We mention several consequences of this identity which, in case M =M*is not spherical, allow us to express the parameters (a, 1,e, r) via the matrix U and thus obtain the proof of the theorem in question. In the fqrmulations, we shall set XJ = c and oA2 = p for brevity.

. (1) We have the identity (see (4.12))

(Uz, uz> = o-(z, z>,

which allows us to express G via U. (2) If Fk is the first non-zero term of F, then Ik-2F:(Uz, o&m) =

oFk(z, Z, u). This identity allows us to express i. in terms of U and 6. In order to write down the succeedi.:g relations, we introduce a notation. If P

is a polynomial or a matrix, we shail denote by [P] an ordering of the coefficients of P.

(3) We’have the relation (see [2], [26]): I

= -+6 6, cul, cFk+ll, [C+,ll, where a is a square matrix depending linearly on [Fi], with det a = 0 if ana only if Fk = 0; t1 is a group of variables, depending on the coe.@ients of h; and A is a polynomial of degree k - 1 in i whose coefficients depend linearly of [U], i!Fk+ 11, and C&F+ 1].

(4) We have the relation :

B(CFd) $ 0

= B@,‘a, 0, tul, CFk+ll, CV+11), fiCFk.21, Lf’,“A 2

where /3([Fk]) is a square matrix depending linearly on [Fk], whose determinant vanishes only for F, = 0, l2 is a group of variables depending on the coefficients of h; and B is a polynomial in rZ and a of degree k in I and degree 2 in a whose CoeffiCients depend linearly on [U],[F,+ 1], [Fki2], [Fj+ 1], [F,*, 2].

’ Let us return to equation (9.51). Along with the mapping h, we consider the

mapping &which is the automorphism of the hyperquadric v = (z, z) given bp the formulas from (1.2). where as initial data we choose the initial data of h.

IV. Holomorphic Mappings a’nd the Geometry of Hypersurlaces 201

Analogous Jo-(9.5.1), we obtain the identity

tmimC?+ (.lJ)jlw=~+i(~,~) E O* (9.5.2)

Substracting (9.5.2) from (9.5.1), denoting by Q, the left member of the resulting identity, and identifying terms of both sides, we obtain:

\ @*(z, 2, u) E 0, s=O,l,... 1

Let h(z, w) = f h,(z, w)

s=o

be the power series expansion of h in z and w, where

h&z, t2 w) = tshs(z, w), h, =(fs, gs).

Further, suppose F f 0, i.e.

F = f F,, F* = i F:, where F,, Fz. # 0. S=p s=p*

Here, we formally allow p* = co, but it will be shown below (see (9.5.8) that

P *= p. We set k = min(p, p*). The s-th term of (9.5.1) allows us to define x3- i and gs (see [2] and [26]), but for s = 0, 1, . . . , k - 1, the s-th term of (9.S.l). agrees with the s-th term of (9.5.2). Thus, we conclude that

j,=ss, fors=O,l,..., k-2 (9.5.4)

ss = !L fors=O,l,.. .,k-1.

Our immediate goal is to calculate Ok, ok + i and ok + 2. For the calculation it is convenient to introduce the following notation,In calculating terms of weight s, we shall write a -+ /I, meaning that all terms of weight s which appear in a also appear in /I. Furthermore, we denote Ah = h-K=(AJAy), \t= u+i(z,z),

q = C-ix and $ = p- lg. We represent @(z, 2, u) in the following form:

cp(~,Z,u)=Rei(gJ,=,+~~-gl~=~)+ReiAgl,~=.f

+K”fJ>l,,,+j~- (f;f>I,=,)+2Re(A~f)+((Af,Af)l,=a)+

+F:(f,~Rey)+F:,,(f,~Reg)+F:,,(~~Reg)+ . . . .

In this representation we shall write the terms in the form: (I), (II), (III), eJc. We need explicit formulae for the first few components of the mapping h and

its derivative (&)/(a~). These can be obtained immediately from the formulas for representing quad&, namely:

fjo=O,~, =z,4,=2i(z,a)z+aw,

ij3 = -4(~,a)~z+(r+i(a, a))wz+2i(z, a)aw,

$, = 0, I$, = 0, 3, = wJ (5, = 2i(z, a)w,

IJ~ = -4(~,a)~w+(r+i(a,a))w~,~

202 A.G. Vitushkin

(go = 6 gQl = 2~(z,a)a+(r+i(a,a))z, (9.5.!)

($). = 1: (z)l =2i(z,a),

ali;

( ) c7w = 2(r+i(a,u))w-4(~,~)2. .

2

Calculation of cPk:

Here, we have used the fact that

($). = (g). = P (see (9.55));.

i(F,+...)+... IV=6 w=J’ -

However, (f, (aflaw)> does not contain terms of degree 0, and SO (III) + 0.

(IV)-,2Re(4f,-l,~)lw-J=2pRe(Arpk-t,z>l,,,

WI + F,*(fl ,A, Reg,) = F,*(Cz, CZ, pu).

Summing, we obtain:

9 = pRe(iA$,+2(Ac+-i, z>)l, + F*(Cz, b, pu) - pFk(z, 5, u).

If we denote Re(iAq,- i + 2(A&- i, z))l, _ + by L,(h), we have

L,(h) = F,(z, 5, u) - p- ’ F,,(Cz, c’r, pu). (9.5.6)

In [2], [26] it is shown that the. equation L,(h) = G (mod R)’ has a uniqut solution for Ap,- *, A$, in the class of mappings (Acp, AJI) for which the initial data is (E, 0, 1, 1,O). R is the collection of equations written in the normal form. The right member of (9.5.6) belongs to R. Thus, L,(h) = 0 (mod R) and by the above assertion, we conclude that

@k-l =0, Atir -0, (9.5.7)

and also .

F,*(cz, cf, pu) = pF,(z, Z, u). . (9.5.8) .

. Thus, (III) + -pF,ZRei(z, a), and

(IV) -+ p2Re(A4$ ‘z>li di. ’ ‘. .

(v) = Flf(f,E Reg) = pF&‘, 5, Reti)= pFt(z + 4’2 + - . -9

z+&+. . .,u+Reti3+. . . ) = p(Fk+dFk{,(cp,+.‘. .,cpi,+. . ., -

3Fk Retis+. . .I+ . . .JItZ,ruJ-,p2Re ~(2i<r,a)z+$a)+i(z,u)wx .

(Vi) = FL*,: ,U;x Reg) --) Ft+ l(Cz, CZ, pu). %

Simming, we obtain:

+ We( - 2i(z, u)FI -k 2i(z, ,

Thus,

L,+,o+T(F,,~)=F,+,(z,~,u)-p-‘F:+,(CZ,CZ,pu), (9.5.9)

where

T(F, a) = Re - 2i(z, a)F + 2i(z, a& aF(z)+iGz a aF( )+i(z,u)Eg .

We remark the functional T is real linear in each of its variables. Calculation of @k+z.

i(Fk+Fk+1+Fk+2+. . .)+ . . . + w=*

+ -pFk+z-pF,t,~2Rei(z,n)-pFk2Re((r+i(a,u))rt-2(z,a)2).

A.G. Vitushkin

(III)~2pRei((w,~))2-pFk.+,2Rei((8,~~))~ ,,

-pF,+zZRei K >>

,

~~~~)2.=~~*,~~o)~~~~,(~)~)=

=2~(z,u)*+w~u,u)-2i<z,u)(u,z)+(r-i(u,~))(~,~),

((%gg, = (%(~>,> = <z,uh

acp , K >> (P?aw 0 = O*

Thus,

WI) -+ pFk2Re(2(z, a>’ - 2(z, a> (a, z)) - pF,+ 1 ZRei(z, a).

(IV)~ZpRe(Acpk+,,z)lw_.+2~Re(Acpk,~,)lw=,. W = F:(f;x Red = pF,(cp, 6, Re$) =

= P~&+cP~+. . .,Z+G2+. . .,u+Re#,+. . .)+

-+ -P~~hh 63, Rell/,)+id2Fk(cp2, (P2, Ret),).

In order to isolate the coefficient of r, we write the first term in detail:

pRe 2$%3)+$ti4 ( > (

= pRe 22(‘4(z, a)*z+

. +(r+i(u,a))~z+2i(z,u)~u + >

+3-4(‘.u)*G+(r+i(u,u))wt).

Hence, the coefficient of r is

pRe 2$2(z)+G*$$ ( >

.

(VI)=F:,,(f;f;Reg)=F:,,iCz+f,+...,~~+f,+...,pu+Reg,+. . .)=

=F?+,+dF,*,,(f2+ . . . . &+ . . . . Reg,+...))

-+ dC+ 1 (if; Reg3)l(cz, ci, pu). (Cz, 9 pu) +

WI) --) Ft+ ~(CZ, Cz, pu).

IV. IbIomorphic blappiw and the Mry of Hypmuffaccs

Summing, we o&aim

p-“&+2 = ~k+2(h)+jf-iF~+2(c~,~~,~u)-Fk+2(z,~,u)+ (9.5.10)

+P,(p-‘F~+,(c~,~~,~~),u)+F,+,P,(u)+rP,(F,)t

+ p,(F,, a)+ P,(Fk, U, a) + p&h, a),

where each of the expressions P,, . . . , Pa is real linear in each of its variables. We notice that

Thus, the proof is complete. .

.

$10. Smooth Hypersurfaces

Some of the results stated above, regarding real-analytic hypersurfaces, have been developed subsequently for other classes of surfaces. There is also a series of works on real-analytic manifolds of high codimension (see Tanaka [36], Moser and Webster [2g], A.E. Tumanov and G.M. Khenkin [22]). Limitations of space force us to leave the exposition of these works to authors of other papers.

10.1. Invariant Structures on a Smooth Hypersurface. As in the case of a real- analytic hypersurface, on a smooth hypersurface, there exists an invariant family of curves called a chain and on each chain a certain class of distinguished parametrizations. On .real-analytic hypersurfaces, geometrically, these curves coincide with the chains introduced in $3. However on such surfaces, the parametrization differs from those described above (see $$3,6). In addition to families of chains, other invariant structures can’be defined on hypersurfaces with the help of the Cartan-Chern theory and these can be useful in studying mappings of hypersurfaces.

In [ 11 J Chern constructed a certain principal bundle Y over a hypersurface Fi having a non-degenerate Levi form. On Y he defined a connection form II with the.property that two hypersurfaces M, and M, are CR-diffeomorphic if and only if there exists a diffeomorphism of the bundle Y, over Mi onto the bundle Y, over M, which preserves the connection form.. The connection form x corresponds to the curvature form, R=dn - II A A.

Let o be a real l:form on M which is non-vanishing and which at each point of M annihilates the complex tangent space to M. It is known that locally on M we can find smooth complex l-forms oi,. . . ,o”-’ which at each point are a basis for the (1, O)-forms on the complex tangent space and for some real l- form

where {gij} .is the Levi-form. _ From the definition of the bundle Y in [ll], the choice of the collection of

.forms (iu;c)‘,-. . .~,g”-i. , cp) satiSfying (10.1.1) gives a section a: M -+ Y.The forms a*(n), o*(a) are smooth forms on M and hence can be represented .via o, wj, and 5~‘. The coefficients in these representations consist of variables defined on M and depending oh the choice offorms satisf$ng(lO.ll). One such variable is the so-called pseudoconformal curvature tensor S = (S,,,,,) (1 = a, /I, y, 6 < n - 1). For n 2 2, if S=O on M, it follows that the curvature form:a vanishes on M and this means that M is locally CR-diffeomorphic to the (2n- l)-dimensional quadric (see [ 1 l] with its appendix and [7]). A point M at which S = 0, is called an umbilical point.

If a real-analytic hype&face M’ is given by equations in normal form

i... u= C&Z) + 1 F,,(z,i;u), k.I>Z

and iffor the form& we take ia(u- (z, z) LX&.,), and ifthe’forms ~9, : .‘. ,m”-i coincide at the origin witfi‘dz’, . . ; ,dz”- i, then the components of the tensor S, corresponding to this, choice ‘of forms, are the same as the coefficients of the polynomial -4Ri, (assuming that the polynomial iS written in SymmetIiC form, i.e. its coefficients are symmetric with respect to a permutation of variables)..

Set

where (tip) is the inverse matrix of (g&. The following fact was noticed and exploited by Webster.

Lemma. The form llSl/o has continuous coefficients and is,independent of the choice of forms (w, oj, cp) satisfying (10.1.1).

Thus, on each hypersurface there exists an invariant real l-form t? = llSl10, which vanishes only at those points where [ISll = 0. In particular, ifM is stiictly pseudoconvex;‘then 8 vanishes only at umbilical points.

We restrict our attention now to the open set of those points of M where llSll#O. On thisset, tIis a smooth l-form. At eachpoint theformdegivesandln-: degenerate real.2-form, on the complex tangent space, which corresponds to a certain Hermitian metric ds’. Moreover, at each point of M, there is a unique tangent vector- x, transversal to the complex tangent space, and such that 6(x) = 1 and d&x,.0) = 0. Thus, on M, an invariant vector field is defined. Using this field and the metric on the complex tangent space, we’ ciin’cbtistruct an* invariant indefinite metric tensor on M.: In case the hypersurface is strictly’

IV. Holomorphic Mappugs and the Geometry of Hypersurfab 207

pseudoconvex, this tensor can be chosen positive and such that it yields an invariant Riemannian metric on M.

The above invariants, defined eon the set of non-umbilical points, were used by Webster [41] to construct a special normal form for a real-analytic hypersurface in the neighbourhood of a non-umbilical point. A system of coordiantes corresponding to this normal form is defined uniquely up to unitary transformations.

In [8] Burns and Shnider, using these invariants, showed that in the space of plurisubharmonic functians, there exists a family of second category such that for functions of this family, the different level sets are not CR-diffeomorphic.

In case. n = 2, the tensor S is identically zero on any hypersurface and in its place one considers the variable denoted by Q in [ll]. For a real-analytic hypersurface given in normal form, Q is precisely (up to a multiplicative constant) the coefficient in Fb2. If Q E 0, then M is spherical. Points at which Q = 0 are also called umbilical points. The form IQl”‘*)a, turns out to be invariant and permits us to define an invariant vector field and an invariant metric on the set CQ # 01. These invariants were constructed by E. Cartan [lo]. Moreover, for n = 2, we can invariantly choose a basis vector, uniquely up to . sign, for the complex tangent space. If the hypersurface is given in normal form, then this corresponds to choosing the coordinate z1 so that the coefiicient in Fb2 is equal to one. Thus, for n = 2, on hypersurfaces without umbilical points, we have a smooth field of rep&s, and the question of CR-equivalence of two such hypersurfaces reduces to the possibility of mapping one hypersurface to another while preserving this field of rep&es. ,

The existence of such a rep&e field on M means that wt! section Q: M + Y. Thus, the coefficients in the expression of the a*(Q) turn out to be invariant functions on. M. Making use of thig, E. ‘Cartan [lo] defined nine functions ,on a three-dimensio+l hmurface such that two hypersurfaces are in general CRdiffeomorphic if and only if there exists a diffeomorphism which identifies these nine-tuples.

The invariants which we have considered are *finedand non-trivial only for non-umbilical points. However, in studying mappings of hypersurfaces, it is not always possible to avoid considering umbilical points. Thus, those invariants, which can be continuously extended to the whole hypersurface, present particular interest. As we already mentioned, the form 0 is such, an invariant and Webster [41] used the continuity of its coefIicients to show that the component of the identity in the group of CRdiffeomorphiams of a compact hypersurface M is compact if M is not everywhere locally CRdiffeomorphic to the sphere.

The Hermitian metric ds2 defined on the complex tangent space also.extends continuously to umbilical points. However, the invariant vector field and consequently the invariant metric tensor do not have this property. On the other hand, ,the (2n - 1)dimensional measure on M, corresponding to the Riemannian metric in the strictly pseudoconvex case, is so constructed that, in any coordinate system its density with respect to real measure extends continuously to all

‘OX A.G. Vitushkin

points. This was used by Burns and Shnider [S] in studying proper mappings of strictly pscudoconvcx domains. In addition, in the same paper, Burns and Shnidcr constructed a certain semi-metric on 111 with the help of#. We shall discuss this in the next section.

10.2. Compactness of the Group of Clobd Automorphisms. Let M be a smooth strictly pscudoconvex hypersurface. The,Hermitian metric ds2 is defined in’the complex’tangent spaces at non-umbilical pointsand extends continuously to zero at umbilical points. Thus, we define the length of any piecewise smooth curve on I\! whose tangent vectors lie in the cotiplex tangent bundle to M. For a pair of points X. J’E M, we define the number 4(x, y) as the minimal.length of all such curves joining .Y and 4’. r’t is not difficult to see that:

I. q: M x A9 -+ W’ IS equality,

a semi-metric on M. i.e. s,atisfies the triangle in-

2. If the point s is no]!-umbilical, and 4‘ # X, then y(.u, y)>O, 3. If ri. ‘44 . ‘ x M -+ R’ is an arbitra;y smooth metric on M, then for each

compact set k’ c itI, there is a c such that for .u,s~K, we have q(.~, ~3 5’ c(d(s, y))“2.

N.G. Kruzhilin [23] showed that from the existence af such a semiimetric on M it immediately follows that the group of global CR-diffeomorphisms is compact provided M is a compact smooth hypersurface, not everywhere locally CR-diffeomorphic to the sphere. Indeed, let I’.be the open subset consisting of non-umbilical points of 44. For some E > 0, the set ‘V, = (x E V: 4(x, M\ V) 2 E} is a non.-empty compact subset of r. Each automorphism of M is an isometry of I’, with respect to the invariant smooth Riemannian metric oi V introduced above. This isometry -maps E’, to V,. Sihce the group of such isometries is compact, it turns out that the group of CR-automorphisms ‘of M is also compact.

. . 10.3. Chains on Smooth Hypersurfaces. The definition of a chain on a

hypersurface has no connection0 with umbilical points. For this reaso,n, chains also prove to be useful in studying mappings of liypcrsurfaces having umbilical points. However, to obtain similar applications, we should clarify some proper- ,ties of chains. Just as in the real-analytic case, it turns out that, on strictly pseudo-convex hypersurfaces, chains which form small angles with the complex tangent spaces are similar to chains on spheres. This is related IO rhc gcomctric behaviour of a chain as well as td its parametrization. We recall that on each chain, there is a distinguished family of parameters and we pass from .one set of such parameters to another via a fractional-linear transformation. A segment of a chain,, forming a small angle with the complex tangent space, and for which dne of the distinguished parameters varies from o to II is close to a circle of small radius. As before, we shall say that such a segment ‘performs one loop. From fhe definition of chains and there parametrizations in terms of the bundle Y over M


! - -- and its connection form II, one ca$ give estimates on the above-mentidned closeness. ,We shall not introduce these es&&es here. Let us merely remark that, using these Gtimates, one can prove the following assertion:

Lemma. Let x E M. Then each point of Msuficiently close to x can be joined to x by a chain segment making less t@n one loop (N.G. Kruzhilin [24]).

_..

10.4.~ Properties of the Stal$lity Group. Let M be a smooth strictly pseudoconvex hypersurface and x E M. As in the real-analytic case, to each local automorphismf, defined in a ne$hbouihdod .of the point.‘% we may associate a set of initial data o =( U, a, . For this we will suppose that, the hypersurface is given by an equation i h the terms sf small degree satisfy the same condition as the corresponding terms of an eq;ia_tion given”in norma-i forin. As initial data pf an automorphism of a hypersmface, we ta@ the parameters of the correspondin,g automorphism of the hyperquadric. f, ./-

Theorem. If M is not CR-difieomorphic to the sphere near x, and f and g are local automorphisms of M with initial data ojf = (U,, af, As, rf) and w, = (Ue, a,, A,, re) and if U, = U,, thea fi= g (N.G. Kruzhilin [24]).

To prove the theorem, it is sufficient to show that if U, is the identity matrix E, fdr a local automorphism,f, then i., = 1, af = 0;~~ = U and, consequentl~;f is the identity Mapping..

.I

The iroof follows if we show that the f&owing Eissertitins hoid ,for an Mzitomotphism’-6f a hypeisurface which is not equivaient to the sphere:

’ I- 1. if = 11

e ‘.

2. from af = 0, it follows that r; = 0, 3. from U, = E,it follows that a, = 0.

The proof of these assertions is based on Lemmas 10.1 and 10.3. If some automorphism does not satisfy one of the properties 1,2,3, then we consider a chain segment, which starts at x and completes naless than one loop. It turns out that’ if this segment lies in a sutiiently small neighbcmrh@0d &f the point x, thti on the segnient, we may define iterations of themappin@If or of the inverse mapping f - ‘. The length of the image of the segment, after these iterations, tends to zero (in the proof of 1 and 3- at the-expense, of making the ande small between the chain rind the complex tangent spa&and in the proof of 2- at be expense of changing the parameters of the chosen quantities). However, by Lemma 10.1, the integrals of the invariant form 8 along the segment ‘dr akng its image coincide. Since 8 has continuous coefficients, this means that 8 = 0 at each point of the original segment. Lemma 10.3 shows that all points sufficiently close to x are umbilical and hence, M is locally CRdiNhie- t0 the sphere.

In contrast to the real-analytic case (see 9.2) the coadirion of strict pseudoconvexity turns out, in the case of a smooth hjrpers&dAc% to be essential, as the following example shows.

210 A.G. Vitushkin

Let A4 be the hyperquadric in the space C3(z,, z2, w = u + iv), given by the equation 0 = z19 ‘+ @I, and let B be the unit polydisc in C3 centered at the origin.

Consider the open sets:

V,=41-kVI,k~N,andset V~={(Z~,Z~,W):(Z~,Z~,U)EV~).

The fractional linear mapping f, taking the point (zr , z2, w) to

((Z1 + w)/(! - 2izz), z2/(l - 2izzX w/(1 - 2izz)),

is an automorphism of M. We have

f’(Zl, Z2, w) = ((Zi + ?mv)/(l-2imzz), zz/(l-2imzz), w/(1 -2imzz)).

Let us denote by Vr the image S”( VE), m E Z. Then, for m # 0, the inequality

bll > 4]ml- 1

Im)+4k+ 1’

is satisfied on F$‘, and consequently, for m # 0 and n 2 k, we have Vr n V,” # 0. This means that the sets V;( are pairwiedisjoint. Moreover, for each S we can find a natural number Ark such that for /ml > &, v;l lies outside of B.

We mwe the hypersurface M onto the set V, so that the Levi form remains non-degenerate and we change it on the intersection with the sets VT, Irnl sN,, so that the mapping f is also a local automorphism on the hypersurface M, so obtained.

By Mz we denote a hypersurface agreeing with Mi outside of

ij vy, m= -Nk _-

and, in the interior of this same set, differingfrom a quadric, and such that f is a lm automorphism of M2. Continuing this &cess and choosing the deformation sufficiently small at each step, we obtain, in the hmit, a smooth hypersurface @ in 3, with non-degenerate indefinite Levi form, which is not equivalent to a hyperquadric in any neighbourhood of the origin and for which the mapping f is a local automorphism.

The restriction of the differential off at the origin to the hyperplane (w = 0) is the identity mapping; at the same time, f itself is not the identity.

105. Li~rization of Local Automorphisms. Local automorphisms of a real analytic strictly pseudoconvex hypersurface not locally equivalent to the sphere can be simultaneously linearized via a holomorphic system of coordinates

” (see 9.3). Generafly speaking, this is not the case for smooth hypersurfaces,

IV. Holomorphic Mappings and the Geometry of Hypersurfaces 211.

because the domain of definition of an automorphism can be arbitrarly small. If we restrict our attenti.on to automorphisms defined in a large neighbourhood, then their linearjzation is possible.

Suppose we are given some family of local automorphisms of a hypersurface M which fix some chain y passing through a point x. By assertion 2 of the preceding section, it is not hard to show that if M is not CR-diffeomorphic to the sphere in a neighbourhood of x, then all points of y are stationary with respect to the automorphisms under consideration. From the existence of’such a chain follows:

1. There exists a system of coordinates ,yith respect to which the initial data for each automorphism f of the given famrly is U, A = 1, a = 0, r -0.

2. For each neighbourhood I/ of x lying within the common domain of definition’of all automorphisms of the family, there exists a neighbourhood V, c V, which is mapped-into itself by each of these automorphisms.

The group of automorphisms of the neighbourhood V, which fix s and have as initial data the unitary matrix U and the parameters I =0, a=O, r=O is compact and consequently can be linearized. Hence, in order to linearize the local automorphisms of a hypersurface M it is sufficient to find a chain which is invariant for all of these.

1

,

Lemma. If a hypersurface M is not CR-diffeomorpkic to the sphere in a _ neig,hbogrhood ofa point x, then clJ1 the local qutomorphisms of M have a comnion

inoariant chain 1241. -. 8

First of all, it happens that each local automorphism of the hypersurface has

an invariant chain. The proof of this fact broceeds similarly to the proof of l-3 in the preceding section. Taking into account the earlier remarks, this m’eans that eadh local automorphism generates a compact subgroup (the closure of the iterations) of the group of all local automorphisms.

Let us denote the group of local automorphisms of,M by (0. Theorem 10.4’ shows that there is an embedding of 0 in the group of unitary matrices. Let us denote by CD0 the connected component of the identity in the group (0, with respect to the topology induced by the representation.-Then cP” is compact, and therefore all automorphisms in a0 have a.common fixed chain; The remainder of the proof is based on the fact that the factor.group @/a0 is isomorphic to some linear group, all of whose elements are of finite order, and which therefore contains a normal commutative subgroup of finite index. v From the above lemma5 we obtain the fohowing assertion: : ’

*

Theorem. Let M be a strictly pseudo&vex hypersurface which in w sigh-

bourhood of the point x is locally CR-difiomorphic to the sphere. Then, for each neighbourhood V of x,, there ex@ts V, c V and local coordinates, (u, Z, r (z,,.. z- 1)). on V,, such that each .automorphisms ‘of M whqse domain of I definitiok iontains V. is a unitary transformation in the variable z[24].

212 kG. Vitushkin

References*

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2. (1974). Zbl. 272.32006 (Zbl. 281.32019) Beloshapka, V.K.: On the dimension of the group of automorphisms of an analytic hypersuiface. In. Akad. Nauk SSSR, Ser. Mat. 43.243-266 (1979); Engl. transl.: Math. USSR, Izv. 14,223-245 (1980). Zbl. 412.58010

3. Beloshapka, V.K.: An example of non-continuable holomorphic transformation of an analytic hypersurface. Mat. Zametki 32, 121-123 (1982); Engl. transl.: Math. Notes 32.540-541 (1983). Zbl. 518.32011

4. ‘Beloshapka, V.K., Vitushkin, A.G.: Estim&s of thk radius of convergence of power s&es defining a mapping of analytic hypersurfaces. Izv. Akad. Nauk SSSR, Ser. Mat. 45,962-984 (1981); Engl. transl.: Math. USSR, Izv. 19, 241-259 (1982). Zbl. 492.32021

5. B+hner, S.: Compact groups of differentiable transformations. Ann. Math,, II. Ser. 46,372-381 (1945)

6. Burns, D., Diederich, K., Shnider, S.: Distinguished curves in pseudoconvex boundaries. Duke Math. J. 44,407-431 (1977). Zbl. 382.32011

7. Bums, D., Shnider, S.: R& hypersurfaces in complex manifolds, Several complex variables. Proc. Symp. Pure Math. 30, Pt. 2, 141-168 (1977). Zbl. 422.32016

8. Bums, D., Shnider, S.: Geometry of hypers&fac& and mapping theorems in c”. Comment. Math. I-!elv. 54, No. 2, 199-217 119791. Zbl. 444.32012

9. Bums, D., Shnider, S., Wells Jr., k.O.:‘Deformations of strictly pseudoconvex domai Math, 46, 237,253 [1978). Zbl. 412.32022

s Invent. 7’

10. Carts& E.: Sul: la g&[email protected] des hypersurfaccs de Oeuv

t”

eux variables

1953 compktes, Pt. II, Vol. 2,12?1-1304, Pt. III, Vol. 2,1217-12 s

omplexes.

Pt. II). Zbl. 58.83; 1955, (Pt. III). ZBI. 59,153 8, Gauthier-Vi lars, Paris

11. Cherb, S.S., Moser 1

, J .I& Rdl hypenurfaces in complex manifolds. Acta Mafh. 133, 219-273 (1974). Zbl. 302.32015 i

12. hhov. V.V.: Asymptotic behaviour of a strictly pseudoconvex surface alon k its chains. IZV

Akad. Nauk SSSR, Ser. Mat. 47, 856-880 (1983); Engl. transl.: Math. USSR,‘Izv. 23, 149-170 (1984). Zbl. 579.32034 I

13. &ho;, V.V., Kruzhilin, N.G., Vitushkin, A.G.: Extension of l&/mappings of pseudoconvex surfaces. Dokl. Akad. Nauk SSSR 270, 271-274 (1983) Engl. tr 58Q-583 (19Q). Zbf. 558.32003”

L nsl.: Sov. bath., Dokl. 27,

14. Ezhov, G.V., iruzhilin, N.G.! Vitushkin, A.G.: Extension of holomorphic maps ;along real- analytic hypersurfaccs. Tr. Mkt. Inst. Steklova 167, 60-95 (1985); Engl. trahsl.: Proc. Steklov Inst. Math. 167, 63-1Oi (194 Zbl. 575.32011

15. Faran, J.J.: Lewy’s curves and chains on real hypersurfaces. Trans. Am. Math. Sot. 265,97-109 (1981). Zbl. 477.32021

16. Fefferman, C.L.: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math. 26, l-65 (1974). Zbl. 289.32012

17. Fefferman, CL.: Monge-Amp&e equations, the Bergman kernel, and geometry of pseudoconvex

18. domains. Ann. Math., II. Ser. 103, 395-416 (1976). Zbl. 322.3zb12 Feflerman, C.L.: Parabolic invariant theory in complex analysis. Mv. Math. 31,13l-?62 (1971). Zbl. 444.32013

19. Grlffiths, P.A.: Two theorems on extensions of holomorphic mappings. Invent. Math: 14.27~2 (1971). Zbl.‘22&32016 .. .

._. * For the convenience of the reader, ieferences to rev& in Zentralblatt fiir Maihematik (&.), ’

compiled using the MATH database, havei as far as poasibk, been included iti thb bibliography.

IV. Holomorphic Mappings and the Geometry of Hypersurfaccs 213

20. Ivashkovich, 5.G.: Extension of locally biholomorphic mappings of domains to a compkx nroiective soace. Izv. Akad. Nauk SSSR, Ser. Mat. 47, 197-206 (1983); Engl. transl.: Math. r.-,~ or USSR, Izv. 22, 181-189 (1984). Zbl. 523.32009

21. Ivashkovich S.M. Vitushkin, A.G.: Extension of holomorphic mappings of a red-adytiC hvnersurfad to a iomulex nroiective space. Dokl. Akad. Nauk SSSR 267, ?79-780 (1982); En@. trynsl.: So;. Math., Dbkl. %, 682-683(1982). Zbl. 578.32024

22. Khenkin, ‘G.M., Tumanov, A.E.: Local characterization of holomorphic automorphisms of Siegel domains: Funkts. Anal. P$&h. 77, No. 4,49-61 (1983); En& transl.:,Funct. Anal. Awl. 17, 285-294 (1983). Zbl. 572.32018

23. Kruzhilin, N.G.: Estimation of the variation of the normal parameter of i chain on a @ctiY pseudoconvex surface. Izv. Akad. Nauk SSSR, Ser. Mat. 147, 1091-l 113 (1983); End. tnnsI.: Math. USSR, Izv. 23, 367-389(1984). Zbl. 579.32033

24. Kruzhilin, N.G.: Local automorphisms and mappings of smdorh.strictly pseudoconvex hw surfaces. Izv. Akad. Nauk SSSR, Ser. Mat. 49, No. 3,566591(1985); Engl. transl.: Math. US!% Izv. 26, 531-552 (1986). Zbl. 597.32018

25. Kruzhilin, N.G., Loboda, A.V.: Linearization of local aufomorphisms of pseC )cUnVeX SU~iW8.

Dokl. Akad. Nauk SSSR 271,280-282 (1983); Engl. transl.: Sov. Math., Dokl. Z&70-72 (1983). Zbl. 582.32040

26. Loboda A.V.: On local automorphisms of-real-analytic hypersurfaks. Izv. Akad. Nauk fSS@, Ser Ma;. 45 620-645 (1981); Engl. transl.: Math. USSR, In. I8,537-559 (1982). Zbl. 473132Oj6

27. Lewy, H.: dn the boundary behaviour of holomorphic mapping?. Acad. Naz. Lincei 35, &8 (i97jj

28. Moser, J.K., Webster, S.M.: Normal forms for real surfaces in C2 near complex tangentsjand hyperbolic surface transformations. Acta Math. ISO, 255-296 (1983). Zbl. 519.32015

i 29. Pinchuk, S.I.: H&morphic mappings of real-analytic hypersurfaces. Mat. Sb., Nov. Se . 105, 574-593 (1978); Engl. transl.: Math. USSR, Sb. 34, 503-519 (1978). Zbl. 389.32008

30. Pinchuk, S.I.: A boundary uniqueness theorem for holomorphic functions of several dmplex variables. Mat. Zametki 15,205-212 (1974); Engl. transl.: Math. Notes IS, 116-120 (1974). Zbl. 285.32002

31. Poincart, H.: Les fonctions analytiques de deux variables et la reprksentation conforme. Rend. Circ. Mat. Palermo 23, 185-220 (1907)

32. Rosay, J.-P.: Sur une caractkrization de la boule parmi les domaines de C” par son groupe d’automorphismes. Ann. Inst. Fourier 29, 91-97 (1979). Zbl. 402.32001 @bl. 414.32001)

33. Segre, B.: Questioni geometriche legate colla teoria delle funzioni di due variabili co@plesse. Rend. Sem. Math. Roma, II. Ser. 7, No. 2, 59-107 (1982). Zbl. 5, 109

34. Shiffman, B.: Extension of holomorphic maps into hermitian manifolds. Math. Ann. 194, 249-258 (1971). Zbl. 213,360

35. Tanaka, N I.: On the pseudo-conformal geometry of hypemurfaces of the space of n complex variables. J. Math. Sot. Japan 14, 397-429 (1962). Zbl. 113,63

36. Tanaka, N.J.: On generalized graded Lie algebras and gedmetric structures. I. J. Math. Sot. Japan 19, 215-254 (1967). Zbl. 165,560

37. Vitushkin, A.G.: Holomorphic extension of mappings of compact hypersurfaccs. Izv. Akad. Nauk SSSR, Ser. Mat. 46, 28-35 (1982); Engl. transl.: Math. USSR,. Izv. 20, 27-33 (1983). Zbl. 571.32011

38. Vitushkin, A.G.: Global normalization of a real-analytic surface along a chain. Dokl. Akad. Nauk SSSR 269,15-18 (1983); Engl. transl.: Sov. Math., Dokl. 27.270-273 (1983). Zbl. 543.32003

39. Vitushkin, A.G.: Analysis of power series defining automorphisms of hypersurfaces in Con- nection with the problem of extending maps. International Conf. on analytic methods in the theory of numbers and-analysis Moscow, 14-19 September 1981, Tr. Mat. Inst.%teklova 163, 37-41 (1984); Engl. transl.: Pmt. Steklov Inst. Math. 163,47-51 (1985). Zbl. 575.32010

40. Vitushkin, A.G.: Rezl-analytic hypersurfaces in complex manifolds. Usp. Mat. Nauk 40, No. % 3-3131985); Engl. transl.: Russ. Math. Surv. 40, NO. 2, l-35 (1985). Zb1. 588.32025

i

\ 214 ; A.G. Vitushkm

I 41. $ehster, SM.: Ktihler metrics associated to’s real hypersurface. Comment. ,Math. ~.fefv, 52, 235-250,(1977). Zbl. ‘354.53050 .

42. Webster, SM.: Pseudo-hermitian structures on a, real hypersurface. J. Differ. GeOm, 13, i5-41

I (1978). Zbl. 379.53016 (Zbl. 394.53023)

43. Webster. SM.: Gn the Moser normal form at a non-umbihc point. Math. Ann. 233, 9iLftj2 (1978). Zbl. 358.32013

44. Wells Jr.. R.O.: On the 10~4 hobmorphic hull of a real submanifold in several &mplex

variables. Commun. Pure Appl. Math. /9, $45-165 {1966).-Zbl. 142,339 i 45. Wells Jr., R.G.: Function theory on differentiable submanifold : Contributions to analysis.

Academic Press, 407-441, Neyv York 1974. Zbl. 293..32001. P 46. Wells Jr., R.G.: The Cauchy-,Riemann equations and differential/geometry. Bull.’ Amer. Math,

Sot. 6, 187-199 (1982). 47. !&a& B.: Characterization of the unit ball in C” by it$automorphism group. Invent. Math, 41,

2+257(1977). ,. z * -. ..-

r .

: : i

. :

i. +.

/ : .

i

- - -

( I

V.. General Theory -of Multidimensional Residues

P. Dolbeault

I

Contents I 1 ,’

1 $0. Introduction. . . . . . . . . . . . . . . . . . I . . . , . . . . . . . . . . . . .i. . . . . . . r216

0.1. I-Dimensional Formula of Residues . . . . . . . . . . . . . . . . . . . . . / zf6

0.2. l-Dimensional Residue Theorem. , ZI I

0.3. Theory of Leray-Norguet ‘. ’ 217 .... : ..............................................

0.4. Grothendieck Residue Symbol ......................... i 219

0.5. Generalizations and Applications ......... t ............. 219

$1. Residue Homomorphism .................................

1.1. Homological Residue. ; ..................... go

..........

1.2. Cohomological Residue. ..............................

1.3. Relation Between the Exact Sequences (1.1) and (1.2) ........ iz

92. Principal Value; Residue Current. ..........................

2.1. Casen= 1.. ....................................... ii:

2.2. Semi-Holomorphic Differential Operators. ................ 222

2.3. The Case of Normal Crossings ......................... 222

2.4. General Case. ............................ : ............ 223

2.5. Residue Current ................. , ..................

$3. Residual Currents. ........................................... 22;

3.1. Semi-Analytic Chains

224

.. + ... I .... , .....................

3.2. Definitions ................................ :. !. ....... 224

3.3.Tubes.. .................................. r.. ........

225

3.4. Essential Intersections of the Family f .................... 225

3.5. Main Results ....................................... 225

3.6. Proof of Main Results ................................ 226

3.7. Properties of Residual Currents. ........................ 227

3.8. Extension to Analytic Spaces and Analytic Cycles ........... 227

. 216 P. Dolbeault

04. Differential Forms with Singularities of Any Codimension ....... 228 ; 4.1. A Parametrix for the de Rham’s Complex ................ 228 4.2. Definitions. .......................... :. ............ 229 4.3. Case of Canonical Injection. ........ ._ ................. 229 4.4. Complex Case; codimc Y= 1.. ........................ 229 4.5. Complex Case; codimc Y 2 1 ... , ...................... 229

§$. Interpretation of the Residue Homomorphisms. .......... :. .... 230 5.1. ComplexCase;codimc Y= 1.. ......................... 230 5.2. Complex Case; codimc Y 2 2. ......................... 5.3. Real Analytic Case: Y Subanalytic of Any Codimension.

231 .

..... 231 5.4, Complex Case Again. ................................ 5.5. Relation Between Homology of Currents an6 Borel-

232

Moore Homology .......................... .I .' .. #6. Residue Theorem; Theorem of Leray

i. .... 232 ............... .‘. .......

6.1. Residue Theorem. 232

.................................... 6.2. Friffiths Residue Theorem :

232 ...........................

6.3. Converse of the Residue Theorem: Theoreb of Leray. 233

....... 6.4. Converse of Griffiths’ Residue Theorem

233

‘$7. ResidueFormulae.. ... 234

..................... :::::::::::‘I:::: I 7.1. The Case of Y be Oriented C” Subvariety of X

234 ............

7.2. Kronecker Index of Two Currents 234

...................... 7.3. The Case of Y be a Closed q-Subanalytic Chain of X

235

$8. A New Definition of Residual Currents : 1’: ‘1 : : : 235

............... 236.

References.......................................... . ..I.. 238

$0. Introduction

‘j ti.1. l-Dimensional Formula of Residties. Let X be a Riemann surface and

let a, be a meromorphic differential form of degree 1 on X; in the neighborhood of a point where, z is a local coordinate, we have w =f(z)dz, where f is a meromorphic function. Let Y = (al},,, be the set of poles of o,.and res Uj(o) the Cauchy residue of o at Uj* Then, for J,a finite subset of I, if for any jeJ, yj is a positively oriented circle whose center is uj and such that the closed disk of which y, is the boundary does not meet Yin any point different from a. and if (nj)j,, is a family of klements of Z, (w or @, we have the reSidue formulu:

I

f , (0.1)

V. General Theory of Multidimensional Residues 217

0.2. l-Dimensional Residue Theorem. Moreover if X is compact and connected, the following residue theorem is known: For any discrete (hence finite) set Y = (u~)~,~ of points X, and for any meromorphic l-form o whose set of poles isY,

& res,,(d = 0 (0.2)

Cor&rsely, for any such subset Y and any subset (aj)j., of complex numbefs

such that c aj = 0, there exists a meromorphic l-form o on X having simple

pples exacii; on Y and such that, for any jeJ, aj = res,,(o). H. PO&ark was the first to give a convenient generalization of the notion of

residues for closed meromorphic differential forms in several complex variables (1887) [54]. Afterwards E. Picard [53] (1901), De Rham [58] [59] (1932,1936), and A. Weil[70] (1947) obtained results on meromorphic forms of degree 1 or 2.

0.3. Theory of Leray-Norguet. Starting from PoincarC’s work, Leray (1959) [45], then Norguet (1959) [49,50] generalized the situation of 0.1 to the case of a complex analytic manifold X of any finite dimensiun, where Y is a complex submanifold of X of codimension 1 and o is a closed differential form of degree p on X, of class C” outside Y and having singularities of the following type on Y: every point XE Y has a neighborhood U in X over which a complex local coordinate function s is defined such that Y n U = ( y E U; s(y) = 0} and w( U is equal to a,fsk where a is C m on U and k E N. More generally, a semi-meromorphic differential form on X has a local expression a/f where a is a C” differential form and where f is a holomorphic function.

(a) The topological situation studied by Leray and generalized by Norguet is the following: let X be a topoiogicui space and Y be a closed subset of X, we have the following exact cohomology sequence with complex coefficienti and com- \ pact supports

. . . -+ H,P(X) + H,P( Y) fl Hf+‘(X\Y) + H:+‘(X) + . . . (0.3.1)

If Y and X\Y are orientable topological manifolds of dimension m and n respectively, then the duality isomorphism of Poincari defines, from 6* the homomorphism

s: H’,-,(Y) + II’.-,- ,(X\ Y).

Hence, from the universal coefficient theorem, we have the residue homomorphism

r = !d:HneP-‘(X\Y) + HmeP( Y)

and the residui formula ’ - (Sh, c) = (h, rc)

in which hLHF,-,(Y) and c~ff"-~-'(X\Y).

(0.3.2)

218 P. Dolbeault

Going back to formula (O.l), it is easy to see that the homology class of

zjj h n y is t e image by 6: H’,(Y) + H’,(X\Y) of the homology class h A Cn,a,

of Y and aj --+ 21tires,,(o) is the cohomology class imagd by r: Hl(X\ Y) + Ho(Y) of the: I-cohomology class c of olX\ Y. So when we consider only homology and cohomology classes, formula (0.1) is a

“case of formula (0.3.2). particular

” (b) Suppose now that X is a complex analytic manifold of complex dimension

n and Y a complex analytic submanifold of X of complex dimension (n - 1,). Given a closed semi-meromorphic differential form o on X, C” except on Y, we say that Y is a p&r set ofw; if, moreover in the notation of the beginning of 0.3,

w = f, we say that Y is a polar set with multiplicity 1. Then, locally, z LC) = (ds/s) A $ + d6’ where $ and 0 are C”i $1 Y has a global definition and is called the residue form of w and its cohomology class in Y is the image by I, of the, cohomology class defined by olX\ Y on X\ Y.

(c) Theorem of Leray. For n = 1 and X compact connected; consider the exact sequence

I .I

H’(X\ Y) I, Ho(Y) ‘i, H’(X). (0.3.3)

By Poincare duality j’ is transformed into j”: H,(Y) + H,(X); the cohomology

class of H”( Y) c; aj + 27r resaj is transformed into C” = z(2ni res,,)ujEIHO( Y); its i

image byj” has the same expression in H,(X); j”c” = 0 if and only if c res,, = 0. _

The residue tl&orem 0.2 results from the exactness of (0.3.3). j

More,generally, the exactness of the cohomology sequence

. . . 4p-qx\ Y) : Hz”-P-*(y)i; HZ”-P(X)+ . . .

where j’ is induced by the inclusion Y + X means, in particular, that an element CEH*“-~-*(Y) b 1 e ongs to Imr if and only if j’c = 0. This is, trivially, a cohomological residue theorem. The converse of the residue theorem is a particular case of the following: I

Theorem. Let X be a complex analytic manifold and Y a submanifold of X of codimension 1, then every class of cohomology of X\ Y contains the restriction to X\ Y, of a closed semi-meromorphic differential form having Y as a polar set with multiplicity 1.

(d) Composed residues. Let Y,, . . . , Ys .be complex submanifolds of co- dimensron 1, in general position, and let y = Y, A . . . n Y,. Then by com-


position of the-residue homomorphism, f0r.p 2 4,

HP(X\Y1uY2u.. . u Yq)+Hr-1(Y17Y2u.. .u Y,)-

+ HP-*( Y1 n Y2\,Y, u . . . u Y,) + . . . --, HP-q(y),

we obtain the composed residue homomorphism. ^

0.4. G&thendieck i&due Symbol ([29], cf. also [24], [67]). The Poincare residue in n variables can be considered as a generalization of the residue in one variable, the singularity being’ol’codimension 11 Another possible generalization is to consider a singularity of dimension zero. This leads to the point residue or Grothendieck residue symbol: let f,, . . . , fn be holomorphic functions in a neighborhood of 0 in @” such that 0 is their only common zero; let w be tn n-meromorphic differential form o = f; 1 . . . fi ' g dz, A . . . A dz, where g 1s holomorphic in a neighborhood of 0, and I = { 1 fi 1 = . . . = If;,1 = 6) for 6

” small enough, then the point residue of o at 0 is resO) o =

to duality theorems

Lo. This leads

0.5. Generalizations and Applications. Since 1 4 68, generalizations of definition (a) of the residue homomorphism have been used for spaces more general than complex manifolds; (b) semi-meromorphic differential forms with polar set having arbitrary singularities and generalizations-of such forms have been used to give interpre&tions of the residue homomorphism; the theorem of Leray (c) and composed residues (d) have been ‘generalized; finally residue formulae of increasing generality have been’obtained..

In fact, a small part of this program. was realized before Leray’s work (1951-1957) [41, 42, 66, 143. But for a more complete realization the following newer techniques were used: Borel-Moore homology ([8], chapter 5); local cohomology [26, 273; Hironaka’s local resolution of singularities of complex analytic spaces [34, 2-J; and properties of semi-analytic and sub-analytic sets (Lojasiewicz [47], Herrera [30], Hironaka [35]).

The Grothendieck residue symbol, introduced in algebraic geometry, has a meaning in analytic geometry and was studied later by Griffifhs [24,25]. It was used. by Bott and others [6, 7, 4, 31 for the singularities of holomorphic and meromorphic vector fields [6, 7, 31. The relation between the Poincare residue and Grothendieck residue is made explicit by the residual currents introduced , by Cole&Herrera [ll]. These currents appear as an interpretation of the composed residue homomorphism (see d), see also [68] and they have applications in a duality theorem [65].

An extensive bibliography on the theory of residues before 1977 can be found in [l], see also the previous survey [203.

220 P.‘Dolbeault

5 I. Residue Homomorphism

Hoinology and cohomology groups are taken with coefficients in C. F

1.1. Homological Residue [ 15, 331. Let X be a locally compact, paracompact space of finite dimension and Y be a closed subset of X. In the Bore&Moore homology (homology of locally compact spaces) [8], we have the following exact honGlOgy sequence ,i:: r

. . . -+i,+,(X)+H,,,(X\Y) 3 H,(Y)4,(X)+. . * (1.1)

It is transposed from the exact cohomology sequence (0.3.1). When X\ Y and Y are orientable topological manifolds of dimension n and m respectively, we Gave the following diagram

ff:(yj 1 H,P+~(x\Y). . ‘/ _^

I (Q) I

HP(Y) 2 4+1(X\ Y)

(Q') PY 1 px\ul

II”--P( Y) : w-p- ‘(X\ Y) ’ .

where the vertical lines mean the pairings between cohomology with compact supports and*homology with closed supports, 6, = ‘6*, and P,, Pxiu are the

I duality isomorphisms of Poincar.6 Moreover, the square (Q’) is anticommutative and (Q) gives a residue formula. Note that 6, is the connecting homomorphism of (1.1); 6, is defined under niuch more general hypotheses than T, so it gives a good’generalization of r and will be called the homological residue homomorphism.

1.2. Cohomological Residue. For any topblogical space X and any closed subset Y of X, we have the exact sequence of local cohomology

. . . -P HP(X) + HPIX\ Y) : Hf”( Y) -+Hp+l(X)_, . . .* (1.2)

where H;+*(X) is the (p+, )- g 1 th rou of cohomology of X with support iti p Y [27]. When X\ Y and Y are topological manifolds of dimension n and m respectively, the following diagram is commutative

HP(X\Y) 1: H;+“(X)

\ r x

/ tip-(n-m)+l(y)


Thus, the hQmomorphism p is a generalization of r and will be called the cohomological ‘residue homomorphism [SZ].

1.3. Relation Between the Exact Sequences (1.1) and (1.2). Suppose that X is locally compact, of homological dimension n, then Y and X\Y are locauy compact. Let (X) be thefundamental class of X; then applying the cap product by (X) to (1.2) we get the commutative diagram

(C) :. . + HP(X)-+ HP(X\Y) : HP,+l(X)+. . *

In(X) lnW\Y) in(Y)

(D) . . . -+H,~,(X)-4&(X\Y) + H,-,-,(Y,C)+. . .

When X is a manifold, X\ Y is also a manifold, then n(X) and n(X.\ Y) are inverse to PoincarC isomorphisms, and by the five homomorphism lemma, n( Y) is also an isomorphism.

92. Principal Value; Residue Current :

In this sectibn, we suppose that, X is a redyced complex ar/alytic space (complex space), and Y a closed cogplex analytic subvariety of X, of pure codimension 1. Note that holomor$ic functions, C” (or smooth) differential forms, and currents can be defined on X ([S], [33]).

For the sake of simplicity we shall assunie X to be a complex manifold when giving sketches of proofs.

_I 2.1. Case i = 1. First consider the case where X is a coordinate domain U

of a Riemann surface and let o be a meromorphic 1 -form on Cl having only one pole P in ZJ. Choose the coordinate .Z on U such that z(P) = 0,

Consider the current 0 on U\kP) such that, for every $GS(U\(P})

is a curreni on U whose restrict&n to U\(P> is 0; I+(w) is.indekndent of the choice bf the coordinate z and of the representation &the merogorphic form o.

222 P. Dolbeault

‘1t is called the Cauchy prir?cipal value of o. Moreover,

d VP(W) = d”Vp(w) = 2xi res,(o) 6, + dB

where 6, is the ‘Dirac measure at P and B a-current of type (0,l) aind support {P>. If P is a simple pole of w, then B = 0. The current d VP(o) will be called the residue current of o.

We shal! geneialize the construction of the principai value and of the residue current to the case of semi-meromorphic differential forms on a complex space X.

2.2. Semi-Holomorphic Differential Operators. On a complex manifold X, a differenti,@ operator D on the space of currents 91(X) is said to be semi- holomorphic if, for every x E X, there’exists a chart (U, zl, . . . ,,z,,) at x such that on U, for every-current T, the coefficients of DT are linear combinations, with C” coefficients of the partial derivatives

&, (I E FV’), of the coefficients of T [66].

Let Y be a complex analytic subvariety of X of codimension 1, and let 9”; be the vector space of semi-meromorphic forms whose polar set is contained in Y. Then D operates also on.9; and more generally on the space Y’(X) of the semi- meromorphic differential forms on X. The se{ A of the semi-holomorphic differential operators is a ring and L@:(X), 9”‘; and 9’“‘(X) are ‘&modules. ’ .’

I 2.3. The Case of Normal Crossings. A complek analytic subvariety Y of X of

codimension 1 has normal crossings if, for every point x E Y, there exists a chart (z,, . * *, z,) on an bpen neighborhood U of x such that *

U~Y={~~U;Z~(Z)...~,(~)=O;~I~} (2.31)

A semi-meromorphic differentialform o defined on such a neighborhood U of x in X is called elementary if it has a polar set contained in such a subvariety Uhr;theno=ti/z”wheres=(s, ,..., sq,O ,..., O)EN”.

2.3.1. Theorem. Let (U, zl, . . . ,z.) be a chart of a complex analytic manifold X of dimension n and Y be a complex subvariety of U defined by (23.1), then

(a) There exists a unique C-linear map T9’; -+ 9:(U) such that: (i) if WEY; has locally integrable coeficients, then T(o) = 0, the current

defined by o; I (ii) T is A-linear; I (b) T is a equal to the Cauchy principal value (2.1) with respect to the coordinate

functions [ 161.

2.3.2. We shall give an expression of the current T having an easy generalization. With the notations of 2.2 and 2.3, in thd open set U, let g = z’g,.& any function’such that I, . : . . ?Iq ‘f 0 and go is holorhorphic without zeros’ in*U.


Then for k&y ci, E .9’( U), the formula

defines the current T(o) = VP(o) of 2.3.1. Note that IgI 2 6 is a semi-analytic set. This definition of VP(O) is due to Herrera-Lieberman [33]. One establishes the existence of VP(o) for a particular representation of o and a particular g and one verifies that VP(o) sat&&es the conditions (i) and (ii) of 23.1; for (ii), it suffices to

show: - vd-,($, = VP&) WX using the Stokes formula for se@-analytic j \--*I

set. Thus, we obtain the jnbep&dence of VP(o) with respect to g and to the representation of o. Moreover, from 2.3.1 (iii Vp is A--linear, the idea of this proof is due to G. Robin [ 17,lS). 1.

2.3.3. When Y is any subvariety of codimension 1 of X with normal crossings, and o E 9’,, we build up VP(o) locally and glue the local currents by a partition of unity. This makes seqedue to the invariance of the del&&km of Vp.

2.4. Ceacnl Case. Let X be a complex space and Y a complex subvariety of codimension 1 containing the singular locus of X, globally de&d by an equation g = 0 wJ~re g is holomorphic on X. Consider the semi-analytic set

X( > 8) = (x E x; lg(x)/ > t-5) and the integration current I[X( > 6)J over X &fined by the semi-analytic chain [X(>S),e(>6)]wheree(>6)isthefundamentalclassofX(?6~Letfbea holomorphic function on X such thatf = 0 implies g = 0 and let w = a/f~Y; where a is C” on X. We set, for every rpcS(X)

After shrinking X if necessary, there exjsts a proper morphism lI:X’ + X such that X’ is a manifold and Y’ = lI - 1 Y a subvariety of codimension 1 of X’ with normal crossings and lIlX’\Y’ is biholomorphic (Hironaka [34]). Then

Vd4k4 = fy fCx’( > 41 (n ( * w A cp)) = Vp(l-I*;u)(II*‘~) (2.4.2) -D

where X’( > 6) = {x’EX’; IlI*g(x’)I > S}. For lI fixed, the last member-of (2.4.2) is independent of the representation of o and of g from 2.3.2; so (2.4.1) makes

’ sense and, from its expression, it is independent of II. The use of a partition of unity allows one to define VP(o) for every semi-meromorphic form o on X.

24.1 Tbeorem. When X is a complex manifold, the map

Vp: 9; --, .9:(x)

is A-linear, A being the ring of semi-holomorphic differential operators. [ 18, 193

224 I? Uolbeault

2.5. R&he Current. We consider the following residue operator

Res = dVp - Vpd

In the notation of 2.4, it has the following IFcal expression

Res (0) (I& = t; l[X( = Sk] (w h $) (2.5)

where Z[X( = S)] is the integration current on the semi-andytic set {xEX; Ig(x)l = S> with the orientation opposite to that ofJ[X( > $), e( > S)]. From the d-linearity of Vp (when X is a manifold), we get :

Res = d” Vp - Vpd” i *

Moreover, the operator Res: 9; --+ 9:(X) is LI-&te~r~ ard. from (2.9, if w E Y;, Supp~ Res (0) c Y.

-- ‘,~. I

§3. Residual CurfedS ” y

CoIeff and Herrera [Ii] gave a g&e&zation of the &nstruction of the opektors V;i &id l&s of n.2 it is related k the composed residue homomai- phism (0.3,d) and allows us to give an interpretation of this homomorphism. Preliminary results have been obtajned by Herrera (1972) [3,1,32). li

3.1. !Scmi-Analytic Chaks [S]. Let M be a kally cIosed semi-analytic set; its boundary is defined as bM = A$, M; a q-kechaia (M, c) is a pair where M is as above, dim M I q and c E H,(M, Z). The eoefkknt ting Z may be replaced by W‘or C. The boundary of the prechain is i?(M, c) = (bM, a,,&)) where a,.,, is

‘the homomorphism H,(M) qH,-,(bMt The sum of 2 prechains (Ml,cl), (Ml, c2) is (A4? c; + c;) where M = M, u M, - bM, v bM, and ci is the image OfCj(j= 1,2). Let L be the equivaleacc relation: (Ml,cl) N (M,,Q) means (M,, b,) + (M,, - 6,) = (M, Ok ‘the Hass of the q-pmhain (M, c) is denoted [M, c] and is called a stmkznalytic g-chain. The I;et &the q-chains is aZ&esp. R-, CL) module denoted a,(X).

3.2. Defiaitkns. Let X be a complex manifold of complex dimension n. Let

~={L..., Y,+,}(P20)

be a family of hypersurfaces 0f.X. We set v 9 = Y, u . . . u Yp+‘l; ns=

Y, n . . . f-7 YP+l; dim&ng) 2 A - p - 1. When dima: = n - p w 1, n 9 is a complete intersechon.

Let R, be the set of strictly positive real riumbers. Let 4 be a continuous map:

]o,lC+W<+l; PEN*

6 -@,(a . . . , ~,+,@N.

V. G&al Theory of Multidimensional Residues 225

Thema 4 is c&e! admissible if lim 6,+,(S) = 0 and for every Jo [l, . . . , p] d+0

andevery qElR,, _’ ; ,_

lim (djj/@+ 1) (6) = 0 i d-+0

3.3. Tubes. Let X small enough so that every Yj(j = 1, . . . , p + 1) is the zero set of a holomorphic function gj pn X. Then

9 = (Sl, * - ; 7 &+A: x + Cp+ l is a holomorphic map

and Id = (IslL . . . ,(gp+ll): x\“LRiwp=l is a C” (real analytic) map. Con- sider the sets

IT~+l(g)I = {XEX; /gjl= dj; jE[l, . . . ,p+ 11) .

l~~+‘@)l={X~X; 1~jl=sj~~~C1,~~~~Pl~l~~+~l>5~+1}

For every 4 E rW<’ 1 such that,

diml7J”(g)l < 2n - p - 1 (3.3.1)

dimlD$+‘(g)l <2n-p, (3.3.2)

we consider the semi-analytic chains, with c = (6,) ., .‘. , 6,), -

T,p+w=(- IF+’ ~~]E02n-p-1(X)

P(P+ 1)

Q+‘(g)= f - ‘)--(lgp [S’] x Lx,+, > 8p+l])Eo2n-p(X),

where CSI, WI, Lx,+ 1’ a,+ 1 ] denote the points 4, s_l and the open interval 18 P+ 1, + oo[ with their canonical orientation and lgl-l the inverse of the map )gl defined on semi-analytic chains. The chains Ti+ ’ (g) and &(g).are tubes around 9. When n 9 is a complete intersection, then conditions (3.3.1) and (3.3.2) are satisfied.

3.4. Essential Intersections of the Family 9.. Let B = ( Y, , . , . , Y,,, 1 ); we define V,(9) by i?duction on jE [0, 1, . . . , p]. Let Yb = X; suppose that Yj- 1 (j 2 1) is defined, Then let Yj be the union of the irreductible components of Yj- 1 n Yj not contain_ed in Yj+ 1. We call essential inte’rsections of 9 the two complex subvarieties V,(S) = Yi; V,(F) = Yin YP+l; then dime PC(S) = n - p and dim@ V,(9) = n - p - 1.

We shall use the following notation S(j) = ( Y,, . . , ~j, . . . , YP+ 1 >. Then, if n 9 is a complete intersection,

.

Fe(5) = nF(p + I); V,(9) = n9.

.- 3.5. Main Results. We shall uSe the notation of n.3 and the following ones:

84,(* u 9) is the sheaf of semi-meromorphic q-forms whose polar set is

226 P. Dolbeault

contained in u 3. For every semi-analytic chain p, I[p] is the integrati A current on p [30]; it is a generalization of the integration current defined by a complex variety [44].

Theorem. Let W c X be an open set small enough for Yj & W be defined by a global equation gj = 0, then: the tubes Tg+ ’ (g) and Dg ’ l(g) are dejned for 6 small enough. Moreover:

(1) v aEr,(W&-P (* u9)),‘RpPpr1(E) = fin; L[D$“(g)] (6) exists,

(2) v j-k r,(w, dp-p-1 (* us)), Rp+’ (8) = iimO L[Tf”(g)] ($) exists,

(3) the limits (1) and (2) do not depend on the choice of the admissible map &and of the holomorphic map g,

(4) the limits (1) and (2) are zero if bi and $ do not contain terms of type (n, n - p) and (n, n - p - 1) respectioely; and if v’~r,(W,&‘:“-~-‘(* Us)), then RPPP+’ (d”f) = (- l)‘+’ RP+‘(9).

(5) For XE IJ W, JJ$-~(* u S)), the following linear forms RPPP+ ‘[I] and RP+‘[x]

.9-(w) + @

,~I+R~P~+‘[I] (a) = RPPP+‘(X A a) .

p-q-y W) * @

/3t-+Rr+l(X/\ j?) = RP+‘[l] (j?)

are currents; ; (6) the supports of the currents RP Pp+ l [I] and Rp+’ [I] are contained

respectioely in the complex subvarieties vJ9) and V,(F); (7) for every open set W c X, and every admissible map &into IV’, there exists

a;+ 1 > 0 such that, for every 6, + r ~10, SF+ l [, lim I[Pf$ + I (g)] (Z) exists and d-*0

RP pP+ ‘(&) =r lim 6

lim IIDg.tjp + , (g)] (2). ,,,+O d-0

3.5.2. Consequences. The locat Annions of RP Pp+’ and Rr+ ’ respectively, glue together on X (from l”’ .IIU define the currents R Ps [I] and RS [I] called respectivJ*. I i -. p of 1 and (p + I)-residue of x. . . .._ I

bi hen p = 0, we obtain again Vp and Res of n.2.

3.6. Proof of Main Results. To prove 3.5, one shows first, thanks to the properties of the admissible map &that for 6 small enough, the tubes T,P+ ‘(8) and D!(g) have dimension (2n - p - 1) and (2n - p) respectively and; con- sequenily ‘are semi-analytic chains. Then, as in section 2, using Hironaka’s morphism, we come back to the normal crossing case. The proof, in this last case rests on the following result:

V. General Theory .of Midtidithensional Residues 227 I

3.61. Ler’% &I= [ki, , :. , kp] c [l, ..f . ,+$I; :‘: ,$(;a) = (zj,< :i$..:;zjm;A;)~ u&h

J = [j,, . ., . ‘$::p] & [I’: I .:.,,njA;‘zA r’n ‘:z,: &.= $z”“; (L-j,,.‘l’... p’f’l, . kksA

where h;is-an invertible holomdrphic function ,in the unit ball of C”. We sett

D%&+ I (4 = Df.:; + , (9); for h =( 1, . . . , l), we omit h; do+ = dz!\ c dz,, A . . . A .dz,‘ ~d.Zt,; [Il. . . , , l,] = M c [l, . . . , n].

3.6.2. Main Iem&. Let fif(z(A); 5((A))& i(C), then the iterated&t ’

,+,-ro d-o 4~;,+;&fi z,V&, A.do, lim lim 6

(3&j

exists, is inilependent~of the chosen admis’+’ r.f$d and depends cbntiin&uslj’on f ,fot the ,usual topology of &((@“). When Lne limit is’ &t zero, -then (3.6.2) is,equnJto

f ,‘., - , :. L

.‘- +(ini)P lim j z(A)Ly fdwMM‘ *, 60 B(AQ

where B(A); = (z(A); Ijz(il)ll < 1; IIz(A)I(~ > S}.

. . ‘. I’

:’ ~’ 3.7. Properties of Residual Currents . ,

3.7.1. The definition of the operators RPPP+ 1 and Rr+ ’ is local,‘so they define morphisms of sheaves of semi-meromorphic differential forms into sheaves of currents whose support is the essential intersection of the family 9:.

3.7.2. From Stokes formuli, we get bRP + ( - l)p+ ’ RPd = R.

3.7.3. When n@ is a complete intersection, ,’

(1) R depends in an alternating fashion on the order of’the sequence f. : (2) Invariance of tubes: Let 9”’ = ( Y;, . . . ,-Yh+l} such that Yk c Yi, where ::

n gG’ is a complete intersection, then the residue operators relative to 9 and f’ are equal.

(3) Regularity property: let XEIJ W, Bxp(* up(j)) for jo[l, . . . , p + 11, then R[I] = 0.

Properties (2) and (3) have been extended with more general hypotheses concerning the essential intersections in [9].

3.8. Extension to Analytic Spaces and Analytic Cycles

3.8.1. The definitions and results of sections 2 and 3 are valid when X is any paracompact reduced complex analytic space and when the families 9 of complex analytic hypersurfaces Y, satisfy the following condition: for every j = l,,. . . , p + 1, Yj is locally principal, i.e. every z E Y, has a neighborhood U, on which there is a holomorphic function g, such that Y,n U, 7 {XE U,; Bj(X) = O}.

228. P. Dolbcault

More generally, we consider a complex analytic n-cycle i.e. a pair y = (X, c) where X is a paracompact reduced complex space of pure dimension n and CE H,,(X, z). Let Vk)k,, be the family of irreducible components of X and let

yk = (X,, st) wheia: sk denotes the fundamental class of X,, then y =

where nk E Z.

” Then, the tubes of section 3.3 have to be replaced by TfG’ . = & .nk T;,; ;

DP+’ v:_a

p+ i = k& nk D;&i where TV,,;* 3 D$ are relative to X, ,

3.8.2 Point .Residue. Let n”,(* u 9) be the sheaf of meromorehic n-forms having their poles in u 9. When p + 1 = n, then V,(Sr) is a discrete set of X; let 1 F T(X, @,(a u 9)), for y E V,(9), Wan open neighborhood of y in X disjoint from V,(9)\{ y}, or~g’( W) such that Q 5 1 in the neighborhzod of y; let R; be * the operator R” relative to y. We define the point residue of A as

c resi,;3;,($ = R;[x] (a).

If n 9 is a complete intersection in y, if X is smooth and II W = i g; 1 . . . g; ‘; 1~ I( W, a’&) it can be shown that res.~:.p:V(~) is equal to the Grothendieck residue symbol Res,[,, . :. e.].

I

3.83. In the notation of 3.8.1., let 9 = ( Y,, . . . , Y,+ i}, Y = n F(p + 1); O<p<n,such thatdim n@(p+ l)=n-p;dimn.F<n-pi 1.

Let T be a reduced complex analytic space of pure dimension n - p, oriented by its fundamental class and 7z:X + T a morphism; let i: Y + X be the canonical embedding and assume dimr II - l(t) = p; dim& 0 i)- ’ (t) = 0 (t E T). Then, for every t E Reg T(set of regular points of T), II; ’ (t) makes sense and is a’complex analytic p-cycle. Let. CE l-(X, &pX(* u 9)). Then, for every such t, the point residue has a meaning and defines a semi-merom&phic function on Y; it is called the jbered residue of 2.

$4. Differential Forms with Singularities of Any Coditkension

The aim of this section is’ to define locally integrable diffe&tial ?orms generalizing semi-meromorphic differential forms with simple poles.

4.1. A Parametrix’ for the de Rham’s Complex. Let X be a smooth real manifold which is denumerable at infinity. Then there exist continuous linear operators on the space of currents Z(X) such that, for every current T,

T=dAT+ AdT+ RT, .

- -- V. General Theory of Multidinmsionrl Residues 229

where R is regularizing (RT is Cm), A is pseudo-local (Sing Supp AT c Sing Supp n moreover if T is O-continuous, then AT is locally integrable. The puw@@wtri~ A 8 different from that of da Rham [61,62]. In I&“, A = 6G for cuntoti) with compact supports, G being the Greertbperator and 6 the codifktwMiakib R”. Then the global construction follows de Rham ([60],#15). In the compkx case, the same formula is. valid for J’ instead of d.

$2 DeMtk~(~61],cf.also[51]). Let XbeartrrJanalyticmanifoldrail Y be a closed oriented s&analytic set of pure codimunsion r 2 1. Then Y Maas an integration current lr. We call kcrncl assocfate~ to Yevery locally intqt8blu (r - l)-differential form K with singular support contained in Y rucb that: I ,=dK+LwhetiLisC~onX.From4.l,‘K~Afuissuchaker& -.-

Let U = X\Y and .rPe(X, * I’) be thesubspace of @(X\Y) whose &natts. are the smooth differential forms q on CJ defined as: rp - K A 9 + tYl.&J whem JI and 0 are C” on X of respective degrees p - r + 1 and p;rp is d to be K- simple. Moreover the current .# = .K A $ + 8 is locally integrable and is the

I simple extension of cp to X.

W. Case uf Cauouical iujectiou. Let i: Y + X be the canonical injection, iheni*$isaC”(p - r + 1 )-form on Y and is independent of the expression of ##. res(cp) = I*$ will be called the residue form of cp; moreover dq is K-simple and res(&) = -d(res cp).

4.4 Complex Case; codimo y = 1. The definitions and results are valid, in particular, when X is a complex analytic manifold and Y a complex subvariety

1 ds of pure codimension. When codimc Y = 1, K can be locally defined by z 7

where s. = 0 is,a minimal.local equation of Y. If moreover ‘Y is smooth, wegetthen results of Leray (0.3.b).

4.5. Complex Case; codimc Y 2 1. Let X be a complix manifold.and Y be a complex submanifold of codimension’k. For k = 1, let APv9($, * Y) be the space

‘dt of locally integrable (p, q)-forms q, ‘C” on X\ Y, such that tf = t A cp + $ on

U\ Y where U is a coordinate patch, U n Y = (t = 0} where t is a co-ordinate function on U and [tjJ the currenj defined by 11 on X. For k 2 1, let 6: X --) X be the blow-up of X along Y and Y = o- ‘( Y). Consider

A”*q(X, * r) = (a&]; qeAAPv9(i, * fi} I

Then, given Y, there exists VE A k*k-l (X, * Y) and ,r E AL*‘(X) such that E Yl = d” Cd + Ctl; tl is a kernel associated to Y. Using the diagonal of X x X, it is possible to construct a parametrix on X and a kernel associated to any subvariety Y of X (J. King [39]).

*: ’

/’ .

k? Dolbeault ,-

.&et S & the46ingular set .of Y and N&:f”“(“tc’,; +S) be the complex of normal ‘tirrents on X, with supports in ,Y, suchthat their restrictions to X\S, are the dire&images, of smooth Eorins on X\S Then .for ZI) F A-(X,. e r), Res w is.defined atid bcloags to N!“( Yd, *S)(Raby~[55]).: . . : . I. . ’ : . . ,, :-. ,*‘,. ’ ‘- ._ .:. _., :,; :

$5. Interpretation of the’I&iche Homomorphisrins .

. :.

”

‘. The aim of this: paragraph -is to give an intarpretation of the ,diagram of sectiad 1.3. isi‘ terms- of di&iential forms having singularities~ on Y and of &&ents. ,TliL will be done under various hypotheses on X and Y. ‘-

‘dJ&lf’.. On. x l& (#ii . resp &(k Yj) -he the’sheaf complex of Cm differential fori&esp. of semi-mcronio~~differeAtia1 formswith polar set Y). Let I& be

!“‘;I I the+quo.&nt,.sheaf defined by’ the following exact sequencg 0 * 8x f 6x(*- ‘Y) -+ ,Qi + 0 where‘.i denotes the inclusion. ?. “..‘_.1.,, . ,-..

Sr1.2. Let 91x be the sheaf complex of currents on X and 91rm be the sflbbc&f of irrehts tithsupports in Y The’foHoPvjtlg$hott’exact shriince (p&$;~Lii,: ‘P .a : . . . . ; 1 ‘. .. ‘?” :- . . 1 ‘:’ i

:. ) (I “+ zy, G 93:x 4&,, -+ 0

Principal values and Res define sheaf homomorphisms /

.: Res:&(w Y)+9;.-p-l,ym

In fact Res(w) depends only on the class of o modulo a$, and hence defines a homomorphism .

Res: ,Qf: 1 a;,-,- r, r’ / .set vp’=j”vp:8,(*y)~~Ix,y”. ‘_

From 2.5, we get dRes = -. Resd, hence a skew complex homomorphism

we have the following cohomology diagram

(A) . . . --) HPI-(X, 8;) + HT(X, a;( * Y)) --) HPW,,Qi)

(*I Pl Gl G l(*) +Hp+‘r(X,cbi)-+. . .

Pl (B) . :: -+ H zn-pr(X,~31x)~H2,-pT(X,~alx,um)-’Hzr-p-,T(X,~Irm)

-) H2n-p-,r(X,9:,)A ,

V. General Theory of Multidimens1onA Residues 231 -I-- Y

where vi:, Rd are induced by Vi,, Res resp. and L” is defined by integration over X. The squares are commutative, except (*) which, is anticommutative.

5.1.3. Finally, we have the diagram

i

where horizontal, arrcws come from morphisms of de Rham’s theorem in cohomology [S] and homology [33]. The squares occurring in the solid diagram are commutative except one out of three in (A) + (‘6) and (A)-+(C), which is anticommutative. Moreover (B) -+ (D) is surjective.

This diagram gives an interpretation of the residue homomorphism for homology or cohomology classes in X \ Y which can be defined by restriction to X\ Y of semi-meromorphic differential forms with poles on Y. Moreover, Herrera-Lieberman [33] have a similar result in which (A) is replaced by the corresponding exact hypercohomoiogy sequence for meromorphic differential forms with polar set on Y.

5.1.4. When X and Y are manifolds, then the morphisms of diagram (5.1.3) are all isomorphisms and if res (0) denotes the Leray residue class of w, then 6(o) = 2ni Ir n res (w). When X\ Y is ‘a manifold, vp’ splits canonically. When X is a manifold, v is an isomorphism and & and vp’ split canonically.

5.2. Complex Case; codimc Y > 2 [ 11,3 1,321. In the situations of section 3, for a family 9 = {Y,, . . . , Yp+r}’ of corn p ex 1 hypersurfaces of X such that nF= Y is a complete intersection, we have codimc = p + 1. Then using residual currents, it is possible to give an interpretation of the residue homomorphisms as in 5.1. -_

5.3. Real Analytic Case: Y Subanalytic of Any Codimension [61J Let X be a real analytic manifold and Y be a closed oriented subanalytic set of pure’ codimension r 9 1 without boundary. Then, in the situation of section 4.2., we consider the sheaf 9”;( * Y) bf K-simple differential forms. Then we have the analogue of the exact sequences (A), (B), (D) where (A) is replaced by

(A’) . . . +Hqr(X,~~))~HT(X,Y~(*Y))-=HP-'+I(T(X,B;))~~ * *

and res is the map defined, by cp H res cp. In (B) and (D), 2n has to be replaced by the nal dimension n of X and Res is

equal to res followed by the homomorphism d&ned by ($ i Ir A $).

232 P. Dolbeault L

5.4, Complex Case Again. When X is a complex manifold and Y acomplex subvariety of pure codimension k, then the Residue defined by King and Raby (section 4.5) leads also to an interpretation of the residue homomorphisms ([39], c551 )a

5.5. Relation Between Homology of Currents and Eorel-Moore Homology. The exact sequences (B) and (D) and the morphism (B) + (D) are defined for any C” manifold X of dimension n and any closed set Y of X.

(B) . . . + H,~pr~X,~:,)-,H,~pr(X;~:~,,r)~Hn~p~lr(X,91um) .

(D) : . . + F,,!,,(X) . 1 +4-,(X\ Y) + H.1-,-,W

-+H,-,,-*I-(X,9’ )+. . . .X

1 -+ H,-,-,(X1--+ . . .

Theorem. When X is a real analytic manifold and Y a closed subanalytic set of atiy codimension the morphism (B) + (D) is an isomorphism [63, 62j.

This theorem is local and easy to pfove when Y is a submanifold. When Y is C? with normal crossings, it is proved usi_ng the Mayer-Vietdris exact sevence.; Whexi Y is a_n$real analytic set, then using a morphism of Hironaka n : X + Y, such_ that Y- n-‘(Y) has normal crossings, it can be._proved that A*: 9(X) i .9:(X) is sujective. The main stip is that 1c*: 8(X) + B(x) is an injective morphism of F&hit spaces. For the case of semi-analytic sets, the Mayer:Vietoris sequence ii used again.

There exists an independdnt and more algebraic proof in the’ ‘more general case of subanalytic sets, using the ‘triangulation theorem of Hirbnaka 1361, in [13]. “k. .

$6. Residue Theorem; Theorem of Leray

. .

6.1. Residue Theorein. diagram

In the notation of 4.2.and. 5.3, we have the following .

HP r(X, Yx( * Y)) .-Z HP-r+ l r(x, 8;) --HP+ l r-(x, 6,) 1 Resl 1

Hn-p ux .9:X,yr) - Hnmpel r(X, g:y-l--+ fb-,- 1 r(x, %X) Gl xl z:1

, K-pW\‘Y) 6.

- H,-,-,(Y)

V. General Theory of Multidimensional Residues 233 e

Let CO be a closed K-simple form of degree p, [o] be its cohomology class and Res [o] b: the image of Co.11 in Hnop- 1 T(X, 91,) (or in H,-,-,(X)). Then, from the exactness of ‘the lines, Res [w] = 0. This is valid, in particular, in the complex case where Y has codimension 1 and Y;(* Y) = 8;(* Y). Thus we get

6.1.1. Theorem. The image, in HnmP- i (X) of the residue of a closed K-simple p-form is zero.

6.12 The same theorem is true fur the residue defined in 4.5 and 5.4.

6.2. Gkiths’ Residue Theorem. Let X be a compact complex manifold, with dimcX=n, and .F=(Yl,..., Y,,) be a family of complex analytic hyper- ‘. surfaces. Then, in the notation of n.3, let resg,r,,(co) be the point residue at y. E V.( 9 ) of an n-meromorphic form o E T(X, Q;( + u 9 ). Then

(6.2)

The first member of (6.2) is R”[o] (1); the result is a consequence of the identity 3.7.2. for p = nandofdo=O,dl=O.

This theoreh is due to Griffiths [24] when n 9 is a complete intersection and has been extended by Coleff-Herrera to the general case [ll].

63. Converse of the Residue Theorem; Theorem of Leray [6 1,63,64]

6.3.1. Under the hypothesis of 6.1, Y being a closed submangold, let w be a _ closed differential form on Y whose image in H,, _ ,,- 1 l”(X, Bix) is zero. Then w is

the residue form of a closed p-form q E T(X, Yi(* Y)). When Y is a closed suboariety of X, there iS a canonical homomorphfsm iy : H’( Y, C) -+ HST( Y, 8,). Let /3eHP-‘+‘(Y, C) such that i*([ fl n/I) = 0. Then any, closed o E r( Y, J;-‘+‘) belonging to the class erp is the residue form of a closed K- simple form.

6.3.2 (Generalization of the theorem of Leray). Under the hypothesis of 6.1 and if Y is a closed subpariety, the following conditions are equivaient (for p + q = n):

(i) a E H,(X\ r; C) contains u closed K-simple p-form on X\ r; (ii) 6,a = [Y]nfl whereBEH#-‘+‘(Y,C).

When Y is a submanifold, condition (ii) is always satisfied.

6.3.3. Let X be a complex manifold, and Y be a complex subvariety of X whosesingular set is S. Let 6’: X’ + X\S be the blow-up.of X\S with center Y\S and d be itserestriction to X’\ Y’, with Y’ = a-‘( US). Then every class of HP(X\ Y; C) contains a smooth closed form cp on X\ Y such that: .Q is thi restriction to’X.\ Y of a locally integrable form of X such ihat dp is O-continuous and a*~ is the restriction to X’\ Y’ ,f a semi-meromorphic form on X’ with-

234 P. Dolbeault

simple poles along Y’, (cf. 4.5). This is a theorem of Leray more satisfactory than 6.3.2. in the complex case [SS].

A more precise result is obtained when X is a complex surface and Y a complex subvariety of codimension one [ 561.

See also [57].

6.4. Converse of Criffiths’ Residue ThBorem. Let X be a compact complex manifold with dim,X = n and let f L. ( Y,, . . . , Y,,) be a family of complex hypersurfaces such that I (i) n p is a ~omplete.+ersection; (ii) forj = 1, . . . ,

is positive. n, the holomorphic line bundle [ Y,] defined by the drvisor I’, ,

Let n9F(pY)v=i,,..,q and (c,) be a fami!y of complex ‘numbers such that

It C” = 0. Then there exists a meromorphic n-form q on M whose polar divisor Y’l is $ Y, and whose point residue in P, is c,, V==jl,..., 4. The main tool of the

p&1$ the, Kodaira vanishing theorem [24].

$7. Residue Formulae

7.1, The Case of Y be Oriented C” Subvariety of X.~ ‘Let X be a C” manifc’d, Y an oriented Co subvariety of X of pure dimension q, and K a kernel associated

. to .Y (4.2). ..I

Let I be a (p + lksubanalytic chain with faithful representative (2, c) with compact support 2 in X which properly intersects Y (i.e. such that dimYnZ<dimY+dimZ-dimX) and such that bZnY=@. Then for any K-simple form cp in X (4.2) whose differential dq~ is Cm in X, the following formula is valid

(7.1)

where 2’ l 0 is the subanalytic intersection of the chains I and %, the chain aY having ( Y, q) as a faithful representative and q being the fundamental class of Y.

This formula is a generalization of the classical formula (0.1); it is due to Poly [62]. Particular cases of (7.1) are due to Griffiths [23], King [38] and Weishu Shih [71]. /

Formula (7.1) is proved by methods of algebraic topology and uses tri- arigulation of subanalytic sets [36], the representation of homology by classes of


subanalytic chains [S], [21] and a theory of intersection of subanalytic chains ([62], ch. 4)

7.2. Kronecker Index of Two Currents

7.2.1. Let X bean oriented C’ manifold of dimension n and FL: S* X --) X be the spherical cotangent bundle of X. Every current Tt ‘I:“( X) has the following expression over the domain U of a chart on X. with local coordinates (x1,. * * 9 x,):

where T, is a distribution on X. ‘I The subset WF( T) of S* X such that

WF(T)nl-I-l(U)- WF(TIU)= u!,,=,WF(T,),

where WF( T,) is the wave front of the distribution T, (Hormandti [37]), is called the watle front of T. Denote by WF( T) the antipodal of WF( T); let S, TEE” such that

Wi( T)n @F(S) = 0

then the wedge product T A S is defined.

(7.2.1)

7.2.2. Let (R,), , O be a regularizingfamily of operators on the currents on X [60], [43]. According to de Rham ([60], (j20), we say that the Kronecker index .X( T, S) of two currents T, S on X of respective degree t, s, with I + s = n, is defined if for any regularizing family R,(resp. R:.), (R, T A R;, S, I ) has a limit independent of R,, R:. when E, c’ tend to zero.

7.2.3. Theorem [43]. Let T and S he CMV currents us in 7.22. such thut Supp Tn SuppS is compact. If WF( T) n %‘F( S) = 0, then .iv’( T, S) e.xi\fs cmd ;V(T.S)=(TAS,~).

7.3. The Case of Y be a Closed y-Subanalytic Chain of X. Let X, 9 be as in 7.1,9Y be a closed q-subanalytic chain. K be a kernel !rssociclted to !Y (an obvious extension of 4.2), and cp = K A t+b +.I1 be a K-,;implc form of degree p. Then (7. I) is a particular case of the formula

~(Cba],KA~+e)=~([~].(L-A~+(-l,.-’-’K A d+

+ do)) + x([p’. [y-j A t,b),

where [‘??!I denotes the integration current defined by the subanalytic cham -?I [g/3 = dK + L where L is C” and when the Kronecker indices under consideration have a meaning ([43], 5).

236 “/ /’

,/ P. Doibeault

58. A New Definition of Residual Currents (i77], [78]) [ ,, ’

8.1. The present section contains new results obtained since the, original Russian edition of this work. We use the notations of section 3. All definitions and results are local. Let f/g = (A/g,, . . . , f, + Jgp + 4 ) be an ordered set of (p + q) globally defined meromorphic functions on a connected complex mani- foldX;fj,gjEU(X),j=l ,..., p+q.Letd=(YI ,..., Y,+,}beanordered family of hypersurfaces of X, where 5 has the global equation gj = 0 in X. The aim of the new definition is:

1. to replace integration on semi-analytic chains by integration on X after multiphcation by smooth functions xi vanishing on shrinking neighborhoods of uj, j = 1.. . , p + q;

2. to replace the use of admissible maps (3.2.) by a mean value on parameters defining the xi;

3. using several successive principal values instead of one at most in section 3.

As a result a rather satjsfactory exterior product on residual currents will be obtained. .

! 8.2. Defipitions and Tools. Let x: R + [0, 1 ] be a C” function such that

I x(x) = 0 for x 6: cr < cl; x(x) = 1 for c2 Ix.

t LetC,=

I SE R”; Sj > 0; f .Sj = 1 be the (m - 1 )-canonical euclidean sim-

j=l I plex in Iw” with Lebesgue measure S, of total mass 1. For E > 0, we set

.x., = X(lgjl/E”‘). Then, there exists a finite number of non zero vectors n,,..., ~,EQ~+~ such that, if SEC,+, is not orthogonal to any of them, then

RPPqU/gl(E”) = cft/gl). . . Up+qlgp+qW’x~ * . . . A d”xpxp+-v . . xp+q (8.1) has, in the sense of currents, a limit RPPP Lflg] (s), when E --) 0, which is locally

constant on Cp+q \

ror {(n,, s) = O}; RPP4Lflg] is theV mean value

j RP~qlY/dWSp+q(4~ E l +*

In the expression for xj, t.he function ir may be replaced by the product of gj by a nowhere vanishing holomorphic function on X, without changing the result. Notations:

RPJ’qLf/sl =d”U/s,l A.. . A d”lfplgplLfp+Jgp+J~ - + Lfp+qhp+ql (8.1)

The technique is parallel to that of section 3 but more simple and transparent: it also depends on Hironaka’s morphisms.


8.3. Results. Let o be a holomorphic differential form of degree r on X. ..T

83.1. Rp-+‘rCf/g] A w& ‘(‘*p)(Xk sptRPP4Cf/g] c ny, with 6’ = (gr,. . . ,gp).

8.3.2. From (8.1), d” acts on RpP4Lflg] as on smooth differential forms.

8.3.3. With the multiplication cf/g] . cf’/g’] = m/r/g’], the principal value currents form a field and f/gwCf/g J is a field isomorphism.

Furthermore, RP~Lf/gl = d”Cfi/s~l A . . . A d”lYp/gplLf,+ 1. . a&+,/ gp+ le.* - gp+q 1

8.3.4. In case q = 0 or 1, and n 9, (resp. n 9( p + 1)) is a complete intersection, the new residual currents coincide with the ones of Coleff-Herrera (section 3).

One of the main properties is: RPPqCf/g] is alternating with respect to the residue factors.

8.4. Application. In the complete intersection case, the following properties are equivalent:

(i) h&O(X) belongs locally to the ideal generated by gr, . . . , gp; (ii) hd”[ l/gr] A . . . A d”[ l/g,] = 0, [76-J, [74].

If X is a strictly pseudocon_vex domain in C” whose defining function belongs to C”(X) and if the gjc 6(X), there is a formula of the type

Wl J’(z) + hRPCIhlW’(. , ~11, where F belongs to the ideal and $ is a certain test form [76].

A similar result, but with growth conditions, holds when X = C” and the yj are polynomials [76].

8.5. Relation with Meromorphic Extension of ISI”, for f Real Analytic ([72],

[73]). Irfis holomorphic, then lim Ifl”/f= vp( l/f) = [l/f] [75]. E-0

More generally, with the notation of 8.1., 8.2., M. Passare [79] proves:

’ RfWlId = ;~n)Wl I”/sl) * . . . * Wls,l”/s,)

x (Igp+ll%Ip+l)~ * * (lgp+qlelgp+q). The steps of the proof are as usual: normal crossing case and general case using morphisms of Hironaka.

Previously, A. Yger used this second definition of residual currents to get applications as in 8.4. [SO].

8.6. Another Application of Residual Currents [74]. Let X ‘)e a complex manifold and Y.a complex subspace of pure codimension p > 0, v, hich is locally a complete intersection.

238 * P. Dolbcault /

:.

A current RE Tr(X. Of’.‘) is called locally residual, if for every sufficiently sniall open set U of X, RI U = RP, [G] where GE r( U, a’(* u 9)) and n9= YnU.

It follows that for every TE rr (X, ~2%“. “) with d”T = 0, we have T = R + S’S, where R is locally residual, and SE r,( X, Vr. p - ’ ). Moreover, R and d”S are

. uniquely determined. Consider the p-cohomology group of the compiex ( TV ( X, Q’*..), d”) (moderate cohomology); locally residual currents arc cane- nical representatives for classes of this group. Moreover holomorphic p-chains are residual currents and there exists a duality theorem for such currents.

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44. Le]ong, P.: Integration sur un ensemble analytique. Bull. Sot. Math. Fr. S&239-262 (1959). Zbl. 79,309 I

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; : 1: : i . , L , , , -

, , : . ” - . .,: Author Index ’

, ; . 1:: ‘, ‘. . I

Ahem,P.R.61 .> I Ahlfom, L.V. 9. 16 Ajrapetyan, R.A. 56. 63, 82, 85,.88,,89,.102., Ajzenbcrg, L.A. 31, 53, 54 0 Aleksandrov, A.B. 14, 34 Alexander, H.J. 5, 6, 139,. 161

Amar, E. 25, 66,19 Andersson. MEL. 51, 74 - Andreotti, A. 4, 11,21.64,?9, 186, 107 Aronov, A.M. 29 Atiyah, M.F. 11, 112 2 : <,

Baouendi. MS. 9,26, 82,83’ ‘ Basener, R,F. 448 Becker, E. 151 r Bedford, E. 13, 86 Behnke, H. 35 Bergman,S.6,24,31-3j’ ‘, ’ ’ Bemdtsson, B. 8,25,51,66,73,80

Bernstein, S.N. 3,X86 Bieloshapka, V.K. 6, 162, 193, 19i Bishop, E. 2,26,95, 140, 143,144, 150 Bochner, S. 3, 6, 7, 8, 24-33, 192 Bogsees, A.I. 89, % Bogolyubov, N.N. 3,86 Bonneau, P. 73 Bony, J.-M. 82

Cartan, H. 4,,1$, 24, 25,‘35, 37-Q 70 '. Charpentier, P. 26, 79 Chirka, EM. 9, !3, 30, 64, 86 . ‘. , ’ Chem, S.S. 5, 142, 161, 162, 173, gO5, : chow, W.L. 133 . Coleff, N.R. 219,224,233 . ._, , Cousin, P. 13, 23-26, 39 Crofton, M.W. 138 Cumcnge, A. 67 (.

Dautov, Sh.A. 26, 51, 14,19, 104 -’ Debiard, A. 48 de Rham,G. 217,229,235 Diederich, K. 10,46, 161 Dirt&let, P.G.L. 34 Dolbeauh, P. 11, 31, 70, I!,98 Dolzhenko, E.P. 9 Draper, R.N. 135 Dyson, F.J. 8 Dzhrbashyan, MM. 78

Epstein, H. 86 Ehrenpreis, L. 12 Einstein, A. 17 El Mir. H. 151, 152 Ezhov, V.V. 6, !92

Bore!, A. 16, 219 Bott, R. 219 Boutet de Monvel, L. 62 Bretnertnann, H. 25,4749,86 Brieskom, E. 6 Bras, I. 81, 82 Browder, F.E. 86 . Brown, W. 22 Bums, D., Jr. 5, 6, 161, 176, 193, 198, 207, 208 Gambaryan, M.P. 154 Buslaev, V.I. 12 Gaveau, B.48

Fantappib, L. 7, 25. 55 Fefferman, C. 5, 6, 62, 161, 176 Fischer, 0. 104 Fore!& F. 53 FolIand, G.B. 106 Fomaess, J.E. 10,46,48 Fredericks, G. 107

Caratheodory, C. 6 Carleson, L. 8, 21, 79 Cartan, E. 5, 32, 161, 173, 205, 207

Gindikin, SO. 8, 27, 54, 60, 110 Gonchar, A.A. 10 Gleason, A.M. 32,68 . Grauert, H. 11, 15, 16,25,49,98,99, 104

L

244 Author Index

Grumett, i. 148, 149 Grothedieck, A. 69.219 Gunning, R. 118

. Green. G. 21 Greenfield. S.J. 95

Lebesgue, H. 27 Grcincr, P.C. 75, 106 Lefschetz, S. 30 GtWtths, Ph.A. 13. 16, Ldmann, H. 8 119,

141, 143, 233.234 149,219. ~eiterer, 25.66 I. Lcjbenson, Z.L. 68

A

~longs P. 5. 13.25,% 147 , 140, 47,70,72,78, 139,

b=v, J. 7, 15,25,41,49,j4,217-219, hi. E.E. 4, @-at, 4144

233

hw, H. l&26,41,42,93, 105 Lie, S. 32 Licb, I. 1425, 26, W-104 65, kh=ub D. 71‘223,231 &w&a, E. 26.62

Lipchits, R. 28’ l-dda, A.V. 6, 162, 197 163, Lojasiewicq S. 219 .

Hakim, M. 48 Haps, M. 87 Hartogs, F. 1, 2, 9, 21. 22 Harvey. R. 13,26, 30,65,82, 143, 154 Hefer. H. 40 Herrera, M.E. 71,219,221224,231.233 Her&, M. 118 Hilbert, D. 8 Hironaka, H. 219,223,232 Hitchin, N. 17 Hoffman, K. 36 H&ntander, L. 11, 25, 65, 73, 82, 101 Hua, L.-K. 31, 33.34 Hedge, W.V. 6

,’ Jagolmtzer, P. 81,82 - Jensen, J.L.W.V. 145, 146 ’ JGricke, B. 63

MQmge, B. 6. ii, 86 M=bau, A. 3,37,54,81 Martinelli, E. 7.2431 Manin, Ju.1. 17 Mergnlyan, S.N. 9 Mel’nikoq M.S. 9

Josefson, B. 13 Jost, R. 8

Kallin, Eva, i

Mills, 17 R.L. Milnpr, J. 6 Minkowski, H. 17 Mittag-teWer, M.G. 23 Mohn, R.E. 148, 149’ Moore, J.C. 219

Kashiwara, M. 11, 82. Kawai, 11, 82

Money, 11, 14.25.49 C.B.

Khenkin, G.M. 7,8,9, 11,25-27,48,x-56,61, 63, 64-68, 74-82, 85-93, 102, 104-107, 109-l 12, 147,205

Keldysh, M.V. 9 Kerzman, N. 26, 62, 65,‘68, 104 King, J. 149, 229, 232;234 Kneser, H. 25

MO=, J.K. 5. 161, i62;165, 173; 176,205 Mostov, G.D. 16 Mwnford, D. 122

Nacinovich, M. 107 Nagel, A. 68

Kobayashi, S. 6 Kodaira, K. 6 Koh J.J. 11, 25, 44, 46, 49, 73, 101, 106 Koppelman, W. 26,28, 102 KnuMin, N.G. 6, 163, 172, 184, 192, 198,208,

209

Napalkov, V.V. 3 Narasimhan, R. 4,49 Naruki, I. 27, 34, 83, 88, 90, 107 Neuman, C.G. 25,72 Nirenberg, L. 11,44,65.82,83,96 Norguet, F. 25,27,47, 56, 100,217

Kuranishi, M. 16 Kytmanov, A.M. 29

Laplace, P.S. 34 Lavrentiev, M.A. 9 hwxt, H.B. 13, 26, 30, 154

Oehme, R. 86 Oka, K.4,13,16,24,25,3741,4449,65.71,72 Gvrelid, N. 26, 48, 75, 104 .

Palamodov, V.P. 11, 12 .

Passare, M. 69 Penrose, R. 17,27, 110

. . .

Petrosyan, AI. 9,63, 65

PllU& P. 48 Fhong, D.H. 61 Fiatetski-Shapiro, 1.1. 59 Picard, E. 217 . I’iachuk, S.I. 5. 6,86, 156, 161, 192 Pins, J. 96 Plateau, J. 154 Poinca& H. 1.5, 13,22, 161, 166.217

Spenecr, D.C. 11,25,72 Stein, E.M. 26.60.61.75, 106 Stein, K. 14, 35, 72, % Stokes, 0.3 1’ Stall, W. 13, 25. 78 Stolmberg, G. 154 Stout, E.L. 66 Stutz, J. 128

Poisson, S.D. 34 Polk@, J. 89 Poly, J.C. 234 Polyakov. P.L. 26.27,67, 76, 77, J 34, 109 Pomp&t, D. 21

Szeg6, ci. 26.32

. Raby, G. 230,232 Rado, T. 119 Radon, J. 27

Author Index

Ramires de Arellano, E. 25,48, 53 Range, R.M. 26,82, 102-104 Reinhardt, K. 5 Remmett, R. 14, 150 Riemann, B. 5,21 Robin, 0. 223 Romanov, A.V. 26,51,53,74, 106 Rosenthal, A. 9 Rossi, H. 4, 11,26,34,36,“43,48, 100, 118 Rothstein, W. 99, 103, 152 RouchC, E. 130 Rudin, W. 53, 67, 134, 149 Runge, C. 8,21,24,38 Rutishiiuser, H. 13, 140

Taiani, G. 94,95 Tanaka, N. 5. 161,205 Taylor, B.A. 140 Taylor, J.G. 86 Temlyakov, A.A. 54 Titullen. P. 2.35.47 Tomaasini, G. 81 . T*es, F. 9.26,82,83,87, 105 Tsikh. A.K. 31, 130 Tumanov, A.E. 93,205

Ullman, J.L. 140

Varopoulos, N.Th. 79 Varchettko, A.N. 6 Vesentini. E. 100 Vergne. M.R.H. 34 Vitushkin, A.G. 6, 7, 9, 12, 21, 36, 64, 65 Vladimirov. V.S. 3, 8, 34, 86

Sadullaev, A. 13,48 Sato, M. 7, 11, 81, 82, 106 . Schneider, R. 61 Schwartz, L. 71

. Segre, B. 161 Sergeev, A.G. 34,63, 75 , Sent, J.P. 4, 13, 16, 72 Shccrbina. N.V. 4 Schmidt, W. 34 Shiffman, B. 143, 148, 149, 151 Shih, W. 234 Shilov, G.E. 4 . :

Waelbroeck, L. 49,,. 50 Walsh, J.L. 9 Webster, SM. 161, 198,205208 Weierstrass, K. 5, 12, 13,21-23, 119 Weil, A. 3, 7, 24, 25, 41, 65, 217 Weinstock, B.M. 29 Wells, R.O. 5, 26, 65, 82, 83, 95, 161, 166. 198 Wermer, J. 2, 65, 82 Whitney, H. 14, 123, 127, 128, 131, 135 Wirtinger, W. 135 Wolff, T. 79

Yang, C.N. 17 Shnider, S. 5, 6, 161, 176, 193, 198, 207, 208‘~ Yau. S.-T. 6, 154 Sibony, N. 26,48,77,93, 148, 149 Yuahakov, A.P. 31, 130 Siu, Y.-T. 6. 26, 82, 102, 104, 144 . . SjBstrand, J. 62 Zcmer, M. 86 Skoda, H. 26, 106, 147 Z&phi, A. 67 Sokhotskij, Yu.V. 29 Zharinov, V.V. 3, 8 Sommer, F. 96 Znamenskij. S.V. 54

245

_, . _

:

I1

. ‘. .i

Subject Index . .

Barrier for a domain local q-dimensional 103 strong 43

Boundary, Bergman-Shiloii 4, 3 1 Bundle

dual 97 holomorphic vector 16,97

’ Chain

* holomorphic 132 intersection 133 Moser 173 semi~analyric 224 / I

Characteristic, non-sphericity 185 ChWS

Chern 141 fundamental 221

Cocycle, holomorphic 37 &dimension 120 Compact set

holomorphically convex 38 strictly linearly convex 54 strictly pseudoconvex 65

Component, irreducible 124 Cone

tangent 126 Whitney 127

Conjecture, Weil 37 i Continuation of CR-functions 87 Converse of Griffiths’ Theorem 234 Coordinates, circular normal 179

@ Covering analytic 122 generalized 122

CR-dimension 15 1 CR-forms lo5 CR-function 80

andYtiC representation 81 CR-manifold 80

q-concave 90 standard 88

Crossings ,

complete 41,224 essential 225 normal 222 proper 132 tranversal 133

Current [A] 137 ’ closed 137 positive 137 pseudo-local 229 1’ residue 222

Cycle, maximally complex 153

dlcohomology, Dolbeault 97 a &condition of Neumann-Spencer 29 Degree of algebraic set 134 * &equations 69 ; &problem of Neuman-Spencer 72 Dimension 120 ’ Divisor 72

of a meromorphic function 144 Domain

Classical 32 of type I-IV 32-33

linearly convex 53 of holomorphy 3.22 pseudoconvex 22

Levi 42 strictly 4

Qconvex (q-concave) with piecewise smooth boundary 103

q-linearly concave (convex) 107 Siegel 59 strongly (strictly) q-concave 99. 100 Riemannian 47

Equation PoincarUelong 25 tangential Cauchy-Riemann 29

Extension, analytic 152

,

.c

4

/

_ Form s&Kd 69 eirctilar normal 178 elementary 222 Fubini-Study 135 fundamental 135 K-simple 229 Levi,

for a function 42’ for a CR-manifold 87, 90

linear normal 164 semi-meromorphic 2 17

-&h values in a bundle 97 Formula

Bochner-Martinelli 28 qf&y 21 t%tchy-FantappibLeray 50 for solving &equations 74 residue 216,217.234 Weil 35

Function characteristic 145 defining

canonical 123 ,’ minimal 121

holomorphic 2 1 p-convex 98 plurisubharmonic 5, 46

strictly 46

Index intersection 131, 133

complete 132 Kronecker 235

Indicator, FantappiC 54 Initial data

of a normalization 164 of a parametrization 182

Integral, Cauchy-Fantappie t T1

Kernel, Aociated 229 Kernel function, Bergman 31

Manifold generic 80 holomorphically convex 49 incidence 142 Levi-flat 88 maximally convex 153 Stein 14 strongly p-convex (pAncave) 99 strongly pseudoconvex 49 totally real 80. 151

Subject Index 247

Mapping admissible 225 Fantappli 37 Gauss 142 nornxhzmg 164 proper 122

Multiplicity of - analytic set :30

holomorphic mapping I29 pole 23 zero 23, 127

n-cycle, complex analytic 228 Normalization 164, 179

of one-dimensional analytic set 126 ,. standard 172 Number, Lelong 139

Operator Cauchy-Riemann 21

tangential 105 Hanke161 Neuman-Spencer 73 ’ semi-holomorphic differential 232 Taplitz 61

Order of growth of analytic set 78 holomorphic function 77

Parameter local normalizing 125 normal

circular 180 linear 175

Parametrix 228 Parametrization of a chain, normal 180 Point

regular 120 singular 120

Polyhedron analytic 35 complex non-degenerate 36 strictlypseudoconvex 57 weil3,35

Polynomial Levi 43 Weierstrass 21

Prechain 224 p-residue 226 (p+ I)-residue 226 Principle of continuity, Hartog 22

.

248 Subject Index \ /

\ Problem

Cousin 24 Corona 79 Levi 22 Naruki 90 Poincare 24 Weierstrass 22

Projection Bergman 31 Szegi, 32

Regularizing operator 229 Residue

cohomological220 composed 218 homological220 point 219, 228

Residue form 2 18 Residue homomorphism 217

cohomological221 homological220

Section of bundle 97 set

algebraic 133 analytic 120

irreducible 124 at a point 124 locally 120 *

locally 124 purely p-dimensional 120 , with border 137 ,-

locally removable I22 ’ polar 218 totally phiripolar 150

Spaa complex tangent 42 local cohomology 219 separable cohomology 100 tangent 128

Straightening a hypersurfaa along a chain 176

Symbol, Grothendieck residue 219

Theorem Fantappie-Martinegu 54’ Griffiths 233 Leray 218 233 Oka-Weil approximation 38 residue 217,232,233

Theory of Leray-Norguet 217 Transformation, Radon-Penrose 108 Tul< 225

Value, Cauchy principal 222 Vector, tangent 126

Wave front or current 235

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