Proceedings Symposium on Value Distribution Theory in Several Complex Variables

Proceedings Symposiumon Value Distribution Theory in

Several Complex Variables

Notre Dame Mathematical LecturesNumber 12

Proceedings Symposiumon Value Distribution Theory in

Several Complex Variableson the occasion of the inauguration of Wilhelm Stoll as the

Vincent F. Duncan and Annamarie Micus DuncanProfessor of Mathematics

April 28/29, 1990University of Notre Dame

Edited byWilhelm Stoll

UNIVERSITY OF NOTRE DAME PRESSNOTRE DAME, INDIANA

Copyright © 1992 byUniversity of Notre Dame Press

Notre Dame, Indiana 46556All Rights Reserved

Library of Congress Cataloging-in-Publication Data

Symposium on Value Distribution Theory in Several Complex Variables(1990 : University of Notre Dame)

Proceedings : Symposium on Value Distribution Theory in SeveralComplex Variables : on the occasion of the inauguration of Wilhelm Stollas the Vincent F. Duncan and Annamarie Micus Duncan professor of math-ematics. April 28/29 1990, University of Notre Dame.

p. cm. — (Notre Dame mathematical lectures ; no. 12)ISBN 0-268-01512-01. Value distribution theory—Congresses. 2. Functions of several com-

plex variables—Congresses. I. Stoll, Wilhelm, 1923-n. SeriesQA1.N87 no. 12[QA331.7]510 s—dc20 91-42751[515".94] CIP

Manufactured in the United States of America

CONTENTS

FOREWORD vii

PROGRAM OF THE SYMPOSIUM ix

HIGH POINTS IN THE HISTORY OF VALUEDISTRIBUTION THEORY OF SEVERAL COMPLEXVARIABLES

Wilhelm Stoll, inaugural lecture: 1

THE NEVANLINNA ERROR TERM FOR COVERINGS,GENERICALLY SURJECTIVE CASE

William Cherry 37

ON AHLFOR'S THEORY OF COVERING SPACESDavid Drasin 54

C"—CAPACITY AND MULTIDIMENSIONAL MOMENTPROBLEM

Gennadi Henkin and A.A. Shananin 69

NEVANLINNA THEOREMS IN PUSH-FOREWARDVERSION

Shanyu Ji 86

RECENT WORK ON NEVANLINNA THEORY ANDDIOPHANTINE APPROXIMATION

Paul Vojta 107

DIOPHANTINE APPROXIMATION AND THE THEORY OFHOLOMORPHIC CURVES

Pit-Mann Wong 115

SOME RECENT RESULTS AND PROBLEMS INTHE THEORY OF VALUE DISTRIBUTION

Lo Yang 157

FOREWORD

In 1988 Vincent J. Duncan and Annamarie Micus Duncan spon-sored a chair at the University of Notre Dame in honor of WalterDuncan, a 1912 graduate of Notre Dame and a long time trustee ofthe University. In the fall of that year I was appointed to this chair. Inconjunction with the inaugural lecture, the University of Notre Dameheld a symposium on value distribution in several complex variablesApril 28/29, 1990. It was to reflect the growth of this field from its be-ginning about 60 years ago as well as its connections to related areas.The Symposium was solely funded by Notre Dame. Thus only speak-ers present within the USA at the time could be invited and supportwas restricted to them. Professor Shiing-shen Chern, who contributedsubstantially to the field and helped to build up Mathematics at NotreDame, declined to lecture, but he participated actively in the Sympo-sium as an honored guest. Professor Phillip Griffiths made significantcontributions to the field, but in the end, his obligation as Provost ofDuke University prevented him from coining. The names of the otherinvited speakers and the title of their talks can be found in the Sympo-sium program reprinted below. Two significant lectures preceded theSymposium the day before. Sponsored by the Physics Department,Paul Chu gave an University wide address on superconductivity. Pro-fessor Chern attended his son in law's lecture. Dr. Peter Polyakov, aclose collaborator of Professor Henkin and a recent immigrant to theUSA, spoke about a proof of a 1973 conjecture of mine in a mathemat-ics colloquium. The Symposium was well attended, but, unfortunatelyno precise list of participants was kept.

In these proceedings, some speakers wrote on the topics of theirlectures, some others selected different topics and some did not sendcontributions. Serge Lang, who had spoken on William Cherry's re-sults, requested, that in the Proceedings these results should be com-municated directly. My inaugural address in the Proceedings is anexpanded version of the original one.

I thank all those who made this Symposium and these Proceed-ings possible and those who contributed and helped with it. In partic-ular I thank the donors for their generosity.

Wilhelm StollJuly 1991

PROGRAM

Saturday, April 28, 1990

9:00-9:15 am Introducing RemarksTimothy O'Meara, Provost, University of

Notre DameFrancis Castellino, Dean, College of Science

9:15-10:15 am Inaugural LectureWilhelm Stoll, Notre DameTitle: High Points in the History of Value

Distribution Theory of Several ComplexVariables

10:30-11:30 am Bernard Shiffman, Johns HopkinsTitle: Bounds On the Distance to Algebraic

Varieties in Cn

12:00 noon Lunch - Morris Inn

2:00-3:00 pm Yum Tong Siu, HarvardTitle: Some Recent Results On

Non-equidimensional Value DistributionTheory

3:15-4:15 pm David Drasin, PurdueTitle: The Branching Term of Nevanlinna's

Theory

4:30-5:30 pm Paul Vojta, Institute for Advanced StudyTitle: Recent Work On Nevanlinna Theory

and Diophantine Approximations

6:30 pm Dinner - Morris Inn

ix

x Program

Sunday, April 29, 1990

9:15-10:15 am Serge Lang, YaleTitle: The Error Term in Nevanlinna Theory

10:30-11:30 am Pit-Mann Wong, Notre DameTitle: Second Main Theorems of Nevanlinna's

Theory

12:00 noon Lunch - Morris Inn

1:45-2:45 pm Lo Yang, Notre Dame/Academia Sinica, BeijingTitle: Some Results and Problems in the Theory

of Value Distribution

3:00-4:00 pm Gennadi Henkin, Notre Dame/Academy ofScience USSRTitle: The Characterization of the Scattering

Datas for the Schrodinger Operator inTerms of the d - Equation and GrowthConditions

HIGH POINTS IN THE HISTORY OFVALUE DISTRIBUTION THEORY OF

SEVERAL COMPLEX VARIABLES

Wilhelm Stoll

Inaugural Lecture

Timothy O'Meara and Frank Castellino thank your for your kindintroduction. I am deeply moved by your words and by the appoint-ment to the chair. Foremost I thank the donors Vincent J. Duncan andAnnamarie Micus Duncan for their generosity. My colleagues and Iare most grateful for this recognition of our work by the donors andthe administration of the University.

Ladies and gentlemen, colleagues, speakers and participants!This inaugural address opens the Symposium on Value DistributionTheory in Several Complex Variables sponsored by the Universityof Notre Dame. Welcome to all of you. An inaugural address, anAntrittsvorlesung, so late in life seems to be out of place and perhapsshould be called an Abschiedsvorlesung. Yet, hopefully, this is pre-mature and I can be around a few more years. Taking the hint, I willlook backwards and recall some of the high points in the developmentof the theory. Time permits only a few topics.

Looking backwards, out of the mist of time there emerges not anabstract theory but the lively memory of those who taught me math-ematics: Siegfried Kerridge, Wilhelm Germann, Wilhelm Schweizerand later at the University Hellmuth Kneser, Konrad Knopp, ErichKamke, G. G. Lorentz and Max Mtiller. Also there appear those whoinspired me but who were not directly my teachers: Heinz Hopf, Her-mann Weyl, Rolf Nevanlinna and one who is right here with us:Shiing-shen Chern, we all welcome you. Thirty years ago you re-cruited me for Notre Dame. You supported the growth of this de-partment in many ways. Your work on value distribution in several

This research was supported in part by the National Science Foundation Grant

DMS-87-02144.

1

2 High Points in the History of Value Distribution Theory

complex variables counts as one of your many marvelous contributionsto mathematics. Thank you for coming.

The giants of the 19th century created the theory of entire func-tions. In this century, in 1925, with a stroke of genius, Rolf Nevanlinnaextended this theory to a value distribution theory of meromorphicfunctions. His two Main Theorems are the foundation upon whichNevanlinna theory rests.

In 1933, Henri Cartan [8] proved Nevanlinna's Second MainTheorem for the case of holomorphic curves. If we view curves be-longing to the theory of several dependent variables, then Cartan'spaper provides the first theorem in the theory of value distribution inseveral complex variables. Thus let me outline his result. However, Ishall use today's terminology and advancement.

For each 0 < r G R define the discs and circle

(1) C[r] = {z G C | \z\ < r} C(r) = {z G C | \z\ < r}

(2) C<r> - {z G C | |*| = r} C* = C - {0}.

An integral valued function v : C —> Z is said to be a divisor if

(3) S = suppz/ = clos{^ G C | i/(z) i=- 0}

is a closed set of isolated points in C. For all r > 0 the countingfunction nv of z/ is defined by the finite sum

(4)

For 0 < s < r £ IR, the valence function Nv of v is defined by

(5)

If h ^ 0 is an entire function, let /^(z) be the zero-multiplicityof h at z. Then /^ : C — -> / is a non-negative divisor called the zerodivisor of h.

Wilhelm Stoll 3

The exterior derivative d = d + d on differential forms twists to

(6) d"=^(5-0)

on complex manifolds. Define TO : C — > R by TO(Z) = \z\2 for z E C.Define

(7) cr

If r > 0, then

(8) j a = 1.

C<r>

If h ^ 0 is an entire function and if r > o, the Jensen Formula

(9) AW(r,s)= I log/io-- j log/io-

C<r> C<s>

is a forerunner of Nevanlinna's First Main Theorem.Let V be a normed, complex vector space of finite dimension

n + 1 > 1. Put V* = V - {0}. Then the multiplicative group C* actson V*. The quotient space P(V) = K/C* is the associated projectivespace. The quotient map P : V* — > P(V) is open and holomorphic.If M C V, put P(M) - P(M H K). If W is a linear subspace ofV with dimension p + 1, then P(W) is called a p-plane of P(V).If p = n — 1, then P(W) is called a hyperplane. The dw0/ complexvector space V™ of V consists of all C-linear functions a : V — » C.Here ||a|| is the smallest real number such that |a(s)| < ||a|| ||E|| forall g E V. Then || || is a norm on V*. Also write <g, a> = a(j). Here<g5a> = <a,£> indicates (V*)* = V. If a = P(a) G P(V*), thenE[a] = P(ker a) is a hyperplane in P(V). The assignment a — > £7[a]parameterizes the set of hyperplanes bijectively. The distance fromx = P(j) E P(V) to J5[a] is measured by

= s l -


Let / : C -* P(V) be a holomorphic map. A holomorphic mapt) : C — >• V* is called a reduced representation of / if P o b = /.A reduced representation exists. Then to : C —» V# is a reducedrepresentation of / if and only if there is a holomorphic functionh ; C — > C* without zeroes such that to = hti. For 0 < s < r e Rthe characteristic function of / is defined by

(11) T/M*: y log||b||<T- J log||b||<7.

c<r>

By (9), T/(r, s) does not depend on the choice of b, Since log ||b|| issubharmonic, Ty > 0, If / is constant, b can be taken as a constant.Hence T/(r, s) = 0, If / is not constant, then T/(r,a) > 0 and2/(r, s) — * oo for r — > oo. If || || and ||| ||| are two norms on V,there are constants <72 > Ci > 0 such that Ci|||£||| < ||E|| < C72|||£|||for all g e V. Put C = log C^/Ci > 0. If 0 < 5 < r, then

(12) i r / ^ ^ l l l D - r X r ^ J H i i D i ^ aLet / ; C -* P(V) and p : C -» P(V*) be holomorphic maps.

They are called free if f ( z ) £ E[g(z)] for some z e C. Take reducedrepresentations b of / and It) of g, then (/, 5) is free if and only if<b, to> = /i ̂ 0. If so, the intersection divisor -/x/ j f l = ^ > 0 doesnot depend on the choices of b and to. Its, counting function and itsvalence junction are abbreviated by n/jff and Nf# respectively. Thepair (/,#) is free if and only if D/,3, D ^ 0. If so, for r > 0 thecompensation function m/}5 of (f,g) is defined by

(13) mf,g(r)=c<r>

For 0 < s < r, the identities (9), (11) and (13) imply the First MainTheorem

(14) T,(r, a) + Tg(r, s} = AT/j5(r, 5) + m/» - m/^(«).

Cartan [8] considered the case of constant g = a G P(V*) onlywhich yields

Wilhelm Stoll

(15) T/(r, s) = Nfta(r, s) + mf<a(r) - mfia(s)

which Cartan [8] mentions only implicitely. If n = 1, Rolf Nevanlinnaproved (15) in [32] (1925).

If / or g or both are not constant and if (/, g) is free the defectis defined by

(16) = 1 - lim " ™ v » ~ / <~

The map 0 is said to grow slower than /, if T5(r, s)/T/(r, s) — > 0for r- — > oo. By (12), the defect does not depend on the choice ofthe norm on V. Also the defect is independent of s. Observe that

6(g, /). Since most investigators concentrate on constant g or on thecase where g grows slower than /, this symmetry is little known.

Since the choice of the norm on V does not matter, we can choosea hermitian norm which comes from a positive definite hermitian form(•!•) : V x V -* C with ||E||2 - (5(5) for £ E V. Define r : V -> Cby r(g) = ||E||2 for £ E V. Then r is of class <7°°. There is one andonly one positive form fi of bidegree(l,l) on P(V), called the FubiniStudy form such that dcflogr = P*(fi) on K- Let b : C -> K be areduced representation of /. Then / = P o t) implies

If Stokes theorem and fiber integration are applied to (11) we obtainthe Ahlfors-Shimizu definition of the characteristic function of /

r

(18) T/(r, s) = I ! /*(«) y for 0 < s < r.

Here Aj(t) = I f*(Sl) > 0 increases. Put A/(oo) = limy i-KX)

oo. Then

High Points in the History of Value Distribution Theory

(19) lim = Af(oo).r->oo log T

Now / is constant if and only if Af(oo) = 0 and / is rational if andonly if Af(oo) < oo.

Let 31 = {a>j}jeQ be a family of points a, G P(Vr*) representat-ing hyperplanes. If P C Q, define Sip = {a>j}jep. For each j G Qpick Oj G F** with a^ = P(ctj). Our definitions will not depend on thechoice of a,. Put g = #Q. Then 31 is said to be linearly independentif there is a bijective map A : W[l, #] — > Q such that Ct^), . . . , cr\(g)are linearly independent. If so, then q < n + 1. Moreover SI is said tobe basic if 81 is linearly independent and q = n + 1. Moreover 31 issaid to be in general position if Sip is linearly independent for eachP C Q with 0 < #P < n + 1. If N is an integer and if q > TV > n,then SI is said to be in N-subgeneral position (Chen [9]) if for everysubset S of Q with #S = N + 1, there is a subset P of S such thatSip is basic.

Let / : C — > P^F) be a holomorphic map. Then there is aunique linear subspace W of smallest dimension k + 1 of V such that/(C) C P>(W). Then / is said to be k-flat. If k = n, then W = Vand / is said to be linearly non-degenerated.

Take 0 < s G R. Let G : R[0, +00) -> R and tf : R[s, +00) befunctions. Then G<H means that there is a subset E of finite measureof R+ = R[0, +00)' such that G(r) < H(r) for all r G R[0, +00) -E.

Second Main Theorem (Cartan [8] 1933)

V be a hermitian vector space of dimension n + 1 > 1. Letf : C — > P(V) fee 0 linearly non-degenerated, holomorphic map.Let SI = {aj}jeQ te a finite family of "hyperplanes" Q.J G P(V*) mgeneral position with n + 1 < q = #Q < oo. Tizfce s > 0 «nrf £ > 0.

(20)

-n(n + 1)(1 + e) logT/(r, s)

Wilhelm Stoll

As a consequence, we obtain trivially

Defect Relation (Cartan [8] 1933)

Under the assumptions of the Second Main Theorem we have

(21)

If / : C —* P(V) is only k-flat, and if 91 is in general positionsuch that (/, a,j) is free for each j 6 Q, Henri Cartan conjectured in1933 that

(22)

which was proven by Nochka [35] in 1982. Thus if #Q > 2n +1 and/(C) n E[a.j] = 0 for all j € Q, then 2n + 1 < 2n - fc + 1. Thereforefc = 0 and / is constant. Hence

(23)

is Brody-hyperbolic. In fact by a theorem of Chen [9] (22) can beimproved:

Defect Relation of Cartan-Nochka-Chen

Let V be a hermitian vector space of dimension n + 1 > 1. Letf : C —> P(V) be a k-flat, holomorphic map. Let 21 = {O>J}JEQ be afinite family of "hyperplanes" <LJ E P(V*) in N-subgeneral positionwith N > n and N + 1 < #Q = q < oo. Assume that (/, aj) is freefor each j E Q. Then

(24)

An alternative proof of the defect relation (21) was given byAhlfors [1] in 1941. Also he proves a defect relation for associatedmaps. His proof is very powerful and works in more general situations.


Hermann and Joachim Weyl [90] lifted Ahlfors's proof to Riemannsurfaces. It was simplified by H. Wu [92] in 1970, Cowen and Griffiths[17] in 1976 and Pit-Mann Wong [93] in 1976. 1 extended this Ahlfors-Weyl theory to non-compact Kaehler manifolds [65]. However first wehave to inquire how value distribution was extended to functions andmaps of several independent complex variables.

Hellmuth Kneser created such an extension in two fundamentalpapers [23] in 1936 and [24] in 1938. Although these papers are littleremembered today, they still influence the present research in valuedistribution of several independent complex variables. Therefore letme explain his fundamental ideas. Again I will cast them in modernterminology and perspective.

Let M be a connected, complex manifold of dimension m. Let/ ^ 0 be a holomorphic function on M. Take p G M. Let a : Uf — > Ube a biholomorphic map of an open ball U' in Cm centered at 0 ontoan open subset U of M with a(0) = p. Then for each integer A > 0there is a unique homogeneous polynomial PA of degree A such that

(25) foa =A=0

where the convergence is uniform on every compact subset of Uf.Since f\U ^ 0, there is a unique number IJL = p>/(p) > 0 dependingon / and p only such that PM ^ 0 and P\ = 0 for all A £ Z with0 < A < jLt. The number /*/(p) is called the zero-multiplicity of / atp and the function p,f : M — > Z is called the zero-divisor of /.

An integral valued function is : M — » Z is said to be a divisoron M if and only if for every point p G M there is an open, connectedneighborhood U of p with holomorphic functions g ^ 0 and h ^ 0on U such that

(26) i/|tf = /i,-Mfc.

Let S be the support of z/. Then 5 = 0 if and only if i/ = 0. If S ^ 0,then S is a pure (m— l)-dimensional analytic subset of M. Let $t(S)be the set of regular points of S and let £(#) = s - ^(s) be the

set of singular points of S. Then z/|St(S) is locally constant.

Wilhelm Stoll 9

Let r : M —> R+ be an unbounded, non-negative function ofclass C°° on M. If B C M and 0 < r E [R, abbreviate

(27) B[r] ={z E B\r(x) < r2} B(r) = {x E B|r(a;) < r2}

(28) B<r> ={x E J3|r(z) = r2} B, = {x E B\r(x) > 0}

Here r is called an exhaustion of M if and only if M[r] is compactfor each r > 0. Abbreviate

(29) v = ddcr u = ddc log r a = dc log r A a;™"1

Then da = um. The function r is said to be parabolic if and only if

(30) cj > 0 o;m = 0 vm •£ 0.

If so, then t> > 0. More over r is said to be strictly parabolic if andonly if r is parabolic and v > 0 on M. If r is an exhaustion andparabolic, then (M, r) is said to be a parabolic manifold. If so, thereis a constant c > 0 such that

(31) / vm = <;r*m

M[r]

for all r > 0. Then for almost all r > 0 we have

(32) J * = *.

M<r>

In 1973 Griffiths and King [19] introduced parabolic manifolds.The concept was expanded in [75]. If r is an exhaustion and strictlyparabolic function, (M, r) is said to be a strictly parabolic manifold.In [77] 1980 I showed that (M, r) is strictly parabolic if and only ifthere is a hermitian vector space W of dimension m and a biholo-morphic map h : M —> W such that r = \\h\\2. We assume that(M, r) is strictly parabolic and we identify M — W such that h be-comes the identity. In this case c = 1 and M is a hermitian vector


space, which was Kneser's starting point. We assume that m > 1. Ifu : M<1> — > C is a function such that ua is integrable over the unitsphere M<1>, the mean value of u is defined by

(33) 3W(w)= w3W(w)= /

Let V be a hermitian vector space of dimension n + 1 > 1.Let / : M — > P(V) and g : M — > P(V*) be meromorphic maps.Let // and Ig be the indeterminacies of / and g respectively. Then(/, g) is said to be free if there exists z G M — // U Ig such thatf ( z ) £ E[g(z)]. For each "unit" vector 6 e M<1> an isometricembedding j& : C — > M is defined by j&(z) = 26 for 2 G C. IfJb(C) £IfUlg the pull back holomorphic maps /& = jb(/) : C — >P(F) and #, = jb*(0) : C -> P(V*) exist and (/b,0b) is free foralmost all 6 G M<1>. If 0 < s < r the First Main Theorem holds

(34) T/b (r, s) + Tflb (r, a) = Nf^ (r, a) + m/b,5b (r) - m/b,5b (s)

Now Kneser [24] applied the operator Sft termwise in (34) to obtain therespective value distribution functions and the First Main Theorem

(35) T/(r, s) + Tg(r, s) = JV», s) + mfi9(r) - mfig(a)

Of course Kneser considered the case n = 1 only. Then / is a mero-morphic function. Also he assumed that g = a G PI is constant. Hadhe stopped with the above derivation of (35), his result would havebeen worthless. He proceeded and expressed the value distributionfunctions in meaningful analytic and geometric terms. This made thepaper successful.

Let £2 be the Fubini Study form on P(F). For t > 0 define A/by

M[t]

He showed that Aj increases. Hence the limits

Wilhelm Stoll 11

0 < l imA f ( t ) = Af(0) < oo

0 < lim Af(t] = Af(oo) < oo

exist. Kneser obtained the identity

(38) Af(t) = f /*(£)) A um~l + Af(0)

M[t]

for t > 0. Here / is constant if and only if Af(oo) = 0 and / isrational if and only if ^4/(oo) < oo. Kneser proved

(39)

for 0 < 5 < r. Moreover we have

(40) lim = Af(co).r-KX) log r

A holomorphic map t) : M — > V is said to be a reduced represen-tation of / if and only if dimtrÔ) < m - 2 and f ( z ) = P(t)(^))for all z E M - If with ti(z) ^ 0. In fact // = t)"^0)- Reducedrepresentations exist since M is a vector space. If b is a reducedrepresentation of /, any other reduced representation is given by /it),where h : M — > C* is an entire function without zeroes. If 0 < s < r,then

(41) Tf(r,8)= log||b||a- log|N<7.M<r> M<s>

Since (/, g) is free, D/, gO ̂ 0, and for r > 0 the compensationfunction m/jfl of /, g is defined

(42) mf,g(r}=

M<r>

Let v : M — > Z be a divisor with support 5. Fot t > 0 the


counting function nv of v is defined by

S[t] 3(t]

where the limit 71^(0) = limô^^t) exists. Actually since M is avector space, n^(0) = i/(0) (see Stoll [62]). For each b G M<1>with jb(C) ^ 5, the pullback divisor z/& = Jl(v) exists. If £ > 0 then

(44) n

Thus f or 0 < s < r the valence function nv of z/ is given by

(45) A",,(r,a) =

Take reduced resprentations b : M -> P(F) of / and ft) : M ^P(V*) of g. Since (/,^) is free, /i = <b, ft)> ̂ 0. Then ///^ = ^depends on / and g only. Put S = /rÔ). If b G M<1> withjb(C) g S, then M/bj5B = ̂ (/i/J- Hence

(46) A»,«)=

Thus each term in (35) is explicitely expressed.Actually, Kneser [24] provided a more general version of (42).

For t > 0 the counting function of a pure p-dimensional analytic setS in M is defined by

(47) ns(t) = ± l^f = I tf + n5(0),

S[t] S[t]

where

(48) ns(Q)

Wilhelm Stoll 13

exists and is called the Lelong Number of S at 0. Kneser assumed that0^5, then ns(Q) = 0. Pierre Lelong permitted 0 E S and proved (46)in 1957 [26] by the use of currents. Paul Thie [87] (1967) moved thatthe Lelong number is an integer. This result constituted Paul Thie'stheses at Notre Dame and by coincidence Pierre Lelong was presentat the defense of the theses. Of course, if 0 E S then ns(0) > 0. PaulThie's result proved to be most helpful in estimating volumes frombelow. Of course the Lelong number of S can be defined for everyx E M and shall be denoted by Ls(x). Yum-Tong Siu [56] (1974)proved that the sets {x E M\Ls(x) > q} is analytic for every q E M.The proof was simplified by Lelong [28]

Since HS increases, the limit

(49) ras(oo) = lim ra$(t) < oot—*oo

exists, As an application of value distribution theory on complexspaces, I was able to show that S is affine algebraic if and only ifns(oo) < oo ([63]).

This result was localized by Errett Bishop [5] (1964) to extendanalytic sets over higher dimensional analytic sets. His result wasrefined by Shiftman [47], [48], [49].

Hellmuth Kneser did not proceed to a Second Main Theorem anda Defect Relation. Also he did not consider the possible extension ofhis theory to parabolic manifolds or Kahler manifolds. However, heinvestigated another problem: the theory of functions of finite order.He solved the two dimensional case and provided the basic ideas inm-dimensions. Later he assigned the completion of these investigationsto me as my thesis topic [62], [63].

Again let (M, r) a strictly parabolic manifold of dimensionsra > 1. Thus M is a hermitian vector space of dimension ra > 1and r is the square of the norm. If £ E M, ty E M, then (j:|t)) isthe hermitian product of £ and t). If u : IR+ —> R+ is an increasingfunction, its order is defined by

(50) 0 < Ord u = lim sup lofu^ < oo.r->oo logr

If v > 0 is a non-negative divisor, define Ordz/ = Ord?v Then


Ordz/ = Ord-/VI/(-,s). If / : M — » P(V) is a meromorphic map,define Ord/ = OrdT/(-, s).

If q is a non-negative integer, the Weierstrass prime factor isdefined for all z G C by

q I

P=i p(51) E(z,q) = (l-

For all z G C(l) the Kneser Kernel is defined by

1 dm~^(52) em(^) = ^— —

where log £?(0,g) =0.Let / : M -» C be an entire function of finite order with /(O) =

1. Let S be the support of the zero divisor v — fif of /. Trivially5 = /-1(0). Assume that 5 ^ 0 . Then there exists a largest realnumber s > 0 such that S(s) = 0. Since / has finite order, there is asmallest, non-negative integer q such that

(53)

Then q < Ord / < q + 1. Also there exists a holomorphic function Fon M(s) such that F(0) = 0 and f\W(s) = eF. By the First MainTheorem the following integral converges uniformly on every compactsubset of M(s) and defines a holomorphic function H on M(s) with

(54)

for 5 G M(s). Kneser [24] shows that there is a unique polynomial Pof at most degree q with P(0) = 0 such that

(55) F = P\W(s) + H f\W(s) = ep+H .

Hence h = e~pf is an entire function with fih = v = /// and

Wilhelm Stoll 15

h\W(s) = eH. Thus h depends on v only.Given a divisor v > 0 on M of finite order with S = supp v ^ 0,

there is a largest real number s > 0 such that S(s) = 0 and a smallest,non-negative integer q such that

(56)

Then q < Ordz/ < q + 1. The integral (53) converges uniformly onevery compact subset of W(s) and defines a holomorphic function Hon M(s) with #(0) = 0 and /-tjr(O) > q + 1 by (53). Does there existan entire function h on M such that h\W(s) = eH, such that //& = z/and such that Ord /i = Ord vl In his earlier paper, Kneser [23] (1936)proved the existance of such a canonical function if m = 2. It was mythesis problem to solve the case m > 2. His method required to showthat a certain closed form was exact. If m = 2, this lead to a solvableordinary differential equation. If m > 2, it took me two weeks to writeout the system of partial differential equations to be solved, which Icould not do. I asked him for advice. He said he had gone throughthe same terrible calculation and had been unable to solve the system.Then he threw away his notes. I followed his advice, but I found an-other proof ([62], [63]). Independently, Pierre Lelong ([25] 1953, [27]1964) proved the existence of the canonical function h by another in-tegral representation. Both solutions coincide by a uniquenen theoremof Rankin [42] (1968), who provided a third integral representation.In [64] 1953 I showed that the canonical function h of a 2m-periodicdivisor is a theta function for this divisor and that any 2m-periodicmeromorphic function is a quotient of two theta functions (Appell [2]1891 if m = 2 and Poincare [40] 1898 if m > 2). In 1975, HenriSkoda [58] and Gennadi Henkin [20] showed independently, that anon-negative divisor v on a strictly pseudoconvex domain D in Mwith bounded valence N» is the zero divisor v = /^ of a holomor-phic function h on D with bounded characteristic. Later Henkin [21](1978) showed, if Ord v < oo then there is a holomorphic functionh on D with v — /^ and Ordz/ = Ord ft. Recently, Polyakov [41](1987) extended this result to the polydisc. Skoda [60] (1972) solved


the problem for analytic sets of higher codimension in a complexvector space. For more details see [73].

The integral means method of Kneser fails on complex mani-folds. Also he did not attempt to prove a Second Main Theorem anda Defect Relation. From the theory of holomorphic curves there areavailable the method of Cartan [8] and the method of Ahlfors [1]which was extended to Riemann surfaces by Hermann and JoachimWeyl [90], improved later by H. Wu [92].

In 1953/54 I extended the theory of Ahlfors- Weyl to meromor-phic maps / : M — > P(V), where M is a m-dimensional, connected,complex manifold of dimension m > 1 endowed with a positive formX of bidegree (m — 1, m + 1) such that dx = 0. Here V is a hermitianvector space of dimension n + 1. Again the targets are the hyper-planes in P(V) and / is linearly non-degenerated. Let 21 = {Q>J}JZQbe a family of hyperplanes aj G P(V*) in general position. Then,under suitable assumptions a defect relation

(57)

was obtained. Also a defect relation for associated maps was proved[65]. I cannot go into details here. The extension to m > 1 is basedon two ideas:

(1) Let © be a set of open, relative compact subsets G of M withC°° -boundary such that g e G f or all G G ©, where g is open witha C°° -boundary. Assume that for each compact subset K of M thereis G G © with G D K. There the Dirichlet problem ddcV A x = 0 issolved for G - g with ^\dG = 0 and ̂ \dg = 1.

(2) The associated maps are defined by the use of a holomorphicdifferential form B of bidegree (m — 1, 0) such that

(58) 0 < mim-\B A B < Y(G)x on G

where Y(G) is the smallest possible constant.On parabolic manifolds the proof has been greatly simplified

by Cowen-Grifflths [17] (1976), Pit-Mann Wong [93] (1976), Stoll[80] (1983), [82] (1985), [86] (1992). The definitions and identities(34)-(46) also hold on parabolic manifolds except, of course, for the

Wilhelm Stoll 17

slicing jb and the equality n,/(0) = ^(0) and (41) may be vacuous,since / may not have a global, reduced representation on M. Thedefect of (/,#) is defined as in (16). For an exact statement of thedefect relation I refer to the papers mentioned before, but I will statethe defect relation in a special case with a new variation:

Let M be a connected, complex manifold of dimension m > 1.Let W be a hermitian vector space of dimension m. Let TT : M — >W be a surjective, proper, holomorphic map. Then r = ||7r||2 is aparabolic exhaustion of M and (M, r) is called a parabolic coveringspace of W. Take any holomorphic form £ of bidegree (m, 0) onW without zeroes. Then the zero divisor ft > 0 of 7r*(C) does notdepend on the choice of £ and is called the branching divisor of TT.Put B = supp/3. Then TT is locally biholomorphic at z E M if andonly if z £ M — B. Since TT is proper and holomorphic, Bf = 7r(J3)and B = n~l(Bf} are analytic and TT : M — B = W — B' is a coveringspace in the sense of topology. Its sheet number c is given by (31).

Let V be a hermitian vector space of dimension n + 1 > 1.Let / : M — > P(V) be a linearly non-degenerated meromorphic mapof transcendental growth (i.e. A/(oo) = oo). Assume that the Riccidefect

(59) Rf = lira'

.Tp(r, s)

Let 21 = {dj}jeQ a finite family of hyperplanes a^ G P(V*) in generalposition. Then we have the Defect Relation

(60) J] «(/, aj)<n + l + ±n(n + l)Rf.

A meromorphic map h : M — » P(V) is said to separate thefibers of TT, if there is a point x eW -Bf such that Tr"1^) A /& = 0and such that /i|?r~1(x) is injective. If so, and if s > 0, there is aconstant C(s) > 0 such that

(61) Np(r, s) < 2(* -

for all r > 0 (Noguchi [38], Stoll [83]). Define


(62) ^ = ^J{/i|/i : M — > Pfc meromorphic, separates fibers of TT}fc€N

Then the separation index of / is defined by

m\ • tr(63) v 7 - mf km sup

If / separates the fibers of TT, then 7 < 1. We obtain the DefectRelation

(64) <5(/, a,-) < n + 1 + n(n + !)(*- 1)7

If n = 1, that is, if / is a meromorphic function with transcendentalgrowth separating the fibers of TT, then

(65)

which, in the case m = 1, was already proved by H. Cartan [8] (1933).In 1977 Al Vitter [89] proved the Lemma of the logarithmic

derivative for meromorphic functions on a hermitian vector space Wand derived the defect relation for meromorphic maps / : W — » P(V)by Cartan' s original method. For a detailed account see also Stoll [79],1982. E. Bardis [3] (1990) extended the result to parabolic coveringspaces of W.

In 1973-74, Carlson and Griffiths [16] and Griffiths and King[19] invented a new method to prove the defect relation. In keepingwithin [19], the result shall be stated only in the case of a paraboliccovering space (M, r) of a hermitian vector space of dimension m >1. The advantage of the new method is, that it applies to holomorphicmaps / : M — > JV, where N is a connected, n-dimensional, compact,complex manifold. A positive holomorphic line bundle L spanned byits holomorphic sections is given on N. Then N is projective algebraic.The disadvantage of the new method is, that we have to assume thatthe map / is dominant which means that rank / = n. The vector spaceY* of all holomorphic sections of L have finite dimension k + 1 > 1.If o ± a e y*, the zero set EL[a] = {x £ N\a(x) = 0} depends on

Wilhelm Stoll 19

a = P(a) 6 P(y*) only. Let Y = (Y*)* be the dual vector space ofY. If x G N9 the linear subspace <&(#) = {a G V*|a(#) = 0} hasdimension k. Thus one and only one tp(x) G P(y) exists such thatE[(p(x)] = P(*(x)). The holomorphic map y> : N -> P(y) is calledthe dw0/ classification map of L. The value distribution functions of/ are defined as those of (p o /. First Main Theorem holds but thedefect relation so obtained is not optimal. As before we assume that/ has transcendental growth and that there is given a finite family31 = {a,j}jcQ of points a,j G P(y*). However we have to consider thegeometry of {EI\Q,J\}J^Q and not the geometry of {E[Q,J]}JEQ. Define

(66) EL[W\=\jEL[ai].

For each j G Q take a, G V* with aj = P(a,-). Take x G EL\%\.Then

(67) P = {j G Q\x G EL[aj]} = {j G Q\aj(x) = 0} ^ 0

Put p = #P. Take a bijective map A : N[l,p] -» P. There is an open,connected neighborhood U of x and a holomorphic section b : U — > I/such that 6(2) 7^ 0 for all 2 G C7. For each j G M[l,p], there is oneand only one holomorphic function hj on U such that d\(j)\U = hfo.Then SI is said to have strictly normal crossings at x if and only if

(68) dhi(x) /\...f\dhp(x)^ 0.

The definition is independent of the choices which were made. SI issaid to have strictly normal crossings if 31 has strictly normal crossingsat every x G EL[^] , which we assume now.

Let K be the canonical bundle of N Let K* be the dual bundleto K. Define

(69) 1^1 - inf{-|i; G N, w G N, Lv ® Kw positive }.\_L \ w

Define Rf by (59) and 7 by (63). With these assumptions and defini-tions, the


Defect Relation of Griffiths-King

(70)

(71)

holds. In [75] (1977) the theory was refined and extended to generalparabolic manifolds.

A difficult, major, unsolved problem is the question if "domi-nant" can be replaced by another assumption which does not implym>n. For instance does (70) hold if /(M) is not contained in anyproper analytic subset of TV? As Biancofiore [4] has shown the as-sumption /(M) g EL[a] for all a e P(y*) does not suffice. Can thecondition "strictly normal crossings" be relaxed?

Let V be a hermitian vector space of dimension n+ 1 > 1. Applythe previous theory to N = P(V). Let H be the hyperplane sectionbundle on P(V). Take p e N and choose L = H*. Then K = H~n~l

and LV®KW = #^-™("+i). Thus [^] - **±. Thus (70) and (71)reads

(72)

(73) £«(/,«,)<P

If p = 1, this is sharper than (60) which is due to the dominance of/•

Until now, target families of codimension 1 only where con-sidered. Does there exist a value distribution theory for codimensioni > 1. In 1958, H. Levine [30] proved an unintegrated First MainTheorem for projective planes of codimension t > 1 in P(V). At the1958 Summer School at the University of Chicago, S. S. Chern askedme to find the integrated version. When I left, I told him that there isno such thing. I was much surprised when he published an integratedversion [10] (1960) shortly afterwards. I failed, since I insisted on an

Wilhelm Stoll 21

old version to be obtained and because I had forgotten one of MaxPlanck's admonitions in one of his textbooks: "The energy principleis not a law of nature, but of man. Each time it fails in nature, maninvents a new type of energy to restore the principle." The First MainTheorem is such a principle. In order to retain it, S. S. Chern had toadmit a new, nasty term, later called the deficit, into the equation.

In 1965, Bott and Chern [6] extended the First Main Theorem tothe equidistribution of the zeroes of holomorphic sections in hermitianvector bundles. Thus differential geometry was brought into value dis-tribution theory. Later the theory was expanded to include all Schubertvarieties associated to holomorphic vector bundles. With the work ofH, Wu [91] (1968-70), F. Hirschfelder [22] (1969), L. Dektjarev [18](1970), Michael Cowen [16] (1973), Chia-Chi Tung [88] (1973), andmyself [67] (1967) [68] (1969) [69] (1970) and [76] (1978) a widerange of First Main Theorems for codimension £ > 1 was established.

Mostly, they can be brought under the following scheme

Q

(74)i

M ~- > N

Where M, N and E are connected, complex manifolds of dimensionsm, n and k respectively. Here E is a compact Kahler manifold and Sis an analytic subset of N x E. The projections Q and TT are surjective,open and of pure fiber dimensions q and p respectively with n — p =t > 1 and ra — i > 0. The map Q is locally a product at every pointof S. Since E is compact, Q is proper. Thus

(75) dim S=p-\-k = n-\-q k — q = n — p = £.

The diagram is completed as a pull back by the holomorphic map /:

C~1f\ /^) JY/r y\ I f(rr\ n(y\\


(77) g(x, z) = x f ( x , z) = I ff(x, z) = *(z)

(78) gof = foQ 7 = / 0 7 T .

The map Q has pure fiber dimensions q and is locally a product atevery point of Q. Hence Q has pure dimension ra + q.

For each y £ E, the analytic subset Sy = Q(ir~l(y)} is a purep-dimensional analytic subset of N. The family @ = {Sy}yeE isthe target family for the holomorphic map /. We assume that Ey =f~l(Sy) is either empty or has generically the dimension ra — L Let£ > 0 be the Kahler volume for of E with

(79)

E

Let g* be the fiber integration operator. Then Q, = 0*7r*(£) is a non-negative closed form of bidegree (t, £) and class C°° on N. Here £2 isthe Poincare dual of the homology class defined by @. Take y E E,by Hodge theory or construction (H. Wu [91], Stoll [69]) there is anon-negative form Xy > 0 on E — {y} of bidegree (k — 1, k — 1) withresidue 1 at y such that

(80) ddc\y = £ onE-{y}

Then A.y = g*7r*(Xy) > 0 is a form of bidegree (i — l,i — 1) onN - Sy with

(81) ddcAy = tl onN-Sy

Let (p be a form of bidegree (m — t, m — i] and of class C°°with compact support in M. With proper multiplicities vy, the StokesTheorem, the Residue Theorem and fiber integration imply

(82) I f(Ky} A <UF<p = - I df*(Ky) A <F<p

M M

M

Wilhelm Stoll 23

M

if Ey has pure dimension m — £. As a generalization of the Poincar6-Lelong formula we obtain the Unintegrated First Main Theorem

(83) / /*(«) A (p = I vy(p + I /*(Ay) A dd°<p.

M Fy M

For the integration, we assume that an exhaustion r : M —> R+ isgiven with

(84) w = ddclogr > 0 v = ddcr > 0 a£ = cflogr A wm~*,

Then dat = wm~w. We keep the notations (27) and (28), but do notrequire that r is parabolic. For t > 0 the spherical image function isdefined by

(85) Af(t) = -^-^ j /*(«) A vm~* > 0.

M[t]

For 0 < s < r the characteristic function is defined by

(89)

Take y G J5 such that .E^ has pure codimension £ or is empty. For allt > 0 the counting function is defined by

(90) nf,y(t) = p-y 1 > 0

£„[*]

and for 0 < s < r the valence function is defined by

(91)


For almost all r > 0 the compensation function is defined by

(92) ,vM<r>

For 0 < s < r the deficit is defined by

(93) Df,y(r,s}=1- j /• (A.) A «""*«.

M[r]-M[s]

If t — 1 and r is parabolic, then u;m = 0 which implies Z)/j2/ = 0.However if £ > 1, then this is false even if r is parabolic. The samecalculation as in (82) but respecting boundary terms yields the FirstMain Theorem

(94) r/(r, s) = N/9y(r, s) + m/iy(r) - m/,y (a) - D/lV(r, 5).

A continuous form A > 0 bidegree (k — 1, fc — 1) on E existssuch that x £ E implies

(95) A(x)=

The A = (^*7r*(A) > 0 is a continuous form of bidegree ( ^ — 1 , ^ —on JV. For all x G N, fiber integration yields

(96) A(*)

Thus we obtain

(97) M/(r) = m/iy(r)f (») = /* () A <rt > 0

(98) A/(r, a) = J Df,y(r, s^(y) =\ j f (A) A o;"1^1 >M[r]-M[s]

Wilhelm Stoll 25

Tf(r,s)= J(99)

which implies

(100)

For r > 0 define

(101) B(r) = {y e E \ Ey n Af [r] + 0}

0 < b/(r) = I £ < 1

fl(r)

(102) J3 = {y € E | Ev + 0}

0 < 6 / = Af < 1.

B

Then B = Ur>0 B(r) and 6/(r) —> &/ for r —»• oo increasingly. Now(94) implies

(103) NLy(r, s) < T/(r, 5) + m/iy(a) + £>/iy(r, a).

IfyeE- B(r), then JV/lV(r, s) = 0 and (99) implies

T f ( r , s) =

yeB(r)

(104) < J (Tf(r,S) + mf,y(s)

yeB(r)

< bf(r)Tf(r, s) + I (mf,y(S) + DfiV(r,JyeE

= bf(r)Tf(r, s) + /*,(«) + A,(r, s).

Therefore

(105) 0<(l-bt(r»<V(a+**r>a) ifr>s>0.


Assume that T/(r, s) — > oo and A/(r, s)/T/(r, s] — > 0 for r — > oo.Then &/ = 1. Thus /(M) intersects almost all targets S^. Even forholomorphic curves on C surprising results can be obtained:

Proposition

A holomorphic map f : C — » PG is defined for all z £ C by

(106) /(z)

= e 7f r > |(245 + log7) w 129.3006,intersects at least 99% of all hyperplanes in PQ.

Proof. A reduced representation b of / is defined for all z e Cby

with b(0) = ( ! , - • • , 1). Thus ||t)(0)|| = v/7. We can take s = 0. Thus

:Z)(r,0)= j Iog||b||<7-ilog7

C<r>

Observe that

By Stoll [80] Proposition 15.5 page 201 we have

0<

C<r>

Thus

6r-7rlog7 ^2 ? r :

By Stoll [80] (6.66) page 140 we have ///(s) = \ £)J=1 J = g forall 5 > 0. If r > (TT/G) log?, then

Wilhelrn Stoll 27

0 r0 Then

0 ^jjj, q.e.d.This calculation was made possible by a theorem of Shiffman-

Weyl, The method can be greatly improved, see Molzon, Shiftman,and Sibony [31] (1981), and Lelong and Gruman [29] (1986).

In 1929 Rolf Nevanlinna [33] conjectured that his defect relationremains valid, if the constant target points a,j £ PI are replaced by"target" functions gj : C — >• PI which move slower than the "hunter"function / : C — > PI, that is, if

(107) Tg.(r, *)/T/(r, s) -+ 0 f or r -> oo.

In 1964 Chi-Tai Chuang [14] proved the conjecture for entire functions/ : C -> C and created the basis for the solution of the problem. In1986, Norbet Steinmetz [61] proved Nevanlinna's conjecture. In 1991,Ru Min and I [43] [44] [85] proved the conjecture for holomorphiccurves and solved the case of the Cartan conjecture for moving targets[46], In 1985, Charles F. Osgood [39] claimed that these theorems area consequence of his results in diophantine approximation, but to methis implication is not self evident and still has to be established.

At the end let me state a result at Notre Dame on this subjectmatter, combining the work of Emmanuel Bardis [3], and Ru Min andmyself [44].

At first some concepts have to be explained. Let M be a con-nected, complex manifold of dimension m. Let V be a hermitianvector space of finite dimension n + 1 > 1. Let / : M — > P(V) bea meromorphic map. Take a G V* and 0 ^ b E V*. Put b = P(b).Assume that (/, b) is free. Then there exists one and only one mero-morphic function fa^ on M, called a coordinate function, such thatfor each point p G M there exists an open, connected neighborhoodU of p and a reduced representation t) : U — > V such that


(108) f\U =

Here <b, b> ^ 0 since (/, 6) is free. Let £/ be the set of all thosecoordinate functions of /. Trivially C C @^. Let Sft be the field ofmeromorphic functions on M. Let ^ be a subfield of 3ft. The / issaid to be defined over ® if and only if (£/ C ®. The meromorphicmap / is said to be linearly non-degenerated over ® if and only if(/, g) is free for every meromorphic map g : M — > P(V*) definedover ®. Let © = {pj}jeQ be a finite family of meromorphic mapsg j : M — > P(V*) with indeterminacy I9j. Define

(109) /«

Let ®© = C((£@) be the extension field of ® in M generatedby (£©. The family © is said to be in general position if and onlyif there is a point 2 €. M — I® such that ®(z) = {9j(z)}j^Q is ingeneral position.

Theorem: Defect relation for moving target.Let M be a connected, complex manifold of dimension M. Let

W be a hermitian vector space of dimension m. Let TT : M — > W bea surjective, proper holomorphic map. Then r = ||7r||2 is a parabolicexhaustion ofM. Let V be a hermitian vector space of finite dimensionn + 1 > 1. Let © = {gj}jzQ be a finite family of meromorphic mapsg : M — » P(V*) in general position. Assume at least on k G Q existssuch that Qk is not constant and separates the fibers of it. Let f : M -»P(V) be a meromorphic map which is linearly non-degenerated over$?©. Assume that gj grows slower than f for each j G Q. Then

(HO)

During the time from 1933 to 1960 the foundation was laid. The 1960th

was the decade of the First Main Theorem. The 1970th was the decadeof the Second Main Theorem. The 1980th was the decade of the mov-ing targets. Perhaps the 1990th will be a decade of refinement and of

Wilhelm Stoll 29

value distribution over function fields in conjunction with diophantineapproximation.

References

[1] Ahlfors, L,, The theory of meromorphic curves. Acta. Soc.Sci. Fenn. Nova Ser. A 3 (4) (1941) 171-183.

[2] Appell, P., Sur lesfonctions periodiques de deux variables,J. Math. Pures Appl. (4) 7 (1891), 157-219.

[3] Bardis, E., The Defect Relation for Meromorphic Maps De-fined on Covering Parabolic Manifolds, Notre Dame Thesis(1990), pp. 133.

[4] Biancofiore, A. A hypersurface defect relation for a class ofmeromorphic maps, Trans. Amer. Math. Soc. 270 (1982),47-60.

[5] Bishop, E., Condition for the analycity of certain sets, DukeMath. J. 36 (1969), 283-296.

[6] Bott, R. and Chern, S. S. Hermitian vector bundles and theequidistribution of the zeroes of their holomorphic sections.Acta Math. 114 (1965), 85-121.

[7] Carlson, J. and Griffiths, Ph., A defect relation for equidi-mensional holomorphic mappings between algebraic vari-eties, Ann. of Math. (2) 95 (1972), 557-584.

[8] Cartan, H., Sur les zeros des combinaisons lineaires de pfonctions holomorphes donnees. Mathematica (Cluj) 7(1933) 80-103.

[9] Chen, W., Cartan conjecture: Defect relation for meromor-phic maps from manifold to projective space. Notre DameThesis (1987) pp. 166.

[10] Chern, S. S. The integrated form of the first main theoremfor complex analytic mappings in several variables. Ann. ofMath. 77 (1960), 536-551.

[11] Complex analytic mappings ofRiemann surfaces.I. Amer. J. Math. 82 (1960), 323-337.

[12] Holomorphic curves in the plane, in "Differential


Geometry in honor of K. Yano". Kinokuniya, Tokyo, (1972),72-94.

[13] On holomorphic mappings ofhermitian manifoldsof the same dimension. Proc. Symp. Pure Math. 11 (1968).Entire Functions and Related Parts of Analysis, Amer.Math. Soc., 157-170.

[14] Chuang, Ch. T. Une generalization d'une inegalite de Nev-anlinna. Sci. Sin. 13 (1964), 887-895.

[15] On the zeros of some differential polynomials ofmeromorphic functions. Science Report 89-002, Inst. ofMath. Peking University (1989), 1-29.

[16] Cowen, M. Hermitian vector bundles and value distributionfor Schubert cycles. Trans. Amer. Math. Soc. (180) (1973),189-228.

[17] Cowen, M. and Griffiths, Ph., Holomorphic curves andmetrics of non-negative curvature. J. Analyse Math. 29(1976) 93-153.

[18] Dektjarev, L. The general first fundamental theorem of valuedistribution. Dokl. Akad. Nauk. SSR 193 (1970) (SovietMath. Dokl. 11 (1970), 961-63).

[19] Griffiths, Ph. and King, J., Nevanlinna theory and holomor-phic mappings between algebraic varieties, Acta Math. 130(1973), 145-220.

[20] Henkin, G. H., Solutions with estimates of the H. Lewy andPoincare-Lelong equations. Constructions of functions of theNevanlinna class with prescribed zeroes in strictly pseudo-convex domains, Dokl. Akad. Nauk SSSR 210 (1975), 771-774 (Soviet Math. 16 (1975), 1310-1314).

[21] Henkin, G. M., and Dautov, S. A., Zeroes of holomorphicfunctions of finite order and weighted estimates for the so-lutions of the d-equation, Mat. Sb. (N.S.) 107 (149) (1978),163-174, 317.

[22] Hirschfelder, J. The first main theorem of value distributionin several variables. Invent. Math. 8 (1969), 1-33.

[23] Kneser, H., Ordnung und Nullstellen bei ganzen Funktionenzweier Verdnderlicher, S.-B. Press Akad. Wiss. Phys.-Math.Kl. 31 (1936), 446-462.

Wilhelm Stoll 31

[24] , Zur Theorie der gebrochenen Funktionen mehr-erer Verdnderlicher, Jber, Deutsch. Math. Verein 48 (1938),1-38.

[25] Lelong, P., Sur lf extension auxfonctions entieres de n vari-ables, d'ordre fini, a'un development canonique de Weier-strass, CR Acad. Sei., Paris, 237 (1953), 865-867.

[26] , Integration sur une ensemble analytique complexe,Bull. Soc. Math. France 85 (1957), 328-370.

[27] , Fonctions entieres (n-variables) etfonctions plur-isousharmoniques d'orderfini dans Cn, J. Analyse Math. 12(1964) 365-407.

[28] Lelong, P., Sur la structure des courants positifs fermes,Lecture Notes in Mathematics 578 (1977) 136-158.

[29] Lelong, P. and Gruman, L. Entire Functions of Several Com-plex Variables. Grundl d. Wiss. 282 (1986) pp. 270,Springer-Verlag.

[30] Levine, H. A theorem on holomorphic mappings into com-plex projective space. Ann. of Math. 71 (1960), 529-535.

[31] Molzon, R. E., Shiffman, B., and Sibony, N. Average growthestimates for hyperplane sections of entire analytic sets.Math. Ann. 257 (1981), 43-53.

[32] Nevanlinna, R., Zur Theorie der meromorphen Funktionen,Acta Mathematica 46 (1925) 1-99.

[33] Le Theoreme de Picard-Borel et la Theorie desFonctions Meromorphes, Gauthiers-Villars, Paris (1929) re-print Chelsea Publ. Co. New York (1974) pp. 171.

[34] Eindeutige analytische Funktionen 2nd ed. DieGrundl d. Math Wiss. 46 (1953) pp. 379. Springer-Verlag.

[35] Nochka, E. L, Defect relations for meromorphic curves. Izv,Akad. Nauk. Moldav. SSR Ser. Fiz. Teklam. Mat. Nauk.(1982), 41^7.

[36] , On a theorem from linear algebra Izv.. Akad.Nauk. Modav. SSR Ser. Fiz. Teklam Mat. Nauk. (1982)29-33.

[37] , On the theory of meromorphic curves. Dokl, Akad.Nauk. SSR (1983), 377-381.

[38] Noguchi, J., Meromorphic mappings of a covering space


over Cn into protective algebraic variety and defect rela-tions, Hiroshima Math. J. 6 (1976), 265-280.

[39] Osgood, Ch. F. Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds or better. J. Number Theory 21(1985), 347-389.

[40] Poincare, H., Sur les proprieties du potential algebriques,Acta. Math. 22 (1898), 89-178.

[41] Polyakov, P., Zeroes of holomorphic functions of finite orderinapolydisk, Mat. Sb. (N.S.) 133 (175) (1987), 103-111,114.

[42] Ronkin, L. I., An analog of the canonical product for entirefunctions of several complex variables, Trudy Moskov Mat.Obsc. 18 (1968), 105-146 = Trans. Moscow Math. Soc. 18(1968), 117-160.

[43] Ru, M. and Stoll, W. Courbe holomorphes evitant des hy-perplans mobiles. C. R. Acad. Sci. Paris 310 Serie I (1990),45-48.

[44] The Second Main Theorem for Moving Targets. J.Geom. Anal. 1 (1991), 99-138.

[45] The Nevanlinna Conjecture for moving targets,preprint pp. 16.

[46] The Carton Conjecture for Moving Targets. Pro-ceedings of Symposia in Pure Mathematics. 52 (1991) 477-508.

[47] Shiftman, B., On the removal of singularities of analyticsets, Michigan Math. J. 15 (1968), 111-120.

[48] , On the continuation of analytic curves, Math. Ann.184 (1970), 268-274.

[49] , On the continuation of analytic sets, Math. Ann.185 (1970), 1-12.

[50] , Nevanlinna defect relations for singular divisors,Invent. Math. 31 (1975), 155-182.

[51] , Holomorphic curves in algebraic manifolds, Bull.Amer. Math. Soc. 83 (1977), 553-568.

[52] , On holomorphic curves and meromorphic maps inprojective spaces, Indiana Univ. Math. J. 28 (1979), 627-641.

Wilhelm Stoll 33

[53] , Introduction to Carlson-Griffiths equidistributiontheory, Lecture Notes in Mathematics, 981 (1983), 64-89,Springer-Verlag.

[54] , New defect relations for meromorphic functions onCn, Bull. Amer. Math. Soc. (New Series) 7 (1982), 594-601.

[55] , A general second main theorem for meromorphicfunctions on Cn, Amer. J. Math. 106 (1984), 509-531.

[56] Siu, Y. T., Analyticity of sets associated to Lelong numbersand the extension of closed positive currents, Invent. Math.27 (1974), 53-156.

[57] Skoda, H., Croissance des fonctions entieres s'annulant surune hypersurface donnee de Cn, Seminair P. Lelong 1970-71, Lecture Notes in Mathematics 275 (1972), 82-105,Springer-Verlag.

[58] , Valeurs au board les solutions de Voperateur d",et caracterisation des zeros des fonctions de la classe Nevan-linna, Bull. Soc. Math. France 104 (1976), 225-299.

[59] , Solution a croissance du second probleme Cousindaus Cn, Ann. Inst. Fourier (Grenoble) 21 (1971), 11-23.

[60] , Sous-ensembles analytiques d'ordre fini ou infinidaus C71, Bull. Soc. Math. France 100 (1972), 353-408.

[61] Steinmetz, N. Eine Verallgemeinerung des zweiten Nevan-linnaschen Hauptsatzes. J. Reine Angew. Math. 368 (1986),134-141.

[62] Stoll, W. Mehrfache Integrale aufkomplexen Mannigfaltig-keiten. Math. Zeitschr. 57 (1952), 116-154.

[63] Ganze Funktionen endlicher Ordnung mitgegebe-nenNullstellenfldchen. Math. Zeitschr. 57 (1953), 211-237.

[64] Konstruktion Jacobischer und mehrfach periodis-cher Funktionen zu gegebenen Nullstellenflachen. Math.Zeitschr. 126 (1953), 31^3.

[65] Die beiden Hauptsdtze der Wertverteilungstheoriebei Funktionen mehrerer komplexen Verdnderlichen. I ActaMath. 90 (1953), 1-115, II Acta Math. 92 (1954), 55-169.

[66] The growth of the area of a transcendental analyticset. I, II Math. Ann. 156 (1964), 47-78, 144-170.


[67] A general first main theorem of value distribution.Acta Math. 118 (1967), 111-191.

[68] About the value distribution of holomorphic mapsinto projective space. Acta Math. 123 (1969), 83-114.

[69] Value distribution of holomorphic maps into com-pact, complex manifolds. Lecture Notes in Mathematics. 135(1970), pp. 267, Springer-Verlag.

[70] Value distribution of holomorphic maps. SeveralComplex Variables I. Lecture Notes in Mathematics. 155(1970), 165-190, Springer-Verlag.

[71] A Bezout estimate for complete intersections. Ann.of Math. (2) 96 (1972), 361-401.

[72] Deficit and Bezout estimates. Value DistributionTheory Part B (edited by R. O. Kujala and A. L. Vitter,HI), Pure and Appl. Math. 25 Marcel Dekker, New York(1973), pp. 271.

[73] Holomorphic functions of finite order in severalcomplex variables. CBMS Regional Conference Series inMath. 21 Amer. Math. Soc. Providence, RI, (1974), pp. 83.

[74] Aspects of value distribution theory in several com-plex variables. Bull. Amer. Math. Soc. 83 (1977), 166-183.

[75] Value distribution on parabolic spaces. LectureNotes in Mathematics 600 (1977), p. 216. Springer-Verlag.

[76] A Casorati-Weierstrass theorem for Schubert zerosof semi-ample, holomorphic vector bundles. Atti. Acad. Naz.Lincei. Mem. Cl, Sci. Fis. Mat. Natur. Ser. VIE m. 15(1978), 63-90.

[77] The characterization of strictly parabolic mani-folds. Ann. Scuola Norm. Sup. Pisa, 7 (1980), 87-154.

[78] The characterization of strictly parabolic spaces.Composite Mathematica, 44 (1981), 305-373.

[79] Introduction to value distribution theory ofmero-morphic maps. Lecture Notes in Mathematics 950 (1983),210-359. Springer-Verlag.

[80] The Ahlfors Weyl theory of meromorphic maps onparabolic manifolds. Lecture Notes in Mathematics, 981(1983), 101-219. Springer-Verlag.

Wilhelm Stoll 35

[81] Value distribution and the lemma of the logarith-mic derivative on polydiscs. Intern. J. Math. Sci. 6 (1983),no. 4, 617-669.

[82] Value distribution theory for meromorphic maps.Asp. Math. E7 (1985), pp. 347. Vieweg.

[83] Algebroid reduction of Nevanlinna theory. Com-plex Analysis in (C. A. Berenstein ed.). Lecture Notes inMathematics 1277 (1987), 131-241. Springer-Verlag.

[84] On the propogation of dependences. Pac. J. ofMath. 139(1989), 311-337.

[85] An extension of the theorem of Steinmetz-Nevan-linna to holomorphic curves. Math. Ann. 282 (1988), 185-222.

[86] Value Distribution Theory in Several ComplexVariables. In preparation. To appear in China.

[87] Thie, P., The Lelong number of a point of a complex analyticset, Math. Ann. 172 (1967), 269-312.

[88] Tung, Ch. The first main theorem on complex spaces. (1973Notre Dame Thesis pp. 320) Atti della Ace. Naz. d. Lincei.Serie VIII15 (1979), 93-262.

[89] Vitter, A., The lemma of the logarithmic derivative in severalcomplex variables, Duke Math. J. 44 (1977), 89-104.

[90] Weyl, H., and Weyl, J., Meromorphic functions and analyticcurves. Annals of Math. Studies 12 Princeton Univ. Press,Princeton N.J. (1943) pp. 269.

[91] Wu, H., Remarks on the first main theorem in equidistribu-tion theory, 7, 77, 777, IV. J. Differential Geometry 2 (1968),197-202, 369-384, ibid. 3 (1969), 83-94, 433^46.

[92] , The equidistribution theory of holomorphic curves.Annals of Math. Studies, 64 Princeton Univ. Press, Prince-ton, NJ. (1970) pp. 219.

[93] Wong, P. M., Defect relations for maps on parabolic spacesand Kobayashi metrics on projective spaces omitting hyper-planes, Thesis Notre Dame (1976), pp. 231.

[94] Wong, P. M. On the Second Main Theorem of NevanlinnaTheory. Amer. J. Math. Ill (1989), 549-583.


[95] On holomorphic curves in spaces of constant holo-morphic sectional curvature, preprint 1980, p. 20, to appearin Proc. Conf. in Compl. Geometry, Osaka, Japan (1991).

[96] Wong, P. M. and Ru, M. Integral points in Pn - {2n + 1hyperplanes in general position) Invent. Math. 106 (1991),195-216.

THE NEVANLINNA ERROR TERM FOR COVERINGSGENERICALLY SURJECTIVE CASE

WILLIAM CHERRY

Nevanlinna theory [Ne] started as the theory of the value distribu-tion of meromorphic functions. The so-called Second Main Theoremis a theorem relating how often a function is equal to a given valuecompared with how often, on average, it is close to that value. Thistheorem takes the form of an inequality relating the counting functionand the mean proximity function by means of an error term. Histori-cally, only the order of the error term was considered important, butmotivated by Vojta's [Vo] dictionary between Nevanlinna theory andDiophantine approximations, Lang and others, see [La] and [L-C] forinstance, have started to look more closely at the form of this errorterm.

Vojta has a number theoretic conjecture, analogous to the SecondMain Theorem, where the absolute height of an algebraic point isbounded by an error term, which is independent of the degree of thepoint. This caused Lang to raise the question, "how does the degreeof an analytic covering of C come into the error term in Nevanlinnatheory?" The second part of [L-C] looks at Nevanlinna theory oncoverings in order to answer this question. Noguchi [Nol], [No2], and[No3] and Stoll [St] are among those who have previously looked atthe Nevanlinna theory of coverings.

As part of Vojta's dictionary, the Nevanlinna characteristic func-tion corresponds to the height of a rational point in projective space.For a number field F, there are two notions of height. There is a rel-ative height and an absolute height. Given a point P = (XQ, . . . , xn)in Pn(F), the relative height, hF(P) is defined by

where S is the set of absolute values on F, and [Fv : QJ is the lo-cal degree. The absolute height h(P) is the relative height divided

37

38 The Nevanlinna Error Term

by the global degree [F : Q] and is independent of the field F. TheNevanlinna characteristic function Tf, as defined in Part n of [L-C],corresponds to the relative height. As such, one wanted a second maintheorem where the degree enters into the error term only as a factormultiplied by a universal expression independent of the degree. Thisis more or less what was achieved when T/(r) was larger than the de-gree, but when T/(r) was less than the degree, we could not get sucha result, and it appeared that the error term depended on the degreein a more subtle way. However, this is to be expected because theclassical second main theorem only holds when Tf(r) is greater thanone, and the condition that the relative Tf be greater than the degreeis precisely the condition that the absolute Tf be greater than one.The main objective of this note is to show that when the Nevanlinnafunctions on coverings are normalized from the beginning by dividingby the degree, then the error term is independent of the degree, com-pletely in line with Vojta's conjecture in the number theoretic case,and all the extraneous terms in [L-C] disappear.

Furthermore, by making two minor changes to the method in[L-C], following Griffiths-King [G-K], we are able to work with non-degenerate holomorphic maps from an analytic covering of Cm intoan n-complex dimensional manifold, where ra > n. This is moregeneral than the equidimensional case treated in [L-C] and shows thatthe error term retains the same structure when the dimension of thedomain space is larger than that of the range.

The main result of this note is the following Second MainTheorem:

Theorem. Let p: Y —> Cm be a finite normal analytic coveringof Cm which is unramified and non-singular above zero. Let Xbe an n-complex dimensional manifold, and let f: Y —> X bea non-degenerate holomorphic map such that the "ramification"divisor Rf does not intersect Y<0>.Let:

D = ]C?=i Dj be a divisor with simple normal crossings ofcomplexity k;

LJ = LDJ be the line bundle associated to DJ;be a hermitian metric on L;

William Cherry 39

fi be a volume form on X;K be the metric on the canonical bundle associated to fi;rj be a positive (1, 1) form on X such that rjn/n\ > fi and

r ) > c i ( p j } f o r a l l j ;Assume that f(y) $ D for all y E Y<0>.Let:

Tl

n and n =

where b is the constant of Lemma 11.7 A in [L-C], and dependsonly on£l,D and 77. Then, one has

NRf(r) - Np

i 1+ 'ry • cm1L " J

r > r*i outside of a set of measure <

Remarks. The symbols above, including the divisor Rf, whichis the Griffiths-King ramification term, will be precisely defined in thesequel. Note that except for an additive term, which can be made todisappear by normalizing /, the Jacobian of / and the Jacobian of p atthe points which are above zero, the error term is completely uniformin the functions p and / as well as in the degree of the covering. Also,the extraneous terms involving the degree which appear in [L-C] arenot present here. Furthermore, when the error term function is ex-panded out, the constant which appears in front of the log T/j7? term is

which is better than the constant n(n + 1) appearing in Stoll [St].The larger constants in Stoll result from his method of summing upprojections onto Grassmannians via the "associated maps." By com-bining the equidimensional method used by Wong [Wo] and improved


by Lang, with the ramification terms in Griffiths-King which dependon the choice of Jacobian section, rather than the Wronskian determi-nant which appears in Stoll, the error term obtained in the genericallysurjective case is identical to that of the equidimensional case, and, inparticular, does not contain the unnecessary factors which arise fromprojective linear algebra.

For the proof of the above theorem, we follow Chapter IV of[L-C].

1. Preliminaries. Let p : Y — > Cm be a finite normal analyticcovering of Cm, and assume that Y is non-singular at the points abovezero and that p is also unramified above zero.Let:

[y : Cm] = the degree of the covering;z = (zi, . . . , zm) be the complex coordinates of (7m;

3=1

= {yeY:\\p(y)\\<r};

= {y£Y:\\p(y)\\<r};= {yeY:\\p(y)\\ = r}.

Consider the following differential forms on C™

J=lThe pullback of these forms to Y via p will be denoted by a subscriptY:

William Cherry 41

Note that ay is closed and C°° away from Y<0> and that

ay = [Y : C™].

y<r>

The following form of the Green-Jensen integral formula will beneeded. For a proof, see [L-C] Theorem IV. 1.2.

Theorem 1 (Green-Jensen Formula). Let a be a C2 functionfrom Y —> C except on a negligible set of singularities Z suchthat Zr\Y<0> = 0. Assume, in addition, that the following threeconditions are satisfied:

i) a<7y is absolutely integrable on Y<r> for all r > 0.ii) da A cry is absolutely integrable on Y[r] for all r.

iii) lim / aery = Gfor all r,

S(Z,E)(r)

where for sufficiently small e, S(Z,e)(r] denotes the boundaryof the tubular neighborhood of radius e around the singularitiesZ fl y [r], which is regular for all but a discrete set of values e.Then

r/ A\ f dt f ir A m_i i r i v—\ x N(A) I — / aaAc^v- =- / a e r y — - > ot(y),

I 4- I ^ O/ O ' -^J v i £1 j £*

andT r

/

ii /• /» ij. r

— / ddca A UY~I + / — 1™ / dca A Uy~l

t J Y J t E~O J Y

a^K"« X) afo)'

Let / : Y" —> Jt be a non-degenerate (i.e. not contained in anydivisor on X) holomorphic map, where X is a compact n-complexdimensional manifold and n is assumed less than or equal to ra.


Remark. It is not necessary for the function / to be definedon all of Y". Everything in the sequel remains true for a function/ : Y(R) -» X provided that r < R.

We define the absolute Nevanlinna functions as follows:

Height

If 77 is a (1, 1) form on X, then define

0 Y(t)

and similarly, given a hermitian metric p on a holomorphic line bundleL on X, define

r

o Y(t)

where ci (p) = ddc log /? is the Chern form of p.

Counting functions

Given a divisor D on Y, let

r1 and ND(r) =D(t) 0

and given a divisor Z) on X, let N/^ = Ar/*^. The counting functionfor the ramification divisor of p, defined locally by the zeros of theJacobian matrix, will be denoted

A volume form fi on X defines a metric K on the canonical linebundle K of X. Since,

William Cherry 43

the height associated to the volume form fi is defined as:

0

2. Ramification. Let $ be the Euclidean volume form on Cm

and let $y = p*($) be the pullback to a pseudo-volume form on Y.Let fi be a volume form on X. Because / is non-degenerate, we canassume that the coordinates on Cm were chosen so that

is not identically zero. Following Griffiths and King [G-K], let 7/ bethe non-negative function such that

Note that 7/ is singular along the ramification divisor of p and vanishesalong the divisor Rf given by the equation

/*nAp*[ FT ^r— dZj/\dZi =0.V .ij; 2?T /\j=n+l /

Remark. When n = m, the divisor /fy is the ramificationdivisor associated to the map /. In general, the divisor Rf depends notonly on the ramification of /, but also on the choice of coordinates onCm. However, this dependence on the choice of coordinates is omittedfrom the notation.

Note that because /*ci(tt) = ddclog7/, one has

r

7

0 Y(t)


Theorem 2. Assume that p : Y —> C™ is unramified above zero,and let f : Y —> X be a non-degenerate holomorphic map suchthat the divisor Rf does not intersect Y<0>. Then

TfjK(r) + NRf(r) -

- With new notation, this is simply Theorem 1 (B) combinedwith the fact that

lim / dclog-yf/\w™-l= t u™'\S(Z,e)(t) Z(t)

where Z is the set of singularities for Iog7/, and then divided by thedegree.

3. Calculus Lemmas. Let ̂ be a positive increasing function,such that

is finite. Such a function is called a type function. Given a positiveincreasing function F, let ri(F) be the smallest number such thatF(r) > e for r > ri(F), and let 61 (F) be the smallest number greaterthan or equal to one, such that

Define the error term function to be

William Cherry 45

Given a function a on Y, define the height transform:

0 Y(t)

for r > 0.

Let a be a function on V such that the following conditions aresatisfied:

(a) a is continuous and > 0 except on a divisor of Y.(b) For each r, the integral fY<r> acry *s absolutely convergent

and r i— > Jy<r> aoy is a piecewise continuous function of r.(c) There is an n > 1 such that .Fa(ri) > e.

Afote: Fa has positive derivative, so is strictly increasing.

Lemma 3. If a satisfies (a), (b) and (c) above, then Fa is C2 and

)= _ 2 _ r ay( m _ l ) j y "[FiC"1]'

- Use Fubini's Theorem and the fact that

as in Chapter IV §3, and then divide by the degree.

The standard Nevanlinna calculus lemma then gives

Lemma 4. If a. satisfies (a), (b) and (c) above, then

log j ap^}<S(FaMF«)^Y<r>

for all r > ri(Fa) outside a set of measure < 2&o(VO-


4. Trace and Determinant. Given a (1,1) form 77 on Y", definethe trace and determinant outside the ramification points of p asfollows:

(ra - 1)! tr(r/)$y = rj A ^T1.

Furthermore, define the n x n trace and determinant outside theramification points of p as follows:

(n - 1)! ttwfojSy = r? A ̂ A p* [ ^ A

In the case when Y is Cm and p is the identity, the n x n trace anddeterminant are simply the trace and determinant of the n x n blockin the upper-left of the matrix corresponding to 77.

The following lemma is simply the pull-back to Y of some re-lations on Cm, which follow immediately from the elementary linearalgebra of hermitian positive semi-definite matrices.

Lemma 5. Ifrj is a semi-positive (1, l)form on Y, then

(det^ry))1/" < itrBfa) and trnfa) < trft)Tb

for the regular points in Y which are not ramification points of p.

Let 77 be a closed, positive (1, 1) form such that

Since

( 'I

one finds that 7/ =

William Cherry 47

Proposition 6. Let 77 = trn(/*7?). Then

/: Let Ty- = tr(/*r/). All the symbols have been defined sothat the proof of Proposition II.6.2 in [L-C], after dividing through bythe degree, gives

But, since trn(/*Tj) < tr(/*rj), one has .Fr/ <

5. Second Main Theorem. Replacing the counterparts to thestatements above in the proof of Theorem IV.4.3 in [L-C] gives thefollowing Second Main Theorem.

Theorem 7. Assume that p : Y — > Cm is unramified above zero,and let f : Y — > X be a non-degenerate holomorphic map suchthat the divisor Rf does not intersect Y<0>. Let Tf^ be theheight associated to the volume form fZ = rjn/n\ on X. Then

+ NRf(r) -

^ y , ml

2/ey<o> *• ' J

for all r > ri(FTf] outside a set of measure < 2&o('0)-

Proof:

= 9 / (loS7/)]^ •/

[Theorem 2]

n f= 2 y-, i/nlog7/ [T : Cm]


'/ [F : Cm]

y<r>[concavity of the log]

oyJ[Y:Cm]

Y<r>

[Lemma 5]

&i(Fr/), ̂ , r) + — log(ra — 1)!

[Lemma 4]

[Proposition 6]

for all r > ri(FTf) outside a set of measure <

Remarks. The term on the right involving Iog7/ in the aboveinequality depends only on the values of /, the Jacobian of /, aridthe Jacobian of p above zero, so if these functions are normalizedat the points above zero, then the right hand side is uniform in thefunctions / and p and in the degree. In the case when ra = n, thenFTf = TfJ(n - 1)!, so n(FTf) = ri(r/|f|/(n - 1)!).

Similar changes give the more general Second Main Theorem.Recall that a divisor is said to have simple normal crossings of com-plexity k if k is the minimal number such that there exist local co-ordinates wi,..., Wn around each point of D, such that D is definedlocally by w\... wi = 0, with I < fc.

For the rest of this section, let:

D = S1=i Dj be a divisor on X with simple normal crossingsof complexity k\

LJ = LDJ the holomorphic line bundle associated to Dj withhennitian metric PJ\

rj be a closed, positive (1,1) form on X such that 17 > CI(PJ)for all j, and rjn/n\ > ft;

Sj be a holomorphic section of LJ, such that (sj) = Dj\

William Cherry 49

Since X is compact, after possibly multiplying Sj by a constant, as-sume without loss of generality that

For convenience, also assume that f(y) $ D for all y E Y<0>, andthat y<0> does not intersect the ramification divisor of /.

If A is a constant with 0 < A < 1, then define the Ahlfors-Wongsingular volume form

and define

Given A a positive decreasing function of r with 0 < A < 1, define

Note that because of the assumption \Sj\. < 1/e < 1, one has 7/ < 7A.

Using the fact that trn < tr and dividing through by the degreein the proof of Lemma IV.5.1 in [L-C] gives the following lemma.

Lemma 8. Let b be the constant of Lemma IL7.4 in [L-C], whichdepends only on£l^D and rj. Then for any decreasing function Awith 0 < A < 1, one has

qbl'n log 22n(ro-l)I

for all r.

Remark. Notice that the degree no longer appears in this esti-mate, and this is why the error term is now uniform. Also note thatthe n\ in the denominator has been replaced with n(m — 1)!.


Let n = ri(F i/n) and let

= {<Ff»(r) -•'-•*constant for r < r\.

Note that since r/n/n! > fi, one has F i/« < T/j7?/n!. Therefore

ri(F i/n) > ri(T/>7?/n!), and hence one has A < 1.

Applying Lemma 8 to the function A proves the next lemma.

Lemma 9. Let b be the constant of Lemma IL7.4 of [L-C] andlet

B = —((q + l)qk/n + ^2+fc/nlog2).TL £i

Then

for r > n.

Lemma 10. One has

/ Ti/yY<r>

for all r > r\9 outside a set of measure < 26o(^)» where

TL" log 2)

bi = bi(Fi/n) and n = ri(F i/«).7/ 7/

- Because 7A > 77, one has

v n >Fi / n and Fn>F\n.

Hence 61 = 61 (F i/n) and ri = n(F i/«) are such that for r >

William Cherry 51

*>(r)>e and bir2n~lF'1/n(r) > e.

'A TA

From Lemma 4, one has

/

/ -i \ i

TA7" rFTT^T ^ ^(-î *i. V', r) + log ̂ ——-^\JL m \^f I 'A Zi

for r > ri outside an exceptional set of measure < 26o(VO- Now fromLemma 9, one has

S(F^t blt ̂ r} + log ̂ ^ < S(BT}+kl\ b^, r)

for r > ri.

Finally, we can prove the general Second Main Theorem.

Theorem 11. One has

n

for r >TI outside of a set of measure <Proof: Let A be a constant with 0 < A < 1. Using Theorem

1 (B), and the fact that ddclog transforms products into sums, oneobtains:

T/>K(r) + (1 - A) J>/lft(r) - (1 - A)

1 v^ Iog7/(j/) , -i~ ^ + ?


i r dt~ \Y : Cml / t

o y(t)

n f i/n c= 2 / 10g7* W:

Because of the assumption that \8j\j < 1, one also has

~2

Also, since A is constant on Y<r>, the function A can replace A inthe above equality. Furthermore, Nf^. > 0 and — ! < — ( ! — A), sothe factor (1 — A) in front can be deleted. When r > ri, one has

3=1

from the definition of A, and from the fact that 77 was chosen so that

T /577>T / jp. for all j.

Finally, by moving the log out of the integral, one has

'A(r) ry . Cmi I •Y<r> ~ ~ \Y<r> /

Applying the estimate in Lemma 10 for A to the term with the integralon the right and collecting terms concludes the proof of the theorem.

References

[G-K] P. Griffiths and J. King, Nevanlinna theory and holomor-phic mappings between algebraic varieties, Acta Mathemat-ica 130 (1973), 145-220.

William Cherry 53

[La] S. Lang, The error term in Nevanlinna theory II, Bull. AMS(1990) pp. 115-125.

[L-C] S. Lang and W. Cherry, Topics in Nevalinna Theory, LectureNotes in Mathematics 1433, Springer Verlag, 1990.

[Ne] R. Nevanlinna, Analytic Functions, Springer Verlag, 1970;(revised translation of the German edition, 1953).

[Nol] J. Noguchi, A relation between order and defects of mero-morphic mappings ofCn into PN(C), Nagoya Math. J. 59(1975), 97-106.

[No2] I. Noguchi, Meromorphic mappings of a covering space overCm into a projective variety and defect relations, HiroshimaMath. J. 6 (1976), 265-280.

[No3] J. Noguchi, On the value distribution of meromorphic map-pings of covering spaces over Cm into algebraic varieties,J. Math. Soc. Japan. 37 (1985), 295-313.

[St] W. Stall, The Ahlfors-Weyl theory ofmeromophic maps onparabolic manifolds, Lecture Notes in Mathematics 981,Springer Verlag, 1981.

[Vo] P. Vojta, Diophantine Approximations and Value Distribu-tion Theory, Lecture Notes in Mathematics 1239, SpringerVerlag, 1987.

[Wo] P. M. Wong, On the second main theorem of Nevanlinnatheory, Am. J. Math. Ill (1989), pp. 549-583.

ON AHLFORS'S THEORY OF COVERING SURFACES

DAVID DRASIN

To Wilhelm Stoll, The Vincent F. Duncan andAnnamarie Micus Duncan Professor of Mathematics

1. Introduction. In [1] (see [3, pp. 214-251]) Ahlfors introducedhis theory of covering surfaces. His approach was combinatorial andgeometric, and showed that R. Nevanlinna's theory of meromorphicfunctions had topological significance, and held in differentiated form.Other accounts are in [2], [5], [9], and [12] presents a very efficientproof using Ahlfors's own framework. See [10] for an independentapproach, where the conclusions are slightly weaker than in [1].

Some years ago, John Lewis asked me if there was a way to de-rive Nevanlinna's value distribution theory directly from the argumentprinciple. Since Nevanlinna's approach is based on Jensen's formula,itself the integrated argument principle, it is clear that the argumentprinciple lies behind the theory, but the connection is, to say the least,highly indirect.

In this paper we show a more transparent connection. Very littleis used that is not in a first course in complex analysis, but the sub-tleties needed to achieve (1.5) and (1.6) show the depth of Ahlfors'sown insights. In retrospect our methods have considerable intersectionwith those of [1], although the orientation is different. I thank H. Don-nelly, A. Eremenko, D. Gottlieb, L. Lempert, M. Ramachandran andA. Weitsman for helpful discussions. The idea for the latter part ofProposition (1.8) was shown to me by S. Lalley. The influence ofMiles's work [8] is also apparent; see (2.23) below.

(LI) Preliminaries. (See [1], [9, Ch. 13].) Let ai , . . . , aq be dis-tinct (finite) complex numbers. We develop two situations in parallel:

q

the "base surface" FQ is either the Riemann sphere S2 or S2\ \J Dk,*=i

where the Dk are disjoint continuua about the a^; we also let ag+i = ooand take Dq+i accordingly. Thus, FQ is either closed or bordered.

We impose a unit mass X(w) on FQ with the properties specifiedin [1, I.I], [9, p. 325]; this allows lengths to be assigned to (Ahlfors)

54

David Drasin 55

regular curves and open sets. The essential property of A is that anisoperimetric inequality hold locally: each point PQ £ FQ has a neigh-borhood U = U(p) such that if 7 is a simple closed curve in U whichbounds the region J7 c U, then

(1.2) A(«) < fcA(7),

where h = h(p). By compactness (1.2) holds on FQ with a universalh so long as the A-area of fi is strictly bounded from one. We also let[•] be the chordal metric on £2; it clearly satisfies (1.2).

Let A(r) - {\z\ < r}, B(r) = 0A(r), &X(U*,TI) = {w;X(W,WQ) < 7?}, Bx(wQ,rj) = dAAj>o,?7), and Tr - /(B(r)). Weconsider maps / : A(-R) -» S2 which preserve orientation, with 0 <R < oo. The most important setting is that / be meromorphic, butit is natural to require only that / be a ramified covering of 52, inthe spirit of Ahlfors-Sario [4]; according to Stoilow [11] such mapsbecome meromorphic if A(jR) is given an appropriate structure. Fora good account of this see, for example, [6, §2].

Consider now f~l(F0) C A(jR); this is a union of components{G}. Let G be one such component. Then for 0 < r < R, set G(r) =Gn A(r), dG(r) = GnJB(r), and define in terms of A the expressionsS = S(r), L = L(r] for the area (including multiplicity) of f(G(r)}and length of Tr = /(<9G(r)), measured by A. In this sense, the image$ of G by / is a covering surface over F0 with TT : §f — > F0 theprojection. Ahlfors also considers any A-measurable subset D C F0,and defines

A(£>)

For example, if / is rational of degree N and D is any open set thenflf(r) = AT + o(l) = S(r, D) and L(r) = o(l)(r -> oo).

Ahlfors 's theory has significance primarily when gf is regularlyexhaustible: there exists an r-set ^4, R a limit point of A, such that

(1.3) L(r) - o(5(r)) (r -> Jf2, r G A)

[5, p. 338]. If / is meromorphic or quasiregular (and nonconstant)in A(oo) and A = [], it requires but a few lines and the Schwarz

56 On Ahlfors's Theory of Covering Surfaces

inequality to see that the full image §f of / over F0 = S2 is regularlyexhaustible (cf. [5, p. 352] and [1, p. 186]). In particular, in this caseA consists of nearly all large r.

We state Ahlfors's conclusions in two forms:

(1.4) THEOREM. (A) Let f be meromorphic in A (.ft), and ai, . . . ,aq be distinct (finite) complex numbers. Then there exists h = h(ai,. . . , aq) > 0 such that

(1.5) £ n(r, oj) > (q - 2)5(r) - hL(r).

(B) Let FQ be S2 or S2\ U Dk, f : A(ft) -» F0, an*/ to A be aunit mass as described above. Then there exists h = /i(-Fo) > 0 suchthat the Euler characteristic of any (finite) covering surface $ over FQsatisfies

(1.6) p+ = max(/>, 0) > pQS - hL.

Remarks. 1. In (1.6) we use the definitions of p and po from [1];cf. [4, p. 55]: x = -F + E - V (F = faces, E = edges, V = vertices).In many contemporary topological texts, what we call p is consideredthe negative of the Euler characteristic.

2. Following [9], we assume that FQ is planar: S2\\JDj. Ahlforsobserves [1, p. 174] that the general case follows from this by anelementary combinatorial analysis.

3. Inequality (1.6) is formally stronger than (1.5). One way to seethis is that when (1.6) is used to derive the differentiated Nevanlinnatheory (cf. [5, p. 148]) there is an additional branching term that isnot apparent in (1.5). However, the arguments used to get these re-finements are somewhat intricate; here we find that a common attackcan yield both.

4. In accord with standard tradition, we use h as a positive con-stant which can be taken to depend only on data of the surface FQ.

For example, Picard's theorem is an immediate consequence of(1.5) or (1.6) together with (1.3); we consider (1.6). Let / be noncon-stant on A(oo) and omit 01, a2, a3. Let F0 be S2 with small disks Dk

deleted about the a/^, so that po = 1- By assumption, no inverse image

David Drasin 57

of any Dk can be compactly contained in any A(r), so that alwaysp = — 1. Thus (1.3) and (1.5) are incompatible. Miles [7] shows thatthe main part of Nevanlinna's second fundamental theorem can berecovered from the Ahlfors theory.

(1.7) Normal values, first fundamental theorem. Since the argu-ment principle gives n(r, a&) - n(r, oo) rather than n(r, a^) directly,we first show that / always has many "normal" values. We have

(1.8) PROPOSITION. Let r < R, WQ E S2 and r/0 > 0 be given.Then there exist K < oo and w* E S2, with [w*,w] < TJQ and w*normal in the sense

further, we may find a line L through w* such that Fr intersects L Pi{[iv, w*] < 770} in at most KL(r) points.

Proof. We assume the elementary first covering theorem ofAhlfors (the analogue of Nevanlinna's first fundamental theorem; cf.[9, pp. 328-9]): if D is an open set in FQ then \S(r) - S(r,D)\ <h(X(D))~lL(r). [Ahlfors also has a variant of this for coverings of"regular" curves, but that is not needed here].

In this proof, we take A to be chordal measure [] on S2, and letS(r), £(r, D) be computed with respect to [].

For a fixed (large) K, let DI = {w E FQ\n(r,w) > S(r) +KL(r)}, and D2 = {w\n(r,w) > S(r) - KL(r}}\ here? n(r,w)is the usual counting function of iu- values in A(r) or G(r). Since5(r,jDi) = (j'Din(r,w)d\(w)){X(Di)}-'L, the first covering theo-rem yields that KL(r)X(Di) < hL(r}\ thus if K is large, A(-Di) isbounded away from 1. The same analysis applies to D%, and hence ifK is sufficiently large, the set W of w* which satisfy the Propositionhas chordal measure at least .9 the measure of the ball {[iu, WQ] < r/o}.

To satisfy the second condition, let us assume that WQ = 0 and,since 770 is small, replace the chordal metric by the Euclidean metric.Write r = Fr, and assume A(F) < oo. By making a rotation, wemay assume that the intersection of T with each horizontal or verticalline contains no segment. We will show that if i(j/o) is the cardinalityof Fr D {Ssz = J/Q} n {|z| < 1}, then there exists a set Y of y,— 5% < y < 2% with Jy dt > .9r/o and


(1.9) i(y) < KL(r), y<EY,

If we grant this, it follows that there exists y$ e Y, |y0| < |%»such that the set {y = 3/0} H {\w\ < |r/o} has nonempty intersectionwith the set W constructed above. We use any w* = XQ + iy^ in W,with |a;o| < |T?, yo € Y, and see that it satisfies both conditions of theProposition.

We now produce yo so that (1.9) holds. By our normalization,T D A (WQ, 1) may be written as an at most countable union of graphsof continuous functions, say y = yj(x), aj < x < /3j, with — 1 <yj(x) < 1. If Vj is the total variation of yj on (QLJ, /3j) and Lj is thelength of the graph of %-, we have that Vj < Lj.

Let ij(y) be the number of points of intersection of the graphof yj with the line {Qz = y}, so that i(y) = £)j*j(l/)"» ên Ba-nach's formula for total variation gives that Vj = Y^-i *j(y)dy. Hence,f-i i(y)dy < L(r), so that (1.9) follows at once.

(1.10) NORMALIZATION. Given a fixed r, we in general take w* =oo in Proposition 1.8.

2. Partitioning of A(r). Given distinct complex numbers ai,...,ag, let 10107j < inf^jA(aâj). By Proposition 1.8, we may,by decreasing 97 if necessary, choose aq+i so that A(ag+i,afc) >1010ry(l < k < q) and then, after a Mobius transformation of /assume that aq+i = oo. This choice of r) is in force for all that fol-lows, so that 77 depends only on F0. Following the ideas of Ahlfors,construct (indexing mod q +1) Jordan arcs /3^(1 < k < q +1) to joinak to afc+i. The /3's divide FQ with two Jordan domains Ff and F",and the preimages of the /3's divide A(r) (or G, as appropriate) intoN domains Ga. We let Fa = f(Ga), so that Fa is contained in F1 orFn. We usually ignore the specific choice of F1 or Ff/, and write thatf(Ga) C F, where F is the relevant choice of F1 or F".

Depending on the context, we may view the domain of / as allof A(^J), or as in a component G of /~1(Fo)fl A(R). Thus, the settingwill determine the relevant collection of (7a's. Similarly, n(r, oo) will

David Drasin 59

be the number of poles of / in either A(r) or G(r). We will developour method so that the reader can readily adapt it to either situation.

We make certain inessential normalizations: the /?& are pointwisedisjoint, Tr meets each /3& at finitely many points, and none of thecountably many branch points of / lies on any /?&. Finally, we assumethat in each ball B\(a^^ 877) there is a line segment Z/& passing throughafc such that relative to this ball, /?&_! Ufa = Lk\a^. When k = q +1,we take L to be the line constructed in Proposition 1.8. By making anarbitrarily small change in r, we may suppose that Fr does not passthrough any of the a^.

(2.1) Princple of the proof. The significance of length-area is seenfrom elementary considerations. The work that follows is to force thehypotheses of Lemma 2.2 to be satisfied.

(2.2) LEMMA. Let the Ga be as above, and suppose Ga meetsB(r) in P(ot) points Cj,a whose images on S2 are separated by some77 > 0. Then

(2.3) L(r)>

Proof. Consider a fixed Ga, and £ i , a j . . . , Cp(a),a< on Ga n B(r),such that A(CijQ!,CjjQ!) > Cty; here the £'s are listed in the order en-countered on circuiting B(r) in the positive direction. Since each Ga

is connected, the £'s are endpoints of P(a) disjoint arcs / of B(r)and hence give a contribution at least hP(a) to L(r). Then if G^ isany other region determined by the {/3j}, Gat must lie in one of thecomplementary domains of A(r)\GQ.

Hence, given an initial choice of Gai, choose Ga% so that GQ2is closest to Gai in one of these domains (there is not a unique suchGC&\ in fact there are usually P(OL) such). Then the closures of Ga%and Gai can have at most two points in common on B(r). Thus, Ga%adds a term P(«2) - 2 to L(r), since we are forced to introduce atleast P(ot2) — 2 new arcs / due to G^ We exhaust the {Ga} in thismanner, and (2.3) follows.

(2.4) The argument principle. Now for a fixed &, 1 < k < q, letQ

()(k) be the curve \J(3j, so that /3(k) is a Jordan arc on S whichk


joins ak to oo. Note that /3(1) D 0(2) D .... We also set /?'(&) =fc-i

U { |J fa}. Choose a fixed 0, say 9 = 0, such that /(re**) gi

3?. Consider stopping times 6(fc) : 0 < 0i < 02 < • • • < 6n <3

0i + 27r,n = n(fc), such that f(rei0i) G /3(fc); we do not indicatethe dependence on k of the 0's. This divides Tr into a union of arcsI\ = r*(l < i < n) each of which starts and ends on /?(&); r* is theimage of 9i < t < Oi+\.

We partition the rf into classes (Ik), (life) and (Illfc):

(Ifc) those arcs which lie completely in B\(a^ 2ry),(life) those arcs which lie completely in B\(oo,

the others.

If 7 is any curve (not necessarily closed) which does not passthrough afe or oo, we set

(2.5) PA- (7) = — A7 arg(-u; - a*),

and note that the normalization (1.10) reduces (1.5) to an estimatefrom below of Sjfe^fcO-V)- ^ ^" ~ /~1(j^«)' one °^ ̂ e subregionsof A(r) determined by the {/3j} as at beginning of this §, we let

(2.6)

where the sum in (2.6) is over the /-images 7 of the arcs of dGanB(r)(i.e., the relative boundary of Ga).

It is obvious that for curves Tf in classes (I&) and (II&) there canbe no way to bound Ufc(rf ) in terms of the length L(T^). Howeverwe have

(2.7) PROPOSITION. Suppose f is such that w* = oo satisfies theconditions of Proposition L8. Then for 1 < k < q

(2.8)(4)

Thus, the significant contributions to i/fc(IY) arise from curves

David Drasin 61

whose image winds about a^ and are close to a^. We begin the proofhere, and complete it in (2.13) below,

Proof. The critical case is when i E (I I IK). Choose a (maximal)chain X of length p, i/+i, . . . , it+p such that each F*+ . £ (///&)• LetF be the portion of Fr, which corresponds to X\ i.e., the image of

(2.9) LEMMA. Lef F foe as1 above. Then

(2.10)

Proof of (2 JO). Let S(fc) be S2 with the open disks AA(aA,,r?)and A,\ (00,77) deleted. Then for each k S(k) is compact, so thereexist (T > 0 and M < oo such that if 7 is a continuum which meets/3(k),/3'(k) and intersects 5(fc), then

(2.11) A( 7 )><7

and, for any choice of argument on S(k) n /3(fc),

| suparg(w — ak) — inf arg(^' — a&)| < M (w, wf G S(k) fl

It is clear that by increasing M by at most 4?r, we have a similarbound when w and iu' are in S(k) fl ^'(fc).

Let Q be the number of i such that a subcurve of rf meets /3'(fc).Then it follows from the definition of M that

|i/fc(r)| <2Q + 2M:

we think of F having an initial and terminal portion which does notmeet /?'(&), and then Q intermediate portions which join /3(fc) to itself,passing through /?'(&). Similarly, L(F) > Qcr, so that (2.10) holds inthe weaker form

(2.12) |i^(T)|</iL(r) + M.

It is possible to delete M in (2.12). If L(F) > 77, it is obviousthat M in (2.12) may be absorbed in the term hL(T)\ if L(F) < r?and F meets B\(a,k, TJ) or B\(oo, TJ), then F is a curve both of whoseendpoints are on ̂ n B\(a^ 3r/) or ̂ nBx(oo, 3rj). In either case,


/3fc is a ray emulating from a^ or oo in this region and so ^(F) = 0.Finally, if L(T) < 77 but F n {B\(ak, rj) U Bx(oo, r/)} = 0, we see thatin this case r = Tr, a closed curve, so that ^(F) = 0. Hence (2.10)holds in all cases.

(2.13) Completion of Proof of Proposition 2.7. By the normal-ization (1.10) with Proposition 1.8, it is clear that ]C(/ifc) l^fc(rf )| <KL(r). -Thus (2.8) is a consequence of this and (2.10).

(2.14) An extension of Proposition 2.7. By (2.8), the significantcontribution to ^(rr) arises from portions of Fr which circuit a& ina full revolution, and are contained in B\(a,k) 2r/). This can be madea bit sharper.

(2.15) LEMMA. For each k e {!,...,#}, let Aj be the arcs ofTr which lie in B\(ak^ 2??) and join fy-i and fa. Then

(2.16)

Proof. This follows at once from Proposition 2.7 and the obser-vation that each arc if of that Proposition contains two arcs Af (onewhich is mapped into Ff, one into F") plus, perhaps, additional sub-arcs w;hich start and end on one of fa-i or 0k- Since the /?'s are radialsegments in B\(ak^ 877), the latter arcs contribute nothing to z/fc(rr) or

(2J7J More on ffe ra/e o/oo. We modify (2.5) and (2.6) to

(2.18) i/; (7) = ( ^M if ^ c SA(«*, 2r?)few; 10 otherwise

and a similar interpretation for t>l(dFa) (see (2.6)).Let w* = oo be normal in the sense of Proposition 1.8. We show

that oo is typical in a very strong sense.(2.19) LEMMA (I). Let oo be normal For each a, let n(a) be

the number of poles on dGa, so that dGa is partitioned into n(a)components F(QJ, (3). Then with the exception of a set of B poles with

(2.20)

David Drasin 63

the following is true. For each fe, 1 < k < q, the number of disjointarcs 7 C UpT(a., /?) with

(2-21)

plus the number of solutions to the equation

(2.22) f ( z ) =

equals n(a).(II) Conversely, let Fa = f(Ga) be given, choose k as in (I), and

suppose that Fa <jt B\(ak,3rj). Let the {r(a,/3)} be as in (I). Thenwith the exception of a set of B of (a^fS) as in (2.20), each T(ct,/3)contains a pole.

(2.23) Remark. This lemma complements Miles [8], which coversthe situation that Fa C ^(a^jSry) for many a; then there may bemany a^-values not compensated by poles In this situation, ^(7) > 0for many arcs 7 C Fr, in contradistinction to (2.21).

Proof. Choose fc as above. If the /-image of a F(a, /3) does notpass through a^, then r(a, /3) contains an arc of B(r] whose image 7separates oo from a& in Fa. Unless 7 C B\ (a, 877), (a = a^, oo), theargument of (2.11) shows that A(7) > <TI > 0, independent of a, fc.By our normalization (1.10), the total number of poles so separatedas a, /3, k vary satisfies (2.20). This proves (I).

Conversely, let rai(a, fc), 712(0;, fc) be the number of solutions to(2.21) and (2.22) for a given a, and circuit dFa. The arcs and a^-valuesof (2.21) and (2.22) divide dFa into n(a, k) = m(a, k) + n2(a, k)portions r(a,/3, fc). To each F(a,/3, k) which does not pass throughag+i = oo corresponds a crosscut 7 = 7(0;, /3, k) which separatesoo from afc. Since Proposition 1.8 holds, the argument of the para-graph immediately above shows that the number of such (a, /?) canbe absorbed in (2.20). This completes the proof.

(2.24) COROLLARY. Let TV* be the number of pairs (a, (3) whichsatisfy the hypotheses of Part (II) of Lemma 2.19. Then, ifoo is normalin the sense of Proposition 1.8, we have

(2.25) |A r*-2n(oo)|</iL(r).


In particular, ifP is the number of poles which are taken in theser(a,/3), then

(2.26) P<2n(r,oo) + fcL(r).

Proof. The poles of / correspond to regions Fa which have oo intheir closure. Hence Lemma 2.19 applies. The first part of the corollarynow follows since each pole is on the boundary of two (7a's. Estimate(2.26) is immediate.

3. Proof of (1.5)Let n = n(a) be the number of poles of / on Ga relative to

A(r). Note that

and that the contribution of the exceptional (a, ft) satisfies (2.20). Letthe {r(a, /?)} be as in Lemma 2.19. Choose fc G {1, . . .,q}. If the/-image of a F(a, ft) does not pass through a*, then there is an arc7 C r(a, ft)nB(r) whose /-image separates oo from a^ in FQ. Unlessthe image of 7 lies in B\(a^ STJ) \jB\(oo, 877), the argument of (2.11)shows that A(7) > a\ > 0, independent of a or fc. We now applyLemma 2.2 to each of these n(a) sets F(a, ft). Let P(a,/3) be thenumber of fc G {1, . . . , q} such that, as in (2.15), i/fc(Af ) < 0 for an arcA£ of r(a,/3), andP(a) = E0P(a,P). Theso by (2.16), (3.1), (2.3) and (1.10) we have

a /J

n(a) _ 1 ̂ ^{p(a> /J) - 2} - fcL(r)a a /3

> - n(a) -a a ft

= — 2n(r, oo) — hL(r)

> -2S(r) - hL(r).

David Drasin 65

By the argument principle, this is (1.5).

4. Proof of (1.6). Recall the discussion of Euler characteristic in,say, [5, pp. 135-7], [9, pp. 322-3]. We surround each ak(l < k < q)by a small disk D^ such that A(C, a*) ^ r) for £ G dD& and letFQ = £2\ U Djfc. Thus, there are now q crosscuts /%; what is nowf)q consists of what in § 2 had been a connected piece of f3q and/?g+i which passes through oo. We estimate p($) by the standardcombinatorial inequality [5, p. 137], [9, p. 333]

(4.1) p(%)>n-N

where N is the total number of domains Ga = f ~ l ( F a } and n is thenumber of crosscuts Uj/~1(/?j).

(4.2) Remark. Each crosscut 7 bounds two domains {Ga}\ wewill use the fact, needed for (4.1), that crosscuts 7 which disconnect$ make no net change to either side of (4.1)

Consider the arcs r(o:,/3) of Lemma 2.19. In the context here,r(a,/3) C dGa and we write Fa = f(Ga) (so that Fa C F, withF = Ff or F", where Ff and F" are now bounded by the {/?&} andportions of the \J\dDk.

If T E T(a, /3), let F" = mrr be the portion of T in the relativeboundary of $; this convention of starring will be used below. As in§2, choose r\ > 0 such that, if j ^ k, then A(/?y,/3fc) > lOO/y (distancerelative to $). The number of pairs (a, /3) with A(F*(a,/?)) > 77 isat most hL(r). We place these exceptional pairs (a, /?) into class (7);by Lemma 1.8 and the normalization (1.10) we may also include in(I) all r(a, (3) such that an endpoint of some r*(a,/3) lies in an 77-neighborhood of oo. Thus if{E} is the cardinality of E)

(4.3) #{(a,P)e(I)}<hL(r).

We now introduce a significant set Q of pairs {a,/?}, which arenot in (I). Let c?oS be the outer boundary of $, i.e., the componentof d$ which intersects Fr; thus 9oS is connected and consists of partof Fr and perhaps arcs which are mapped to dD^ for various k. If(a, /3) ̂ (/), let (cf. Lemma 2.2) p = p(a, /?) be a maximum choice of


points {£} on d0gnr*(a,/?) such that A(£, CO > 3rj. By Proposition1.8 and the definition of (I), we may assume that none of the £ arein BX(OQ, Srj). The class Q will consist of all (a, /3) for which p ^ 1and a certain subset of the (a, /3) for which p = 1.

If p = 0 then r*(a,/3) = <^> so that we have q crosscuts of $corresponding to this (a, /?), and none of these crosscuts disconnect

».Now let (a,/3) ^ (/) with p = 1, corresponding to a choice

C = Ci. By hypothesis, F*(a,/3) contains a cross-cut 7* of Fa whichis contained in an ^-neighborhood fi of £i« There are two possibili-ties. Since Fa is connected, it is easy to see that either Fa C fZ orFa D {F\fi}. Since (a, /3) 0 (J), there is some k such that the twoendpoints of each component 7* of r(a, /3) are contained in fa, Dôr .Djb+i. Hence, when Fa c fi, we see from (4.2) that all cross-cuts 'over' U/3& can be ignored in computing (4.1), since each dis-connects 2f. This is important since there can be no upper boundfor the number of such components r(a,/3). The remaining pairs(a, /3) for which p = 1 are assigned to Q. In this case, Fa D JP\fi,and since p = 1, it follows that F(a, /3) will contain g crosscutsof g which terminate at each D&, and so there are at least q — 1crosscuts which do not separate $, since at least q — 1 cannot meetr>,/j).

Finally, if p(a, /3) = p > 2 and (a, /3) £ (J), we see that Fa DF\ U^ {^(Ci? 3ry)}. In this situation, there are again q crosscuts fromF(a, /3), but we are assured only that q — p do not disconnect gf.However, by Lemma 2.2,

(4.4) £ {p(a,(3)-2}+<hL(r).

If £7 is as defined above, it follows from Proposition (1.8) and (1.10)that (3.1) holds. Let ng and Ng be the contribution to n and N in(4.1) which arise from {(a, /?) £ £}. Since oo lies on each T(a, /3) if(a, /?) E £, we deduce from our definition of (I), (4.3), (3.1) and (1.8)that

David Drasin 67

n-N>ng-Ng- hL(r)

> [q*{G H {p = 0}} + (q - 1)*{0 n {p = 1}}

- m) #{6? n {p = m}}} - hL(r)m>2

- 2) -

> (q - 2)n(r, oo) - hL(r]

= (q-1}S(r}-hL(r).

Since /OQ = q — 2, we have proved (1.6).

References

[1] Ahlfors, L. V., Zur Theorie der Uberlagerungsflachen,Acta Math., 65 (1935), pp. 157-194.

[2] , Uber die Anwendung DifferentialgeometrischerMethoden zur Untersuchung von Uberlagerungsflachen,Acta Soc. Sci. Fenn. New Series A, V. 11(6), pp. 1-17.

[3] , Collected Papers, Vol. 1, Birkhauser, Boston,1982.

[4] Ahlfors, L. V., and Sario, Riemann Surfaces, Princeton,Princeton, 1960.

[5] Hayman, W. K., Meromorphic Functions, Oxford, 1964.[6] Lyzzak, A. K. and Stephenson, K., The structure of open

continuous functions having two valences, Trans. Amer.Math. Soc. 327 (1991) pp. 525-566.

[7] Miles, J., A note on Ahlfors' theory of covering surfaces,Proc. Amer. Math. Soc., 21 (1969), pp. 30-32.

[8] , Bounds on the ratio n(r, a)/S(r] for meromorphicfunctions, Trans. Amer. Math. Soc., 162 (1971), pp. 383-393.


[9] Nevanlinna, R., Analytic Functions, Springer-Verlag, NewYork, 1970.

[10] Pesonen, M., A path family approach to Ahlfors' value-distribution theory, Ann. Acad. Sci, Fenn, Ser. A, I Math.Dissertationes 39 (1982), pp. 1-32.

[11] S. Stoflow, Principes Topologiques de la Theorie des Fonc-tions Analytiques, Gauthier-Villars, Paris, 1938.

[12] Y. Toki, Proof of Ahlfors principal covering theorem, Rev.Math. Pures et Appl., 2 (1957), pp. 277-280.

INCAPACITY AND MULTIDIMENSIONALMOMENT PROBLEM

G. M. Henkin, A. A. Shananin(Moscow)

Introduction

Let K be a compact set in the n-dimensional complex space Cn,H(K) be a space of holomorphic functions on K, H'(K) be the spaceof linear continuous functionals over H(K). We will write down thevalue of the functional // E H'(K] on the function h G H(K) inthe form of <//, h>. The numbers of the form C^(/LA) = <^ZV>are called the moments of the analytical functional JJL, where Zv =Z"1... Z%n is a holomorphic monomial of the degree |z/| = v\ + . . . +i/n; Z - (Zi , . . . , Zn) E Cn, i/ - (z / i , . . . , i/n) G Z£.

The problem arising from a number of applications (computa-tional tomography [1], inverse problem of the potential theory [2],quadrature formulae [3], and even production functions theory [4]) isto reconstruct a functional from H'(K] through its moments.

The necessary and sufficient condition of uniqueness of a func-tional p. G H'(K), which has the fixed moments {Ci/(/x)} is polyno-mial convexity of the compact set K, since polynomial convexity ofK is necessary and sufficient in order that any function from H(K]will be approximated by holomorphic polynomials (A. Weil, 1932).

If a functional /J, is given by positive measure on the compactset K c Rn C Cn then the considered problem is called the classicalmoment problem. This classical problem is effectively and completelysolved only for the case n = 1 (see [5]).

In connection with applications the problem of the approximatereconstruction of the functional IJL G H'(K) through the finite num-ber of moments Cv, \v\ < N is of particular interest. In the classicaltheory this problem is called the Markov moment problem. In orderto solve this problem it is necessary to answer at least the followingquestions:

69

70 Cn-capacity and Multidimensional Moment Problem

1. What is a guaranteed estimate of the accuracy of the pos-sible reconstruction of the functional p G H'(K) if the momentsCi/(A*)j H ^ N and certain norm of the functional y, are known?

2. How to find actually the functional /j, G H'(K] with a priorigiven moment CV(IJL),\V\<N and with some suitable norm?

It turned out that these questions are closely connected with sev-eral modern themes from several complex variables.

Namely, for exact answer to the question 1, it is used the resultsof the theory of extremal plurisubharmonic functions and of the com-plex Monge-Ampere equation on the parabolic manifolds obtained inthe papers [6]-[22] and also the theory of the Fantappie-Martineauanalytical functional [23]-[27]. The modern variants of the interpola-tional formulae of the Jacobi type for the holomorphic functions inthe hyperconvex domains [28], [29] are very useful for the answer tothe question 2.

In this article we give a suitable answer to the question 1 andindicate the simplest applications. The constructive answer to the ques-tion 2 will be given in the other paper.

§1. The results.

The compact subset K C Cn is called regular (see [8], [9], [14])if there exists (and unique) a continuous solution UK of the followingexterior Dirichlet problem for the complex Monge-Amp&re equation:UK(Z] is a plurisubharmonic function in Cn\K,

= 0 in

UK(Z)=log\Z\ + 0(l) as|Z|->oo

UK(Z) = 0 if Z G dK.

The compact subset K C Cn is called (see [23]-[26]) linearconvex if for any point W G Cn\K a set of complex hyperplanespassing through W and not crossing K is non-empty and contractible.

The compact K is called strictly linear convex if its boundary dKis smooth and for any point W G dK the complex tangent hyperplane

have the unique point of contact {W} with dK and this

G, M, Henkin, A. A. Shananin 71

contact not higher than the first order. Any linear convex compact setK may be represented in the form of

where KI D K% D . . . is a sequence of strictly linear convex compactsets. Besides, there takes place the monotonic convergence for regularlinear convex compact sets K

Uj(Z) -> UK(Z] for j -* oo, Z E Cn\# , (1.2)

where [/^(Z) = UK^Z}- smooth solutions of the type (1.1) of theMonge- Ampere equation in Cn\Kj. Existence and uniqueness of suchsolutions for strictly linear convex compact sets is proved in [17].

We suppose without loss of generality that a linear convex com-pact K contains the origin of coordinates in Cn. We define a domain/^'dual to the compact set K by the formula

K1 = {p E (Cn)' : pZ + 1 ̂ 0 for Z E K}.

For the domain K1 we have such a representation

where K[ c K'2 C . . . is a sequence of strictly linear convex domainsdual to jKj.

According to Lempert [11], [12] there exist smooth solutionsVj = VK*. of the Monge-Ampere equations in the domains jR^Vj(p) is a plurisubharmonic function in /fj\{0}

Besides, O(l) = 5J(|f)+Oj(|p|), where S'j is a smooth functionon CF1-1, i.e., fiJ(A -p) = S»,VA € C.


The following nice formula is valid ([17], p. 882)

Vjtp) = -Ui(Z(p)), (1.4)

where

-i

is a diffeomorphism of the domain Kj\{0} on Cn\Kjm,

dV^ = (dVj dVAdp ~\dPl^"'dpnJ'

It follows from (1.2), (1.4), in particular, that there takes place amonotonic convergence

VS(P) -> VHp), j-»oo, Pex'\{o}, (1.5)

where VK> (p) is a continuous solution of the Monge- Ampere equationof the type (1.3) in the domain #'\{0}.

For regular linear convex compact sets K so called [16], [22]Robin functions of the compact set K and of the domain K1 aredefined and continuous on CPn~l

A-*oo

sA-»0

where C € Cn : |£| = 1 is identified with a point of CPn~l.Following Lelong [21] we shall call the functions 7(£) =

exp(-S(0) and y(C) = exp(-S"(C))5 C e CP71'1 capacitative indi-catrices of the compact K and of the domain K1 respectively.

Due to the statement of convergence of the Robin functions fromBedford-Taylor ([22], p. 163) it follows from (1.2) and (1.5) that

(1.7)

G. M. Henkin, A. A. Shananin 73

where 7^ and 7^ are capacitalive indicatrices of the compact set Kjand of the domain Kj respectively.

The following explicit relation between indicatrices 7 and 7'implies from (1.3), (1.4), (1.6), (1.7)

. (-.8)

where C G Cn : |C| = 1.The most important examples of linear convex and simultane-

ously regular compact sets are compact sets in Cn, which are closuresof the bounded linear-convex domains in Cn with smooth boundaryor closures of the bounded convex domains in Rn c Cn. In partic-ular, for the complex ball K = {Z E Cn : \Z\ < R} it is wellknown that UK(Z] = In ̂ . W. Stoll [10] obtained necessary andsufficient property of UK(Z] which characterizes the manifolds equiv-alent to the complex ball. For the real ball K = {Z = x + iy E Cn :\x\ < R,y = 0} M. Lundin [19] obtained the following nice formula

The entire function /i(£) of the variable ( E Cn of the form

(1.9)

where £Z = Ciî + - • >+CnZn, is called the Fourier-Laplace transformof the analytical functional fi G Hf(K).

For the functional /j, E Hf(K) where K is a regular compact set,we define semi-norms of the form

6 = ,

\h(Z)\<l, ZeK6, ( '

where K6 = {Z E Cn : UK(Z) < 8}, 6 > 0, UK satisfies (1.1).The following result gives a sufficiently exact answer to the ques-

tion 1 for functional with support on the regular linear convex com-pact.

Theorem. Let K be a regular linear convex compact in Cn and7;(C) be a capacitative indicatrix of the domain K1. Then

A) for any N E /+ any functional // E H'(K) with the moments

74 Cw-capacity and Multidimensional Moment Problem

Cv(n) = Ofor 1 1/| < TV, any £ G Cn, |f | = 1, o/iy A G C and /or6 > 0 tfzere takes place the following inequality

where O#,f (e) -> 0 if e -> 0; d(KiKs) = inf |l+p-*|, z G #,p G J^'B) for any ( G Cn, |(| = 1, any N G Z+ ^rg gjcw/5 the func-

tional p. = ILLN^ G Hf(K) with the moments Cv(p) = Ofor \t/\ < Nand with estimates of the form

(1.12)

s S

where f](Z] is any smooth Cn'-valued function of the variable Z GdKs with the property [26]: for all Z G dKs and W £ K we have

1 + r)(Z) • Z = 0 and 1 + ri(Z) -W ^0; u(Z) = A <j=i

A *For the case when the compact set If is a strictly linear convex

then the compact set K& for any 6 > 0 is also strictly linear convex

[17]. Using in this case S = 0 and r](Z] = d-^j^- / (z • ac/^z)) we\ /

obtain from (1.13) that the functionals //jv,c have a uniformly boundednorm ||//||o-

The theorem, roughly speaking, means that if the momentsCV(IJL)I \v\ < N are known for the finite measure /JL with the supporton the K then its Fourier-Laplace transform /i(£) is reconstructed

with accuracy of the order \\LL\\ \ / fv^ ) and not better, in\7/(jcl)^+1)/

general. It is important to express capacitative indicatrix 7'(C/|Cl) ifl


geometric terms in order to use such an estimate. For general caseit is not simple. However, the following statement is valid for theparticular case when the compact set K and direction £ are real.

Proposition. Let K be a closure of the bounded convex domainin Rn C Cn. Then the following equality is valid

1 = Usup(C • x) - inf (C • x)] (1.14)4 IxzK xtK J

for any real C e Rn, |C| = 1-Remark. If we drop demand of the regularity of the linear convex

compact set in the theorem then the theorem is still valid if we willwrite in the statement that ( E CPn~1\J5 where E is some polarsubset of CP71"1. In addition, instead of the function UK(Z) of theform (1.1) it is necessary to use extreme plurisubharmonic function[8], [9] of the form

UK(Z) = sup{U(Z) : U is plurisubharmonic on Cn\K}

U(Z) < log \Z\ + 0(1), U(Z) < 0 on 6K.

The necessary properties of the Robin function for such extremal func-tions are obtained by P. Lelong [21] and E. Bedford, B. Taylor [22].

This theorem supposes may be more clear interpretation in termsof the best approximations of the function exp(£ • Z) by polynomialson the compact set K.

Let us define the numbers

where PN is a polynomial of the degree TV in Z = (Zi, . . . , Zn).

Consequence 1. The following equality takes place

^ N • EHN(K, e«z) = e • |CI/V(C/IC|)

for any regular linear convex compact set K C Cn and any C £ Cn.Note, that the result of the consequence 1 may be considered as

76 C^-capacity and Multidimensional Moment Problem

complement of the following general approximating result of Siciak[6], [9]. In order that / G H(KS) (see (1.10) it is necessary andsufficient that

Tim

Now we will give an application of the theorem to one of compu-tational tomography problem — to an estimate of the accuracy of theRadon transform inversion through the finite number of directions.

The transform of the typer\ /»

,s) = — /

where 5 € R,u € Sn~l = {u e R1 : \u\ = 1} is called the Radontransfrom for a finite measure ^ with compact support in IP1.

The finite subset fi of the sphere Sn~l is called N-solvable [1]if any polynomial PN(%) of the degree N is represented in the form

where PNJLJ is a polynomial of degree N of the variable u • x. For thenumber of elements fi in fi we have the estimate

Conversely, if the inequality (1.16) is held and elements in fi are inthe general position then J) is JV-solvable (see [1]).

If the Radon transform Rû, s), u G fi is known for the measureIJL and £7 is TV-solvable, then the moments Cv(p) of the order |z/| < Nare known for the measure p, due to (1.15).

Hence from the theorem we obtain the following consequence.

Consequence 2. Let a support of the finite measure p, belong tothe closure of the bounded convex domain K C Rn and let the Radontransform Rp(w, s) of the measure IJL is equal to zero for directions ubelonging to N -solvable subset £1 Then the Fourier-Laplace transformMO for any C £ C71 admits the estimate of the form (Lll).

G. M, Henkin, A. A. Shananin 77

Note, that due to (1.14) we have 7;(C) = 2 for the real unitsphere Kl = {x G Rn : \x\ < 1} and for real directions £ G Sn~l.So, for this case the consequence 2 yields a preciser estimate:

sup

for any 9 < 2/e.It is interesting to associate this result with the following Logan-

Louis estimate (see [1]):under the conditions of the consequence 2 we have

for K = Kl and for any 6 < 1.

§2. The proof of the theorem.

This proof essentially uses the notion of the Fantappie indicatrixof the analytical functional.

The holomorphic function of the type

in the domain K1 is called the Fantappie indicatrix of the analyticalfunctional IJL G Hf(K). Immediately from the definition (2.1) it followsthat the equality C^(//) = <jn, Z"> = 0 for |i/| < TV is equivalent tothe equalities

for |i/| < AT, z/= (z/!,...,^n).The Fantappie transform ^^(p) is simply expressed through the

Fourier-Laplace transform /i(()


(2.3)

Martineau [24] obtained a general formula expressingthrough 3>n(p) on the basis of the Cauchy-Fantappie-Leray formula(see [27], [29]). Here we will have a need of the following elemen-tary formula.

(2-4){AeC:|A|=R}

where R is such that £/A e K' for any A : |A| = R.The formula (2.4) is a simple consequence of the classical

Cauchy formula. In fact, substituting the Cauchy representation

exd\'2=i /XeC:\\\=R

in the equality (1.9) we obtain

I f= ̂ J

The formula (2.4) allows to obtain necessary estimate for /x(^) onthe basis of suitable estimates for ^(C/A). We will obtain estimatesfor $M(£/A) from equalities (2.2) and from the following immediateestimate.

<1+pZ

(2.5)

where ||̂ ||a is a norm of the form (1.10), Ka = {Z € C" : UK(Z) <a}, P€IC.

Suppose, further, ^ = {Z e C" : UK(Z) < 6}, K's = {p €^' : Vjf'(p) + S < 0}, 5 > 0, where UK and VK> are the functionssatisfying (1.1), (1.5).


Consider now the plurisubharmonic function

This function is negative in the domain K'6 c K! due to (2.5). Theestimate ^/s(p) < lnH + O(l)> P G K's also takes place due to(2.2). Due to (1.5) the function VK((P)+8 satisfies the Monge-Ampereequation (1.3) in the domain K'6. As it was shown in [18], [20] such afunction is extremal plurisubharmonic function in the following sense:

VK>(P) + 8 = sup{V/(p) : V is plurisubharmonic

V(p) < 0 and V(p) < In \p\ + O(l) in K's} (2.7)

So, ®6(p) < V^/(p) + <$. From (2.6), (2.7) it follows that

\\tt\\s< "ylL , exp [(TV + l}(VK>(p) + 8)] (2.8)

forpel i f j .Substitute now the estimate (2.8) in the formula (2.4). Taking

into account (1.3), (1.5) we obtain the following inequality

for any C € Cn, 5 > 0 and for such J? that C,R~lei(f e K^ for ally € [0,27r]. Suppose R = N + 1, we obtain

IA(C)I < d(K,Ks)

The estimate (1.11), i.e. the part A) of the theorem is proved.


In order to prove the part B) of the theorem we shall have needof one more formula for the capacitative indicatrix:

sup{FeH(K'):F(0)=Q,\F(Q\<l,teK'}

(2.9)

The proof of (2.9) is based on the Lempert results [15]. Due to(1.3), (1.5) for the solution V(p) of the Monge-Ampere equation inthe domain we have an asymptotic equality

F(AC) = log |A| - log7'(C) + 0(| A|) for A -> 0, (2.10)

where p = AC C £ Cn, |C| = 1; A G C.Further, the following equality is valid (Lempert [15])

V$(p)= sup ln|F(p)|, (2.11)

where functions Vj satisfies (1.3).Taking into account (1.5) from (2.11) we obtain also the equality

= supln\F(p)\. (2.12)

{F G H(K') : F(0) = O, \F\ < 1}

The equality (2.9) follows from (2.12).Now we prove part B) of the theorem. We fix C £ Cn : |C| = 1

and N G Z+. Due to (2.10) there exists a function F G H(K'} withthe property

\F(p)\ < 1, p e K' and F(\(,} = (1(Q}-1\ + OK,((X2)

for A -» 0

Consider, further, a holomorphic function ^(p) = FN+l(p). Wehave

< li P e K1 and

(0) = 0 for M < AT. (2.14)


Due to the Martineau theorem [23], [24] refined in [25], [26],there exists a functional IJL G Hf(K) such that its indicatrix ^^(p)satisfies the equality

(2.15)

where Z>$ = $ +p. It follows from (2.14), (2.15) that Cv(p) = 0for \v\ < JV. We will prove the estimate (1.13).

Let,

A ̂

where Z — > 77 (Z) is any smooth mapping with the property: for anyZ e dK6 and W € if we have 1 + rf(Z) • Z = 0 and 1+ r](Z}W ^0. Let /i be any bounded holomorphic function on KS. Due to theCauchy-Fantappie-Leray formula we have (see [24]-[26]):

= j<p,h>= LpMi. (2.16)

The estimate (1.13) is an immediate consequence of (2.16). We willprove now the estimate (1.12). Taking into account formulae (2.4),(2.15) we obtain the equality

{teC:\t\=R}

Due to (2.13) for the function * f ̂ J and \t\ = N + 1 we have

inequalities

iJV+l

/ A ^ "+1

W(o • *


N+l N+1

Substituting (2.18) into (2.17) we have

A(AC) = (n - l)\(-l)n-l[Ji + J& (2.19)

where

T)Computing exactly J\ and estimating J^ we find

V• I \\I/ I—I / 1

Ti I I V ./V + n + j(]V + n)l

1 ((f)-1 • |A| - e)^1

JV+1

(2-20)

It follows from (2.19), (2.20) that


(N + n)!

The estimate (1.12), and consequently, the theorem is proved.

References

[1] Natterer, F., The mathematics of computerized tomography.B. G. Teubner, Stuttgart, John Wiley and Sons, 1986.

[2] Zalcman, L., Some inverse problems of potential theory,Contemp. Math 6 (1987), 337-350.

[3] Tchakaloff, V., Formules de cubatures mecaniques a coef-ficients non negatifs, Bull. Sci. Math., Ser. 2, 1957, v. 81,N3, 123-134.

[4] Henkin, G., Shananin, A., Bernstein theorems and Radontransform, Application to the theory of production functions.Trans, of Math, monographs., 1990, v.81, 189-223.

[5] Akhiezer, N. L, The classical moment problem and somerelated questions in analysis, Hafner, New York, 1965.

[6] Siciak, J., On some extremal functions and their applica-tion in the theory of analytic functions of several complexvariables. Trans. Am. Math. Soc. 1962, 105, 322-350.

[7] Zaharjuta, V. P., Transfinite diameter, Cebysev constantsand capacity for compact in Cn, Math. USSR Sbornik, 1975,25, 350-364.

[8] Siciak, J., An extremal problem in a class of plurisubhar-monic functions, Bull. Acad. Pol. Sci., 1976, 24, 563-568.

[9] Zaharjuta, V. P., Extremal plurisubharmonic functions, or-thonormal polynomials and the Bernstein-Walsh theorem foranalytic functions of several complex variables, Ann. Pol.Math. 1976/77, 33, 137-148.

[10] Stoll, W., The characterization of strictly parabolic mani-folds. Ann. Scuola Norm. St. Pisa, 1980, 7, 87-154.


[11] Lempert, L., La metrique de Kobayashi et la representationdes domaines sur la boule, Bull. Soc. Mat. France 1981,109, 427-474.

[12] , Intrinsic distances and holomorphic retracts. Proc.Conf. Varna, 1981, Complex Analysis and Applications, 81,Sofia, 1984, 341-364.

[13] Siciak, J., Extremal plurisubharmonic functions in C71, Ann.Polon. Math., 1981, 39, 175-211.

[14] Bedford, E., Taylor, B. A., A new capacity for plurisubhar-monic functions, Acta Math., 1982, 149, 1-40.

[ 15] Lempert, L., Solving the degenerate complex Monge-Ampereequation with one concentrated singularity, Math. Ann.,1983, 263, 515-532.

[16] Levenberg, L., Taylor, B. A., Comparison of capacities inC71, Lecture Notes Springer, 1094, 1984, 162-171.

[17] Lempert, L., Symmetries and other transformations of thecomplex Monge-Ampere equation, Duke Math. Journal,1985, v.52, 4, 869-885.

[18] Klimek, M., Extremal plurisubharmonic functions and in-variantpseudodistances. Bull Soc. Math. France, 1985, 113,123-142.

[19] Lundin, M., The extremal plurisubharmonic functions forconvex symmetric subsets of Rn, Michigan Math. J., 1985,32, 197-201.

[20] Demailly, J. P., Mesures de Monge-Ampere et mesures plur-isubharmoniques, Math., Z., 1987, 194, 519-564.

[21] Lelong, P., Notions capacitaires etfonctions de Green pluri-complexes dans les espaces de Banach, C. R. Acad. Sci.,Paris, 1987, 305, Serie J, 71-76.

[22] Bedford, E., Taylor, B. A., Plurisubharmonic functions withlogarithmic singularities, Ann. Inst. Fourier, Grenoble,1988, 38, 4, 133-171.

[23] Martineau, A., Sur la topologie des espaces de fonctionsholomorphes, Math. Ann., 1966, 163, 62-88.

[24] , Equations differentielles d'ordre infini, Bull. Soc.Math. France, 1967, 95, 109-154.


[25] Aizenberg, L. A., Linear convexity in Cn and the distributionof the singularities of holomorphic functions. Bull., Acad.Sci. Math., 1967, 15, 487-495.

[26] Gindikin, S. G., Henkin, G. M., Integral geometry for d-cohomology in q - linear concave domains in Cpn. Funct.Anal. Appl. 1979, 12, 247-261.

[27] Lelong, P., Gruman, L., Entire functions of several complexvariables. Grundlehren, Math. Wiss., N282, Springer, 1986.

[28] Berndtsson, B., A formula for interpolation and division inCn, Math. Ann., 1983, 263, N4, 399-418.

[29] Henkin, G. M., Method of integral representation in complexanalysis, in Several Complex Variables 7, (Encycl. Math.Sc., 7), Springer-Verlag, 1990, 19-116.

NEVANLINNA THEOREMS IN PUSH-FORWARD VERSION

Shanyu Ji

I. Introduction

We consider a polynomial map / of Cn, i.e., a holomporphic map/ = C:->C£,z = (zi,...,O *-> (/i(*),...,/n(*)), where CJ =C%, = Cn, z = (zi, . . . j Z n ) and w = ( tui , . . . , wn) are the coordinatesystems for C£ and C™, respectively, and /i,..., /„ E C[zi,..., zn].For any polynomial map / : C" —> C™ with det(D/) ^ 0, it isnaturally associated a dominant rational map F : P™ —> P™

defined by [20 : 21 : ... : ^] i-> IXf3^ : îCô* zij • • •»*n) : • • • :^n(^bj • • • j 2n)], where Fj is the homogeneous polynomial of degreedeg / := maxi<t<n deg ft uniquely determined by Ff(l, z\,..., zn) =fi(zi,..., zn) for i = 1,2,.. . , n.

There is the well-known Jacobian problem which was raised byKeller in 1939 [K] and is still unknown (cf.[V]): If / : Cn

z -» C£is a polynomial map with the Jacobian det(Df) = 1, then / has aninverse of polynomial map. In [J, corollary 4], we have proved: Let/ : C" —> CJJ, be a polynomial map with det(Z)/) = 1. Then / hasan inverse of polynomial map if and only if suppF^D^ = suppD^,where DZQ is the divisor given by ZQ = 0 and F^D^ is the push-forward current which is indeed a divisor.

From the above result, it leads us to take attention to the push-forward divisor F^D^. In order to investigate general push-forwarddivisors, in this paper, we establish the Nevanlinna main theorems inpush-forward version which are analogous to the ones in the valuedistribution theory. We shall study any polynomial map / : C™ —> C^with det Df ^ 0, its associated a dominant rational map F : P^ >P^ and any divisor D on P". We shall prove the first main theorem,the second main theorem, the defect relation and some other results.For proving these theorems, besides the modified traditional methodin the value distribution theory, some estimate from [J, theorem 2]about push-forward currents will be used.

86

Shanyu Ji 87

This work is a part of the author's thesis. The author would like tothank his advisor Professor Shiffman for assistance and encouragementabout this work. While preparing the final version of this work, theauthor is partially supported by a University of Houston ResearchInitiation Grant and by the NSF DMS-8922760.

2. Preliminaries

Meromorphic maps Let M and N be connected complex man-ifolds and let S be a proper analytic subset of M. Let / : M — S — > Nbe a holomorphic map. The closed graph G of / is the closure of thegraph of / over M — S in M x N. Let TT : G — > M and / : G — > Nbe the natural projections. The map is said to be meromorphic on M,denoted by / : M --- > TV, if G is analytic in M x N and if TTis proper. The indeterminacy If = {x E M \ #/(TT~I(X)) > 1} isanalytic, where #B means the cardinacy of a set JB, and is containedin S. We know codim // > 2. We assume S = //.

If B C M is a subset, we define the image of B by f is the set

f(B) = J(w~l(B)) = { y € N \ ( x , y ) e G, for some x G B}.

Currents on complex spaces Let X be a reduced, pure n-dimensional complex space and Reg(Jf) be the set of all regular pointsof X. We can define currents on X (cf.[D, p. 14]). Since the problemis local, we assume that there is an embedding j : X — > fi, wherefZ C CN is an open subset (i.e., X is identified with a closed analyticsubset of fi). We define

with the quotient topology, where

*j :

is the usual pull-back map and £p>q(M) is the space of all (p, g)-formson a manifold M. It is known that the definition of £p>q is indepedentof the choice of the embedding j (see [D, p. 14]). Then we define

= {a G £P>T(X) | a has compact support on X} with the

88 Nevanlinna Theorems in Push-forward Version

inductive limit topology. The dual space T>/p>q(X) is defined as thespace of (p, q) bidimensional currents on X.

The operators d, 9, d and push-forward of currents (by properholomorphic maps) then are defined for currents on such complexspaces as defined on manifolds.

Divisors on complex spaces For any reduced, pure n-dimen-sional complex space (X, O), we denote M the sheaf of germs ofmeromorphic functions on X and denote M* the sheaf (of multipli-cation groups) of invertible elements in M. Similary O* is the sheafof invertible elements in O. A divisor D on X is a global section ofthe sheaf M*/O*. A divisor D on X also can be described by givingan open cover Ui of X and for each i an element fa G r(f/^jM*)such that fi/fj G T(Ui n Z7j, O*) for any i and j.

If F : P£ — > P™ is a dominant rational (also meromorphic) map.Let G be the closed graph and TT, F are the natural projections. Forany divisor £> on P™, we can pull back it on G as a divisor ?r*D inan obvious way.

The Carlson-Griffiths singular form Let DI, . . . , Dq be divi-sors on P™ such that suppZ)i, . . . , suppZ)g are manifolds located innormal crossings and each Dj = Dg., where gj G C[ZQ, . . . , 2n] is ahomogeneous polynomial of degree PJ. Denote D = ]CjL=i ^j- EachDJ is also given by the system {U^gj/^}^<i<n, where Ui = {[ZQ :. . . : zn] G Pn | Zi 7^ 0}. The associated holomorphic line bundle LDJ

of .Dj has the Hermitian metric hj = {C/i, /iji}o<i<7i? where

Let L = L^ (g) . . . ® L^. Then the Hermitian metric /i of L is/i = {Ui, hii-...' hqi}Q<i<n. For each Djy as the section fe/^f }of LD., a globally defined function \\Dj\\2 on P^ is defined by

1.2 | rr =

where C > 0 is a constant such that

Shanyu Ji 89

for all ZQ, . . . , zn G C. We know the Chern form c(Kpn) of the canon-ical bundle Kp*

C(#pn) = -(n+l)nPn, (2.1)

which is also defined as the Ricci form of the volume form fipj onP". Let's recall the notions of Ricci form and volume form. Let Mbe any complex manifold. The canonical bundle KM of M is theholomorphic line bundle whose transition functions are the Jacobianof the coordinate change mappings in the intersection of domains in acovering of M, i.e., let {[/«, Wa}a be a coordinate syatem coveringof M, then on Uar}Up, the transition functions gap — det(dw?/dwP).

If 3>Q = n«=i zjrdw% A dw% on Ua is the local Euclidean volumeform, then a positive (n, n)-form £7 which is defined locally on Ua asAa$a is a global form on M if and only if A/? = \gap\2Xa in Z7a Pi Up.Such (n, n)-form O is called a volume form. The Ricci form of fi,denoted by RicO, is defined by Ricfi | Ua = ddclog Xa.

The Carlson-Griffiths Singular volume form * on P" — suppDis defined by (cf. [SHA, p.79])

where the constant C > 0 is determined by the following properties

(2.2) Ric* >0;

(2.3) (Ric *)">*;

(2.5) Ric* | (PJ - suppl>) = c(LD)

3. Push-forward of currents by F

Let / : C™ — » C^ be a polynomial map with the Jacobiandet(D/) ^ 0. Let F : P£ --- > P^, be its associated dominant


rational map. Let G be the closed graph of F, and TT and F be theprojections. G is an irreducible, pure complex n-dimensional analyticsubset in P™ x P™, so G is regarded as an irreducible reduced complexspace. Therefore TT and F are proper holomorphic maps from complexspace onto complex manifolds.

For any divisor D on P™, we pull back it on G as a divisorir*D. Then we obtain a pushforward current F*(n*D) on P™. Wewant to show that this push-forward current is indeed a divisor. Beforedoing that, we need the following lemma. The proof below is due toShiftman.

Lemma 3.1 Let M and N be n-dimensional complex manifolds andlet f : M —> N be a surjective proper holomorphic map. If D is adivisor on M, then the current f*D is a divisor on N.

Proof Let A = suppA and / = / | A : A_-> N. We assume thatcodimf(A) = l._Let S = {x G A \ d i m f - 1 ( f ( x ) ) > 1}. Thenbecause of codim f ( A ) = 1,

codim/(S)>2.

We first show that /,£> | N - f ( S ) G pn>l(n - f ( S ) ) is adivisor. In fact, for any point w G N — /(£), f ~ l ( w ) is a_finite set.Then there is an open neighborhood W(w) of w in TV - 7(5), andfinite disjoint open subsets E/i(iu), . . . , Ur(w) in M such that for eachUi(w), there is a holomorphic function g+ G O(Ui(w)),

f-l(W(w)) n A = U^Mw) H A, and

D | Ui(w) = ddc\og |<fc|2, for i = 1,2,... 3 r,

where the Poincare-Lelong formula is used. Let J/ = {z G M | 2; isa critical point of f}. Then /(J/) C JV is an analytic subset. SinceW(w) — f(Jf) is connected, there are integers AI, . . . , \r such thatfor any u G M^(w) — /(J/), there is an open neighborhood W(u] ofw in W(w) — /(J/), and disjoint open subsets Uiti(u),..., f/ljAl(w)C C/l(^); . . . ;C/r- , l (n) , . . . ,C/rJA r(w) C C/rH, SO that / | I7y(tt) :Uij(u) —> ^(w) is biholomorphic for all i = 1,2,.. .; 1 < j <\i. Then

Shanyu Ji 91

W(u) =t=i

r

= ddclog

>>-i|2

EM"))

where g = Y[ri==j TljLi 9i ° (/l^,j(w)) 1 on W(u). g is a well-defined

holomorphic function on W(w) — /(J/), which can be extended onW(w) holomorphically. Therefore we have proved that f*D is a di-visor on TV — f ( S ) .

Let V = suppf*D | TV — /(£). V has a decomposition V =U f V j , where Vj are irreducible hypersurfaces on N — f ( S ) . Since we

have proved f*D \ N — f ( S ) = Sjnj^'» ^ âs an extension V} inTV for all j. _

It suffices to show the current T = ^ njVj ~ f*D £ Wl'l(N)must be zero. Since T \ TV - f ( S ) = 0, and dimR f ( S ) < 2n - 4.Then the current T = 0 follows from the following lemma. QED

Lemma 3.2 Let 0 fp(£l) is d-closed and of order 0, then

Proof This is a special case of Federer [F, 4.1.20], or cf. [Sh, lemmaA.2]. QED

^x

Now we can prove that the current F*(ir*D) is a divisor. In fact,by Hironaka's theorem of resolution of singularities [H], there is amodification a : G1 — > G, where G1 is a compact complex manifold.It follows that F*D = (F o a)*(a*£>). Thus F*£> is a divisor on P£,by the lemma above.

In [J], we proved that let / : M -- — > TV be a surjectivemeromorphic map, where M and TV are compact connected com-plex manifolds of complex n-dimension. Let £ be a semi-positive


holomorphic line bundle over M with a nozero holomorphic sections. The locus of s on M is denoted by V as an analytic hypersurface.Then the image f ( V ) is also an analytic hypersurface on N.

We take an open covering {Ua} of M and a Hermitian metrich = {ha} of £ such that the curvature of (£, h) is semi-positive. Letthe given holomorphic section s = {sa}. Then we have a globallydefined function on M : ||s||2 = J&0[sa|

2 on Ua. Put <p = — log ||s||2.By [J], /*<£ is the plurisubharmonic exhaustion function of N — f ( V ) .By the lemma 3.1, if we denote Ds to be the divisor determined by s,f*Ds = /*(7r*-Da) is also a divisor. Then for any point w G JVn.F(V),there exists an open neighborhood Ui of w in. N and a holomorphicfunction 5 G O(C/i) such that /*DS = ddc\og\g\2. We notice that/*¥? G (7°°(N - /(V U Jf U //)). Then we can present

lemma 3.3 (See [J, theorem 2]) Let /, M, JV, £, s and <p fee asLet w be any given point in N n /(F) with a neighborhood U\ asabove. Then there exists an open neighborhood Uofw with U C C/iflnrf a positive constant number C = C(w} /, g) such that

for alluCU- f(V U J/ n //).

We can apply this theorem to any dominant rational map F :P^ --- > P™ and any holomorphic section s because of the fact thatany hypersurface V on P£ should be a locus of some holomorphicsection of some positive holomorphic line bundle over P". For thesection s, it is associated a globally defined function (p = \\s\\ on P".It was proved that F*(p is an exhaustion plurisubharmonic fimction forP£, - F(V). Furthermore, the lemma 3.3 said that

FW G rkTO - (3.4)

4. Notations in the value distribution theory

Let / and jP be as before. Assume deg/ > 1. Consider theinclusion map i : C£, ^ P£, (tui,...,ti;n) *-+ [1 : wi : ... : wn],which identifies t(C£) = CJJ,. We use (wi,...,ii;n) as coordinatessystem on i(C^).

Shanyu Ji 93

On t(C£), we let

+ ... + K|2), u = ddclog(H2 + ...+ K|2),

= <flog(|u;i|2 + . . . + |iwn|2) A up; w = wn_i,

Lemma 4.1 (Jenson-Lelong formula) Let T be a real valued functionand T G >Côc(C

n) such that ddcT is of order 0. Then for 0 < r0 < r,

ry * y ^CT A^-1 - i j T/Ô^ - 1 yconstant C is independent ofr.

Proof See [Sh, lemma 2.3]. QED

We define the characteristic Junction of F by

r

, r0) = FAj A fi""1,

where fipn is the Fubini-Study metric form on PJ, and F* =For any positive current % on i(CJJ) of (n— 1, n— 1) bidimension,

we define the counting function of x by

ro J3(t)

Note that if % is d-closed, by Stokes' theorem,

ro B(t)


Abbreviately, we denote

where Dg is a divisor on PJ given by a homogeneous polynomial g.

5. The first main theorem

Let 0 ^ g G Cfco, zi,. •., zn] be any homogeneous polynomial.Denote Dg be its associated divisor on P". Put

where C5 is a positive constant satisfying

for all 2;0, • • • , zn € C. Thus ^ > 0 and (pg € C°°(P™ - suppD5) n

*um-Apply the Poincare-Lelong formula, we see

ddc(pg = degg - QPn - Dg, on P".

Then

F* on P.XV

Since F*, TT* commute with d, dc, we see F* commutes with ddc, thenwe restrict the above relation i(C^) to obtain

F*Dg, on

By (3.4), it follows that

Then by applying the lemma 4.1, we have proved

Shanyu Ji 95

Theorem 5.1 (First main theorem) Let /, F be as before. Then

degs • T,Xr, fo) = N*P(Dg\ r, r0) + ^ / F,yyr + O(l)

5(r)

for r » r0.

Corollary 5.2 (Nevanlinna inequality)

N*p(Dg\ r, r0) < deg0T^(r, r0) + 0(1).

Proof Note (pg > 0, then J5(r) F^fla < 0. QED

We would like to give the following proposition to close this sec-tion. The inequality here is conjectured to be equality which remainsa problem.

Proposition 5.3 ESôo j <

Proof

r-.oo logr

= FtUpnAu

P!

- /TT^pnAF

c

= lim

A(dd"]og(|/i \n-l /logr,


where G is the closed graph of F, and J5(r, C") = {(zi, . . . , zn) £C:||^|2 + ... + |^|2<r2}.

/

Since

ro

<\ f

9B(ro>C")

< (deg / • log r + ^) (rfrfc log |/|2)"-2 A drfc log(l + |^|2) A

<(deg/-logr + A)

where |/|2 = |/i|2 + . . . + |/n|2 and the positive constant A is inde-

pendent of r, we then have

logr

fl(r>c:)

"-2 A

Shanyu Ji 97

c?

QED

6. The second main theoremIn this section, we shall prove the second main theorem. Let /, F

be as before. Let DJF be the ramification divisor of the meromorphicmap F on P", i.e., locally on PJ — Ip, it is given by the Jacobiandeterminant of F, and then it is extended on PJ (cf.[SH, p.73]). LetDJF be determined by a unique (up to a constant factor) homoge-neous polynomial Jp £ C[2o, • - -, zn]- We use A^ to denote the sheetsnumber of F.

Theorem 6.1 (Second main theorem) Let /, F be as before. LetDi,...,Dqbe divisors on P™ so that supp£)i,..., suppDg are man-ifolds located in normal crossings. Suppose each Dj = D9j, wheregj £ C[;&0j • • • ? Zn] is homogeneous polynomial of degree pj, for j =1, 2 , . . . , q. Denote D = £? x Dj. Then

for r >> T*O.

Proof For any w 6 i(CJJ) — F(suppDjF), there is an open neigh-borhood W(w) of w in i(C^) — F(suppDjF) and A^ disjoint opensubsets t/i(«;),..., U\F(w) in C^ such that

the restriction F | f/«(w;) is biholomorphic.

Consider the Carlson-Griffiths singular volume form \P,


F*V | W(w)

\F

where

* =•

and

= | detZ>(F

Put\J? I

n!

Apply the Poincare-Lelong formula, def log \\Dj\\2 = — cso

2 2

= F* Ric * -

Shanyu Ji 99

where the formula (2,5) was used.

By (2.1) we obtain

N(ddc log fc r, r0) + ]T N(F*ddclog(log \\Dt||2)2; r, r0)

j=l- N*F (D\ r, r0) - N*F (DJp ; r, r0) (6.2)

To prove the theorem, it suffices to estimate the left hand side of theabove identity.

Let's estimate 7V(J^ddclog(log ||£y2)2;r,r0) first. Since

then

, e ^ . - . ^ ,^ft = log - j - - - r^ - > 1.

3 b(^0,.. . ,^n)|2

Thus

0 < F* log(log pj ||2)2 < 2F* log ̂ . + 2 log 2. (6.3)

Take any w E i(CJJ,), take W(w) and C/i(ti;), . . . , C/AJT(^) as before,we then have

. | W(w) =v=l

\F

v=l


XF

< XF log ̂ <pgj o (F | Uv(w))~l - XF log XF.v=l

= XF log F*<pgj - XF log XF.

By the proof of the lemma 3.3, we also know

Thus apply the lemma 4.1 and by (6.3), (6.4), we obtain

i=1S(r)

<£ /"(F. logoff + 0(1)J'=15(r)

< AF J^ log(2 deg 9i

(6.5)

for r » T-Q. Here the last second inequality is due to the first maintheorem 5.1.

Next we estimate the term ^(d^log^j^ro) in (6.2). By theprevious argument, we see log£ G ̂ ioc(i(C^)). Then

Shanyu Ji 101

AT(dcf log£;r,ro) = i J logger + O(l)

S(r)

S(r)

S(r)(6.6)

ie definitions of£ and £,

and by ^F <

where c = , L/n. Since

B(t) 0 \S(t)

t(cl*)a= I .— I (cl"}(pn. (6.7)/ 2r2n~* dr /t/ i/

0(r) B(r)

Put

(6.8)

From (6.6), (6.7) and (6.8), it follows


(6.9)

By the classical result in the value distribution theory (cf.[Sha,p.84]), (6.9) implies that for any e > 0, there is S = 6(e) > 0, so that<5(e) — > 0 as e -» 0 and there is a subset £7 = £?(e) C R+ with finite(5-measure, such that

/ ^ x< elogr + O (k>gT(r,r0)J

(6.10)

for all r € IR+ - E.To complete estimating (6.10), we estimate the term T(r, TO). Let

F*V | (i(CJ) - F(suppD U suppL>jF)) = ££fc ^dti;^ A rfi^, wherethe matrix R = (Rjk) is positive definite. Recall the definition of |and (2.3),

Thus | < nldet^ holds on «(C£) - F(supp£> U suppDJj?). ByHardamard inequality: for any positive definity matrix R, (det R)l/n <

Then we have

Shanyu Ji 103

wheresince (d<f (K|2 + . . . + Kl2))"'1 = (n-l)l E"=i (l?

dwi A dwi A . . . (/\dwj)oiaii A (dw])omii /\.../\dwn/\ dw^, jF* Ric *

|u>n|2))n by direct computation.Recall the definition of T(r, ro) and (6.4), we see

ror

*dd° log(log ||^ ||2)2; r,

Together with (6.10), for any e > 0, and r € U+ - E,

N(dd° log ̂ r, r-o) < e log r + O(log T^(r, r0)) + O(l)(6.13)

< e logr + 0(log+logr) + O(l).

Here the proposition (5.3) is used. Also by a classical result (cf. [Sha,remark 1, p.88]), we obtain

N(ddclog& r, r0) < e logr + O(log+ logr) (6.13)

for all r » ro.Combinating (6.2), (6.5) and (6.13), we proved the theorem.

QED

7. Other results

For any divisor D — Dg on P™, we define the defect of Dunder f by


c i v8*F = degg- li

where 8*F(D) is independent of the choice of TQ. By the second maintheorem, we have the following

Theorem 7.1 (Defect relation) Let /, F,Z>i, . . ., Dq be as in thetheorem 6.L Then

n

Theorem 7.2 Let /, F be as above. Let D be a divisor given bya hyperplane on P™ with codim (suppD n F(suppDjF)) > 2. LetF*D = D9, and suppose that suppD5l is smooth and gi divides g,where g\, g G C[^QJ • • • ? zn] are homogeneous polynomials. Then

< \F + n + 1 + deg JF.

More precisely,

logr< AFlimôo— — - r + n + 1 + deg JF.

Proof If we can show that

N*F(D91 ; r, r0) < \FN(D : r, r0) , (7.3)

then by the theorem 6.1, for any e > 0 and 0 <TQ <r < +00,

< N*F(D91 • r, r0) + NtF(DjF ; r, r0) + e log r + O(l)< XFN(D; r, r0) + deg JF - T*F(r, r0) + e log r< XF log r + deg JF • T*F(r, r0) + 6 log +O(1).

Here the hypothesis that D is a hyperplane is used.Now we prove (7.3). By the proof of [Dr, lemma 3.2], we know

Shanyu Ji 105

F*D < [suppF*D] + DJp

holds on i(C"), where [suppF*D] is the current by integration onsuppF*Z). Then if U C i(C%) is an open subset with U fl suppDjp

= 0,

[supp.F*£>] | U < F*D | U < [mppF*D] \ U.

i.e., F*JD | U = [suppF*D] | U. (7.4)

For any w G i(C^) - F(suppDjF). There is an open neighbor-hood W(w) of w in ̂ CJ^-F^supp-Dj^) and A^ disjoint open subsetsUi(w),..., C/A^,(^) in i(C"), such that F~1(W(w)) = (J^ ^(w),and jP | f/j(iy) : Ui(w) —* W(w) is biholomorphic. Then

F*D9l < F*D9 | W(w) = F*[sMppDg] \ W(w)

XF

= XFD.

Thus F*D9 | i(CJi) - F(mppDjF) = XFD. Since codim (supp£>nF(suppDjF)) > 2, then F*Dg \ »(CJ) - A^jD, i.e.,

holds on i(C^). This proves (7.3). QED


REFERENCES

[D] J.-P. Demailly, Measures de Monge-Amp6re et caracterisa-tion geom6trique des vari6tes algebriques affines, Mem. Soc.Math. France (N.S.) 19(1985), 1-124.

[Dr] SJ. Drouihet, A unicity theorem for meromorphic mappingsbetween algebraically varieties, Trans. Amer. Math. Soc.265(1981), 349-358.

[F] H. Federer, Geometric measure theory, Springer-Verlag,Berlin-Heidelberg-New York (1969).

[H] H. Hironaka, Resolution of singularities of an algebraic va-riety, I, II, Ann. of Math. 79(1964), 109-326.

[K] O.H. Keller, Ganze Cremona-Transformationen, Monat-shefte fur Math. undPhys., 47(1939), 299-306.

[J] S. Ji, Image of analytic hypersurfaces, Indiana Univ. ofMath. /., Vol. 39, 2(1990), 477-483.

[Sha] B.V. Shabat, Distribution of values of holomorphic map-pings, Transl. of Math. Mono. Vol. 61, A.M.S., 1985.

[Sh] B. Shiftman, Introduction to the Carlson-Griffiths equidistri-bution theory, Lecture Notes in Math. 981, Springer-Verlag,(1983) 44-89.

[St] W. Stoll, Introduction to value distribution theory of mero-morphic maps, in Complex analysis, Lecture Notes in Math.950, Springer-Verlag, (1982) 210-359.

[V] A.G. Vitushkin, On polynomial transformations of C71, inManifolds, Tokyo Univ. Press, Tokyo (1975), 415 -̂17.

RECENT WORK ON NEVANLINNA THEORYAND DIOPHANTINE APPROXIMATIONS

Paul Vojta

What I will describe here is a formal analogy between value dis-tribution theory and various diophantine questions in number theory.In particular, there is a dictionary which can be used to translate, e.g., the First and Second Main Theorems of Nevanlinna theory intothe number field case. For example, we shall see that the numbertheoretic counterpart to the Second Main Theorem combines Roth'stheorem and Mordell's conjecture (proved by Faltings in 1983).

This analogy is only formal, though: it can only be used to trans-late the statements of main results, and the proofs of some of theircorollaries. The proofs of the main results, though, cannot be trans-lated due to a lack of a number theoretic analogue of the derivativeof a meromorphic function, among other reasons. All that I can say atthis point is that negative curvature plays a role in the proofs in bothcases.

Thus, until recently the analogy was good only for producingconjectures, by translating statements of theorems in value distributiontheory into number theory. But in 1989 it has played a role in findinga new proof of the Mordell conjecture, via the suggestion that theMordell conjecture and Roth's theorem should have a common proof,as is the case with the Second Main Theorem.

We begin by briefly describing this analogy, but only briefly asit has been described elsewhere in [V 1] and [V 2], as well as in thebook [V 3]. Likewise, more recent results will be described in [V 6];therefore we refer the reader to [V 3] and [V 6] for details.

Let / : C —> C be a holomorphic curve in a compact Riemannsurface (which we may assume is connected). Let D be an effectivereduced divisor on C\ i.e., a finite set of points, and let dist(jD, P) besome function measuring the distance from P to a fixed divisor D.Then we have the usual definition

Partially supported by the National Science Foundation Grants DMS-8610730

and DMS-9001372.

107

108 Recent Work on Nevanlinna Theory

2?r

m(D,r} = j -

Assuming that /(O) ^ SuppZ?, the definition of the countingfunction can be rewritten as

\w\

Finally, let the characteristic function be given by the more clas-sical definition:

Note in particular that in the above definitions, we only neededthe restriction of / to the closed disc Dr, of radius r. Thus we areactually regarding / as an infinite family of maps fr : Or —> (7,obtained by restriction from /. In the analogy with number theory, leteach fr correspond to one of (countably many) rational points, so thata holomorphic function / : C —> C corresponds to an infinite set ofrational points on C. For example there are no infinite sets of (distinct)rational points on a curve of genus > 1 (Mordell's conjecture), justas there are no nontrivial holomorphic maps from C to a Riemannsurface of genus > 1. Both these facts follow from the appropriateversion of the Second Main Theorem, as defined below.

To make the number theoretic counterparts to the standard defi-nitions as above, let C be a smooth connected projective curve, andlet I? be a reduced effective divisor on C. Assume that both C and Dare defined over a number field k. For each place v of k (i.e., for eachcomplex embedding a : k —> C and for each non-archimedean abso-lute value corresponding to a prime ideal in the ring of integers of fc),let distv(.D, P) again be the distance from P to a fixed divisor D inthe v-adic topology. These distances should be chosen consistently, asin ([L 2], Ch. 10, Sect. 2). For example, if C = P1 and D = [a], thenthe various distw([a], P) functions can be written as min(l, \x — a\v).

Paul Vojta 109

Then the proximity function is defined as

m(

where the notation v\oo means the sum is taken over the (finitelymany) archimedean places of k. Thus, we are comparing the abso-lute values of / on the boundary of Or, with the absolute values "atinfinity" of a number field.

The formula for the counting function is similar:

This is more clearly a counterpart to the definition in the Nevanlinnacase if we write it as

N(D, P) = ^- ]T ord^(P)

where g is a function which locally defines the divisor D, and p is theprime ideal corresponding to the valuation v. Thus the points insideDr, correspond to non-archimedean places, and the summands (forfixed w £ Dr or fixed v) take on discrete sets of values.

Finally, we again let

= m(D,P)+N(D,P)

= hD(P)9

which is a well-known definition in number theory known as the Weilheight.

As before, we can define the defect S(D) = liminf m(D, P)/]ID(P). The assumption that D is defined over k implies that S(D) <1.

Then the following theorem holds with either set of definitionsabove, replacing "?" by r or P, as appropriate.


Theorem (Second Main Theorem). Let D be a reduced effective divi-sor on a curve C. Let A be an ample divisor on C, let K be a canonicaldivisor on C, and let e > 0 be given. Then for almost all "?",

Of course, in the Nevanlinna case, this is true with (1 + e)log TA(T) in place of eTd(r), but this is only conjectured in the numberfield case.

In the number field case, when g = 0 this is Roth's theorem,which is the following.

Theorem (Roth, 1955). Let kbea number field; for each archimedeanplace vofk let av E Q be given. Also let e > 0. Then for all but finitelymany x E fc,

Here H(x) = Y[vmax:(l, |x|v), so that ho(i)(x) =logJT(x).

To see how this theorem follows from the Second Main Theorem,let A be a divisor corresponding to (9(1), let D be the union of allconjugates over k of all av, and take — log of both sides. For details,see ([V 3], 3.2).

When g(C) > 1, the Second Main Theorem is equivalent toMordell's conjecture. Indeed, take D = 0, so that m(D,P) = 0, andwe can take A = K since K is ample. Then let e < 1; this givesa bound for /ijr(P), which is unbounded for infinite sets of rationalpoints. This gives a contradiction. Conversely, if there are only finitelymany rational points, then any statement will hold up to O(l).

If the genus of C equals 1, then the Second Main Theoremcorresponds to an approximation statement on elliptic curves provedby Lang.

Note that in number theory the Second Main Theorem is viewedas an upper bound on ra(D, P) instead of a lower bound on N(D, P)as is the case in value distribution theory.

PaulVojta 111

The fact that the Second Main Theorem of Nevanlinna theoryhas just one proof valid for all values of g(C) suggests that the sameshould hold for number fields. This led to a new proof of the Mordellconjecture ([V 4] and [V 5]), using methods closer to Roth's. Workon obtaining a truly combined proof is progressing. This new proofled Faltings [F] to generalize the methods to give two new theorems:

Theorem (Faltings). Let X be an affine variety, defined over a numberfieid k, whose projective closure is an abelian variety. Then the set ofintegral points on X (relative to the ring of integers in k) is finite.

Theorem (Faltings). Let X be a closed subvariety of an abelian vari-ety A. Assume that both are defined over k, and that X does not containany translates of any nontrivial abelian subvarieties of A. Then the setX(k] of k-rational points on X is finite.

This is still an incomplete answer, because if X does contain anontrivial translated abelian subvariety of A, then this theorem pro-vides no information. Instead, the following conjecture should hold:

Conjecture (Lang, [L 1]). Let X be a closed subvariety of an abelianvariety A. Then X(k) is contained in the union of finitely many trans-lated abelian subvarieties of A contained in X.

Before discussing this further, let us recall some facts about thegeometry of this situation. For all that follows, assume that X is aclosed subvariety of an abelian variety A.

Theorem (Ueno, ([Ii], Ch. 10, Thm. 10.13)). There exists an abeliansubvariety B of A such that the map TT : A —> A/B has the propertiesthat X = 7r~1(7r(X)) and K(X) is a variety of general type.

The map TT| X is calied the Ueno fibration. It is called trivial ifB is a point.

Theorem (Kawamata Structure Theorem, [K]). There exists a finite setZi,..., Zn of subvarieties ofX, each having nontrivial Ueno fibration,such that any nontrivial translated abelian subvariety of A containedin X is contained in one of the Z{.

The set Z\ U • • • U Zn is called the Kawamata locus of X.Then Lang's conjecture is the analogue of the following state-

ment, proved by Kawamata [K], using work of Ochiai [O]:


Theorem. Let f : C —> X be a nontrivial holomorphic curve. Thenthe image of f is contained in the Kawamata locus of X.

By the Kawamata Structure Theorem, this statement is equivalentto Bloch's conjecture, which asserts that the image of / is not Zariski-dense in X unless X itself is a translated abelian subvariety of A.(Bloch's conjecture was also proved, independently, by Green andGriffiths [G-G], also using Ochiai's work). Similarly, to prove Lang'sconjecture, it would suffice to prove that X(k) is not Zariski-denseunless X is a translated abelian subvariety of A.

For further details on these ideas, see ([L 3], Ch. 1 §6 and Ch. 8§1). For more details on the connection with diophantine questions, see[V 3], especially Section 5.ABC for connections with the asymptoticFermat conjecture.

Bibliography

[F] G. Fallings, Diophantine approximation on abelian varieties.Ann. Math., 133 (1991) 549-576.

[G-G] M. Green and P. Griffiths, Two applications of algebraicgeometry to entire holomorphic mappings. The Chern Sym-posium 1979 (Proceedings of the International Symposiumon Differential Geometry in Honor of S.-S. Chern, held inBerkeley, California, June 1979), Springer-Verlag, NewYork, 1980, 41-74.

[li] S. litaka, Algebraic geometry: an introduction to the bira-tional geometry of algebraic varieties (Graduate texts inmathematics 76) Springer-Verlag, New York-Heidelberg-Berlin, 1982.

[K] Y. Kawamata, On Bloch's conjecture. Invent. Math., 57(1980), 97-100.

[LI] S. Lang, Integral points on curves. Publ. Math. IHES, 6(1960), 27-43.

[L 2] , Fundamentals of diophantine geometry, Springer-Verlag, New York, 1983.

[L 3] , Diophantine geometry. Encyclopedia of Mathe-matics, Springer-Verlag, New York, 1991.

PaulVojta 113

[O] T. Ochiai, On holomorphic curves in algebraic varieties withample irregularity, Invent. Math., 43 (1977), 83-96.

[V 1] P. Vojta, A higher dimensional Mordell conjecture, in Arith-metic Geometry, ed. by G. Cornell and J. H. Silverman,Graduate Texts in Mathematics, Springer-Verlag, New York,1986, 341-353. _

[V 2] , A diophantine conjecture over Q, in Seminairede Theorie des Nombres, Paris 1984-85, ed. by CatherineGoldstein, Progress in Mathematics 63, Birkhauser, Boston-Basel-Stuttgart, 1986, 241-250.

[V 3] , Diophantine approximations and value distribu-tion theory. (Lect. notes math., vol. 1239), Springer-Verlag,Berlin-Heidelberg-New York, 1987.

[V 4] , Mordell's conjecture over function fields. Invent.Math., 98 (1989) 115-138.

[V 5] , Siegel's theorem in the compact case, Ann. Math.,133 (1991) 509-548.

[V 6] , Arithmetic and hyperbolic geometry, in: Proceed-ings of the International Congress of Mathematicians, 1990,Kyoto, Japan, to appear.

DIOPHANTINE APPROXIMATIONAND THE THEORY OF HOLOMORPHIC CURVES

Pit-Mann Wong

In the last few years, due to the works of Osgood [Ol,2], Lang[Ll,2,3], Vojta [Vl,2,3,4] and others, there appear to be evidencesthat the Theory of Diophantine Approximation and The Theory ofHolomorphic curves (Nevanlinna Theory) may be somehow related.Currently, the relationship between the two theories is still on a for-mal level even though the resemblance of many of the correspondingresults is quite striking. Vojta has come up with a dictionary for trans-lating results from one theory into the other. Again the dictionaryis essentially formal in nature and seems somewhat artificial at thispoint, it is perhaps worthwhile to begin a systematic investigation. Re-cently, I began to study the Theory of Diophantine Approximations,with the motivation of formulating the theory so that it parallels thetheory of curves. These notes is a (very) partial survey of some of theresults in diophantine approximations and the corresponding results inNevanlinna Theory.

(I) Diophantine Approximation

The theory of diophantine equations is the study of solutions ofpolynomials over number fields. Typically, results in diophantine equa-tions come in the form of certain finiteness statements; for instancestatements asserting that certain equations have only a finite numberof rational or integral solutions. We begin with a simple example.

Example 1 Consider the algebraic variety X2 + Y2 = 3Z2 in P2, weclaim that there is no rational (integral) points (points with rational(integral) coordinates; on projective spaces a rational point is also anintegral point) on this variety. To see this, suppose P = [x, y, z] bea rational point on the variety with #, y, z E Z and gcd(x, y,z) = 1.

This research was supported in part by the National Science Foundation Grant

DMS-87-02144.

115

116 Diophantine Approximation

Then z2 + y2 = 0 (mod 3) so that x = y = 0 (mod 3). Thus x2 and y2

are divisible by 9 and it follows that z is divisible by 3, contradictingthe assumption that gcd(x, y, z) = 1. This example illustrates one ofthe fundamental tools in diophantine equation:

"To show that a variety has no rational point, it is sufficient to showthat the homogenous defining equation has no non-zero solutions modpfor one prime p"

The converse to this statement, the so called "Hasse Principle " is notvalid in general. The following example, due to Selmer:

has no rational points and yet for any prime p, the correspondingequation mod p admits non-trivial solutions.

Example 2 The algebraic set y2 = x3 + 17 in A2 has many rationalpoints, for example (-2, 3), (-1, 4), (2, 5), (4, 9), (8, 23), (43, 282),(52, 375), (5234, 378661) are integral points; (-8/9, 109/27); (137/64,2651/512) are rational points (unlike the projective varieties, a rationalpoint on an affine variety may not be integral). In fact V(Q) is infi-nite. If we homogenize the equation (replace x by X/Z, y by Y/Z),we get

Y2Z = X* + 17Z3

this defines a variety in P2. It has one point at infinity: [0, 1,0]. Therational points are given by {(x, y) G A2(Q)\y2 = x3+17}U{[0, 1, 0]}.It can be shown that the line connecting any two g-rational pointsintersects the variety again in a (^-rational point. In this way one canshow that there are infinitely many Q-rational points. The variety isan example of an elliptic curve. There are two fundamental theoremsconcerning elliptic curves: The Mordell-Weil theorem asserts that theset of rational points on an elliptic curve is finitely generated. TheSiegel theorem asserts that the set of integral points on an elliptic curveis finite. In this example there are exactly 16 integral points, consistingof the eight points listed above and their negative (negative in the

Pit-Mann Wong 117

sense of the group law of an elliptic curve). For further discussionsconcerning elliptic curves we refer the readers to [Sil].

Example 3 Consider the equation

x3 -2y3 = n

where n is any fixed integer. We claim that such an equation has onlya finite number of integral solutions. First we reduce the problem toan estimate.

The left hand side of the equation can be factorized as

(x - V2y)(x - 9^2y)(x - 92^2y)

where 0 is the primitive cube root of unity. Dividing the equation by2/3 we get

y J \ySince 0 is non-real, the absolute value of the second and third terms onthe left above are clearly bounded away from zero and we can choosethe lower bound to be independent of x and y (take the smaller of thedistances to the real axis from 0^2 and 0*^2 for instance), so that

(1) - -^<^ j3y Iz/rfor some constant C independent of x and y. The problem is reducedto the problem of approximating irrational numbers by rationals. Weshall see shortly that the inequality above can have only finitely manyrational solutions.

First we recall a classical result of Liouville (cf. [Schm 2]).

Theorem 1 (Liouville 1851) Let a be an algebraic number of degreed>2 over Q; i.e., [Q(a) : Q] = d. Then there is a constant C > 0(depending on a) so that for any rational number p/q (p,q integersand q > OJ,

(2)p--aq


Proof. Let f ( X ) G Z[X] be the minimal polynomial (of degree d) ofa. For any rational number p/q, clearly qdf(p/q) is a non-zero integer(non-zero because / has no rational roots). Thus we get a lower boundfor |/(p/«)|:

(3) |/(p/g)| > l/qd

On the other hand, if \p/q — a\ < 1 then we can estimate \f(p/q)\from above:

/(«)! = l/'(c)| IP/? - «| < tflp/9 - a|

where C" = sup|a;_Q!|<1 |/'(x)|. Combining with (3) we get

\p/q-Gt\>C/qd

where C = 1/C", as claimed. If p/q — a\ > 1 then the theorem istrivially verified by taking c = 1 for instance. QED

Remark 1 The assumption that a be algebraic is crucial. In fact, thistheorem is used by Liouville to construct transcendental numbers. Forexample, let

then

2~n! < 2q^k~l < cqld

*k *k

for any given c and d, for all k sufficiently large. Thus a cannotbe algebraic by the theorem. For more details concerning Liouvillenumbers and criterion of transcendence see Mahler [Ma] and Gelfond[Ge].

Remark 2 Liouville's theorem implies the following statement. Let abe an algebraic number of degree d > 2, then for any e > 0 there areat most finitely many rational numbers p/q (p, q integers and q > 0)such that

Pit-Mann Wong 119

(4)

Suppose otherwise, then there are rational numbers p/q with arbi-trary large q satisfying (4). Such rational numbers clearly violates (2)because for any c > 0, g~(d+e) < cq~d for q sufficiently large.

Returning to the example, the number a = v/3 is algebraic ofdegree d = 3. Comparing inequality (1) with (2) we see that Liou-ville's theorem is almost but not quite strong enough to guarantee thefiniteness of integral solutions of the equation in example 3. Beforewe recount the history of the improvements of Liouville's theorem,we should mention that for algebraic numbers of degree d = 2, Liou-ville's estimate is essentially sharp. This is a consequence of a verywell-known result of Dirichlet ([Schm2]):

Theorem 2 (Dirichlet 1842) For any irrational number a there existsinfinitely many rational numbers p/q (p, q integers and q > 0) suchthat

*P- — aq

Remark 3 It follows that there are infinitely many rationals p/q withp and q relatively prime and satisfy the estimate above.

Remark 4 Dirichlet's theorem holds for any irrational number, alge-braic or transcendental.

Thus for algebraic number a. of degree 2, the exponent in (4)cannot be improved to 2. For algebraic numbers of higher degree theexponent d + e was improved to

1 + ]-d (Thue, 1901)

(Siegel, 1921)

V2d + e (Dyson, also Gelfond, 1947)

and finally to 2 + e by Roth (1955). Roth was awarded the Fieldsmedal for this achievement.


Theorem 3 (Roth 1955) Let a be an algebraic number of degreed > 2. Then for any e > 0 the inequality

Pa. > 'holds with the exception of finitely many rationals p/q where p, q areintegers and q > 0.

Remark 5 In the case where the degree of a is 2, Liouville's theoremis stronger than Roth's theorem.

Lang (LI]) conjectured that perhaps the estimate in Theorem 3can be improved to

>q2 Iog1+e q

The conjecture is still open at this time (the corresponding statementof this conjecture in Nevanlinna Theory is due to Wong [2], see also[S-W]). However, this estimate had been verified for some specialnumbers. There is also the theorem of Khinchin that this estimateholds for all but a set of numbers of zero Lebesgue measure (cf.Khinchin [Kh]). More precisely:

Theorem 4 (Khinchin) Let (pbea positive continuous function on thepositive real line such that x(p(x) is non-increasing. Then for almostall (Le., except on a set of zero Lebesgue measure) irrational numbera, the inequality

'P_Q

holds for all but a finite number of solutions in integers p, q (q > 0) ifand only if the integral

oo

/<(p(x)dx

converges for some positive constant c.

Pit-Mann Wong 121

Using his improvement of Thue's theorem, Siegel proved thefollowing finiteness theorem:

Theorem 5 (Siegel) On an affine curve (over any number field) ofpositive genus there can only be a finite number of integral points.

In the projective case, Mordell proved that

Theorem 6 (Mordell) The set of rational points on an elliptic curve(i.e., a curve of genus one) is a finitely generated abelian group.

Mordell also made the famous conjecture (solved by Faltingin the affirmative, cf. Endlichkeitssdtze fur abelsche Varietdten uberZahlkorpern, Invent. Math. 73, 183):

MordelPs Conjecture There are only finitely many rational points ona curve of genus > 1.

Remark 5 Unlike the affine case, there is no distinction betweenintegral and rational points on a complete curve.

Fairing's original proof of the Mordell's conjecture is geometricand did not use the theorem of Thue-Siegel-Roth. Vojta (1988) provedthe Mordell conjecture over function fields and, more recently Faltinggave a proof of the general case of the Mordel conjecture, using theThue-Siegel-Roth's theorem. So far (essentially) all the known resultsin diophantine equations are consequences of the Thue-Siegel-Roth'stheorem,

The extension of Roth's theorem to approximation of p-adicnumbers by algebraic numbers, handling several valuations at the sametime, is due to Ridout ([Ri]) and Mahler ([Ma]). First we recall theproduct formula of Artin-Whaples. Let k be a number field and v avaluation on k. Denote by kv the completion of k with respect to vand by nv = [kv : Qv] the local degree. Define an absolute valueassociated to an archimedean valuation v by

||x||t, = |x| ifKv=R

\\x\\v = \x\ ifKv = C

If v is non-archimedean then v is an extension of p-adic valuation onQ for some prime p, the absolute value is defined so that


r —^v —

if x G Q — {0}. With these conventions, there exists a complete setMk of inequivalent valuations on k such that the product formula issatisfied with multiplicity one; i.e.,

(Artin-Whaples) \\x\\v = I

for all x G k — {0}. Extend || ]!„ to the algebraic closure kv of kv.Let k be a field of characteristic zero, denote by k[X] and k(X)

the polynomial ring and the rational function field over k respectively.Fix an irreducible polynomial p(X) in k(X ), define the order at p ofa rational function r(X) in k(X) to be a if r = pas/t where s and tare polynomials relatively prime to p. A p-adic valuation on k(X) isdefined by

| — p-(ordr)(degp)IP ~~ c

For r(X] in £(J£), there exists polynomials / and g in k[X]9 where3 is not the zero element, such that r = f / g . Define a valuation onk(X] by

Moo = \f/g\oo = e**f-**°.

Denote by fc(-X")oo the completion of k(X) with respect to the valuation| |oo and k(X)p the completion of k(X) with respect to | |p. TheArtin-Whaple Product Formula is satisfied for k(X) (and also its finitealgebraic extension).

We now give an analytic interpretation of the product formula.Consider the special case of k = C, the complex number field, thereis also the field 3ft of meromorphic functions defined on' a domain Gin the Riemann sphere CP1. If G = CP1 then 2ft = C(X] = the fieldof rational functions in one variable. Fix a point ZQ ̂ oo in G and fora function / G Sft, we may write

Pit-Mann Wong 123

where g is meromorphic with g(zo) ^ 0 or oo. For a positive constantc, a valuation is defined by

Thus ord^0 / > 0 if ZQ is a zero of / and ord^ / < 0 if ZQ is a pole.If ZQ = oo, we may write

where g is meromorphic with g(zu) ^ 0 or oo. A valuation is definedby setting

I/loo = C°rd°°'.

In the case of G — CP1, the valuation \f\ZQ coincides with \f\p wherep(z) = z — ZQ is an irreducible polynomial and |/|oo = |/Lo- In thiscase it is clear that

n _ Jt of zeros - # of poles-

and the Product Formula for C(X ) is equivalent to the following well-known theorem in complex analysis:

"For a rational Junction on CP the number of zeros and the numberof poles are equal"

It is understood that the numbers of zeros and poles are countedwith multiplicities.

The generalization of the above statement to meromorphic func-tions is the Argument Principle:

Argument Principle Let f be a junction meromorphic on a domainG containing the closed diskAr or radius r. Assume that there are nozeros nor poles on the boundary dAr of the disk, then

1 r fn(0, r) - n(oo, r) = — / — dz

dAr


where n(0, r) and n(oo,r) are respectively the number of zeros andpoles of f inside Ar.

If / is a rational function then there are only finitely many zerosand poles of /. We may choose sufficiently large r so that all zerosand poles of /, with the exception of the point at infinity, are insidethe open disk Ar. Then

dAT

and

e/(/) = I/U.

With this interpretation, it is clear that we recover the Product Formula.Another way of relating zeros and poles of meromorphic functions isthrough Jensen's Formula which will be discussed in the next section.

Roth's Theorem can be restated as follows:

Theorem 7 Let k be a number field (a finite algebraic extension ofQ), and {av G Q \ v G S} where Q is the algebraic closure of Qand S is a finite set of valuations on k containing all the archimedeanvaluations. Then for any positive real numbers c and e, the inequality

^\x-av\\v}>cH(xY(M*

holds for all but finitely many x in k. Here H is the (multiplicative)height.

The analogue in function fields of Liouville's theorem is due toMahler ([Mai]). He also showed that Liouville's theorem cannot beimproved if the characteristic of the field of constant k is positive.

Theorem 8 (Mahler) Let a = a(X) be an element ofk(X)oo alge-braic, of degree d>2, overk(X). Then there exists a constant C suchthat

pa --

Pit-Mann Wong 125

for any polynomials p and q (q not the zero element) G k[X]. If thecharacteristic ofk is positive, the exponent d cannot be improved.

The p-adic case is due to Uchiyama ([U]):

Theorem 9 (Uchiyama) Let a = Oi(X) be an element ofk(X}p al-gebraic, of degree d > 2, over k(X). Then there exists a constant Csuch that

as

>; \s\p})d

for any polynomials r and s (q not the zero element) G k[X], If thecharacteristic ofk is positive, the exponent d cannot be improved.

However, if the characteristic of k is zero, the function fieldanalogue of Roth's theorem is valid.

Theorem 10 (Uchiyama) Assume that char k — 0. Then

(i)Let a. = a(X] be an element ofk(X)uQ algebraic, of degreed>2, over k(X). Then for any e > 0, there exists a constant C suchthat

>^S-for all but a finite number of pairs of polynomials p and q (q not thezero element) G k[X],

(ii)Let a = a(X] be an element ofk(X}p algebraic, of degreed > 2, over k(X). Then for any e > 0, there exists a constant C suchthat

Cra

s

for all but a finite number of pairs of polynomials r and s (q not thezero element) G k(X).

In the positive characteristic case, Armitage ([Ar]) found a con-dition for which Roth's theorem holds.


Theorem 11 (Armitage) Assume that char k > 0. Then the conclu-sions of Uchiyama's theorem hold for those algebraic a which doesnot lie in a cyclic extension ofk(X).

Remark Armitage actually proved the theorem for fields more generalthan function fields. But the conditions on these fields are somewhattechnical to state at this point. One of these conditions is that theArtin-Whaples product formula holds. In the language of Nevanlinnatheory, this means that the First Main Theorem holds).

Roth's theorem is extended to simultaneous approximations byW. M. Schmidt ([Schml,2]) and later by Schlickewei ([Schl]) to thenon-archimedean case. Fkst we recall some terminologies. For a lin-ear form L of (n + l)-variables with algebraic coefficients, (we shallalso identified L with a hyperplane of jP71), the Weil function XV-L isdefined by

where for a linear form L(x) = Y^Q<j<na3x^\\^\\v — maxo<i<n

{lloiHt,}. Hence \VjL(x) > 0.Given a hyperplane L of F71 and a point x E JP^fc) but x £ L,

the proximity and counting functions are defined by

m(x, L) - J A^(z); N(x, L) = J Xv,L(x).

Note that both the proximity and the counting functions are > 0.By the definition of height we have the analogue of the Fkst

Main Theorem in Nevanlinna Theory ([VI], [R-W]:

Theorem 12 (Fkst Main Theorem) If L is a linear form and L(x) ^0, then

where h(x) is the logarithmic (additive) height.

Pit-Mann Wong 127

The Theorem below (cf. Schmidt [Schml] Theorem 2) is an ana-logue of the Second Main Theorem of Nevanlinna Theory for holo-morphic curves. We shall use the same notation for a linear formand the hyperplane it defines. The following version of the subspacetheorem is due to Schlickelwei [Schl] (see also Schmidt [Schml],Theorem 3) and [Schm2]). The formulation below is due to Vojta([Vol] Theorem 2.2.4).

Theorem 13 (Subspace Theorem) Let {LVj \v£S,l<i<n + l}be linear forms in n-variables with algebraic coefficients. Assume thatfor each fixed v £ S ( a finite set of valuations on k containing allarchimedean valuations) the n + 1 linear forms !/„,!, • • • > A;,n+i orelinearly independent. Then for any e > 0 there exists a finite set J ofhyperplanes ofk71^1 such that the inequality

< ssize fx\\y- V ;/n

«eflf \\Lv,i(x)\\v

holds for all S -integral points x = (SCQ, . . . , xn) E Ogn+i —Here

size (x) = maxuG$maxo^<n{||&j||w.}.

It is more convenient to formulate the Subspace Theorem pro-jectively and express the estimate in terms of height rather than size.

Theorem 14 Let {LVti \ v E 5, 1 0 there exists a finite set J ofhyperplanes ofPn(k] such that the inequality

holds for all points x G Pn(k] — \JLejL.

The Subspace Theorem of Schmidt can be extended to the caseof hyperplanes in sub-general position by using the Nochka weight(cf. [No]). This extension is due to Ru and Wong [R-W]. First werecall the definition of sub-general position due to Chen (cf. [Ch]).


Definition Let V be a vector space over F (a field of characteristic0) of dimension (over F) k + 1. Denote by V* the dual of V. For1 < k < n < q, a collection of non-zero vectors A = {vi, . . . , vq}in V* is said to be in n-subgeneral position iff the linear span of any(distinct) n + 1 elements of A is V*. If n = k the concept coincideswith the usual concept of general position.

Remarks (i) It is clear that {i?i,. . .,vq} is in n-subgeneral posi-tion iff {aiî, . . . , aqvq} is in n-subgeneral position where each QJis a unit of F (i.e., ay G F - {o}). Denote by P(F*) the projec-tive space of V*. Then the elements of P(V*) are identified as hy-perplanes of the projective space P(V). A collection of hyperplanes{ttj G P(V*) | 1 < j < q} is said to be in n-subgeneral positioniff {vij . . . , Vg} is in n-subgeneral position where Vj G V* satisfiesP(VJ) = dj. For n = fc this concept agrees with the usual concept ofhyperplanes in general position.

(i"0 If A = {vi, . . . , vg} is in n-subgeneral position then it is also inm-subgeneral position for all m > n provided that m < q.

(Hi) Let (bj G P(W*) \ 1 < j < q} be hyperplanes in generalposition, where W is a vector space over F of dimension n + 1.Let V be a vector subspace of W of dimension fc + 1; then A ={a,j = bj fl P(V) \ 1 < j < q} is a set of hyperplanes in P(V], notnecessarily in general position but is in n-subgeneral position.

Lemma (Nochka-Chen) Let A = {î, . . . , vq} be a set of vectors inV* in n-subgeneral position. Then there exists a function, called theNochka weight associated to A, w : A — > R and a constant 0 with thefollowing properties:

(0 fc + 1 <g<*±l.W 2n-fc + l~ ~ n + l(ii) 0 < u(a) <0fora£A,

0") Ea£A w(o) = k + 1 + 0(#A - 2n + k + 1),

(iv) /or any swfosef B o/A wM # B < ra + 1, Ea£j4u;(a) ^

= dimension of the linear space spanned by elements ofB.

Pit-Mann Wong 129

The generalization of The Subspace Theorem to the case of hy-perplanes in sub-general position takes the following form (cf. [R-W]):

Theorem 15 Let {LVii \ v E S, 1 < i < q} be linear forms of(k + l)-variables (or hyperplanes in I*) with algebraic coefficients.Assume that for each fixed v £ S, the linear forms A;,i? • • • 5 LVjq <*re

in n-subgeneral position (1 < k < n < q) with associated Nochkaweights ILÎ, . . . , u>v,q- Then for any E > 0 there exists a finite set J ofhyperplanes ofPk(k] such that the inequality

holds for all points x E Pk(k) -

In terms of the proximity function, Theorem 15 takes the fol-lowing forms:

Corollary Let {Li, . . . , Lq} be linear forms in (k + 1) -variables withalgebraic coefficients, in n-subgeneral position ( 1 < k < n and q >2n— k+ 1). Then for any e > 0 there exists a finite set J of hyperplanesofPk(k] such that

i<m(x> A) < (A + 1+ e) h(x)

holds for all points x E Pk(k) — U^jL and where u)i are the Nochkaweights.

Corollary Let {Li, . . . , Lq} be linear forms in (k + l)-variables withalgebraic coefficients, in n-subgeneral position (1 < k < n). Givenany e > 0, there exists a finite set J of hyperplanes ofPk(k] such that

. m(x,Li) < (In- fc + l+e) h(x)

holds for all points x G Pk(k) — Uj^^jL.

Corollary Let {Li,...,I/g} be hyperplanes of P"(k), in generalposition. Then for any e > 0 and 1 < k < n, the set of points of

such that

E m(a?, LA > (2n - k + 1 + e) h(x)l<i<


is contained a finite union of linear subspaces, U^jL, of dimensionk — 1. In particular, the set of points ofPn(k) — Ui<i<qLi such that

is a finite set of points.

For a finite subset S of M# of valuations containing the set SQQ ofall archimedean valuations of k. Denote by Os the ring of S-integersof k, i.e., the set of x G k such that

for all v £ S. A point x = (xi, . . . , o^) G fcn is said to be a S -integralpoint if #i G Os for all 1 < i < n. Let D be a very ample effectivedivisor on a projective variety V and let 1 = ZQ, EI, . . . , XN be a basisof the vector space:

I(D) = {/ | / is a rational function on the variety Vsuch that f = 0 or (/) + D > 0}.

Then P -> (zi(P), . . . , xN(P}} defines an embedding of V(k) - Dinto the affine space kN. A point P of V(k) — D is said to be aD -integral point if x»(P) G 05 for all 1 < i < N.

The following theorem of Ru-Wong extends the classical theoremof Thue-Siegel that P1 - {3 distinct points} has finitely many integralpoints:

Theorem 16 Let k be a number field and HI, . . . , Hq be a finiteset of hyperplanes ofPn(k)f assumed to be in general position. LetD = J2i<j<q Hi> then for any integer 1 < k < n, the set ofD-integralpoints ofPn(k) — D is contained in a finite union of linear subspaces of^(k) of dimension k — l provided that q > 2n — k + 1. In particular,the set of D -integral points ofP" (k) — {2n + 1 hyperplanes in generalposition} is finite.

More generally, let V be a projective variety, D a very ampledivisor on V and let {<po, • • - , VN} be a basis of l(D\ such that

Pit-Mann Wong 131

is an embedding of V into PN with V — D embedded in kN. Weidentify V with its image $(V). As an immediate consequence of themain theorem we also have:

Corollary Let V be a projective variety, D a very ample divisor onV. Let D\,..., Dq be divisors in the linear system \D\ such that E =DI + ... + Dq has, at worst, simple normal crossing singularities. Ifq > 2N - k + 1 where N = diml(D) - 1 and 1 < k < n, then theset of E-integral points ofV — E is contained in the intersection of afinite number of linear subspaces, of dimension k — l, ofPN with V. Inparticular, the set of E-integral points ofV — E is finite ifq > 2N+1.

(II) Theory of Holomorphic Curves

The Theory of holomorphic curves is the study of holomorphicmaps from the complex plane into complex manifolds. More gener-ally, one studies holomorphic maps between complex manifolds withthe case of curves being the most difficult. This is due to the factthat the image of a holomorphic curve is usually of high codimension.Typically results in the theory of maps assert that, under appropriateconditions, every holomorphic map in a complex manifold M degen-erates. The types of degeneracy range from the weakest form: "theimage does not contain an open set" to the strongest form: "the imageconsists of one point". In between we have degeneration at a certaindimension. Namely, the image is contained in a complex subvarietyof dimension p with 0 M is constant is said to be Brody-hyperbolic [B]. If M is compact,then the concept of Brody-hyperbolic is equivalent to the concept ofKobayashi-hyperbolic [Kl]. The following differential geometric de-scription of Kobayashi-hyperbolicity is due to Royden [Roy]. Given anon-zero tangent vector £ E TXM9 the infinitesimal Kobayashi-Royden(pseudo) metric is defined by

0 < MO = mf


where the inf is taken over all positive real numbers r such that thereexists a holomorphic map / : Ar —> M such that f(0) = x and/'(O) = £. Here Ar is the disk of radius r in C. Alternatively,

MO = sup |t| > 0

where the sup is taken over all t G C* such that there exists a holo-morphic map / : A —> Af with /(O) and /'(0) = t£. A complexmanifold M is said to be hyperbolic at a point x if there exists anopen neighborhood U of x and a hermitian metric dsu2 on TC7 suchthat kM > dsu on TU. A complex manifold M is said to be hyper-bolic if it is hyperbolic at every point. The Kobayashi pseudo-distanceassociate to kM is defined by

where the inf is taken over all piecewise smooth curves joining xand y. The condition that M is Kobayashi-hyperbolic is equivalentto the condition that the Kobayashi pseudo-distance is a distance;i.e., d(x, y) > 0 if x ^ y. With this distance function M is a met-ric space and M is said to be complete if M is a complete metricspace.

The infinitesimal Kobayashi metric satisfies kM(t£) = \t\kM(£),hence it is a Finsler metric. It has the nice property that every holomor-phic map is metric decreasing. Namely, if / : M —> N is holomorphicthen £#(/*£) < #M(£)- ^ particular, every biholomorphic self map ofM is an isometry of the Kobayashi metric.

The infinitesimal Kobayashi metric does not have very good reg-ularity in general. In this direction we have the fundamental result ofRoyden [Rl] that the infinitesimal Kobayashi metric is always uppersemi-continuous. If it is complete hyperbolic then the metric is con-tinuous. It is well-known that the poly-discs are complete-hyperbolicbut the infinitesimal Kobayashi metric is not differentiable.

As mentioned above, for compact manifolds Kobayashi-hyper-bolic is equivalent to Brody-hyperbolic. In general, Kobayashi-hyper-bolic implies Brody-hyperbolic but the converse may not be true if Mis non-compact. The example below is very well-known.

Pit-Mann Wong 133

Example (Eisenman and Taylor) Let M be the domain in C2 given by

M= {(z,w) E C2 | \z\ < I9\zw\ < 1 and \w\ < 1 if z = 0}

Then M contains no complex lines; for if / : C —* M is holomorphicthen TTi o / is bounded (where TTI is the projection onto the first coor-dinate), hence constant; now if TTI o / is constant then KZ o / (where mis the projection onto the second coordinate) is bounded and so mustbe constant as well. However M is not Kobayashi-hyperbolic, becausethe Kobayashi distance of any point of the form (0, w] from the ori-gin is zero. This is evident by considering the connecting paths: forany positive integer n, let /;-jn : A —> M, j = 0,1,2, be holomorphicmaps defined by f^(z) = (z,Q),fitn(z) = (l/ra,nz) and /2,n(*) =(l/n + z/29w). Then /0,n(0) = (0,0) =po,/o In(lAO = (l/n,0) =Pl,/l,n(0) = (l/n,0) = Pl,fltn(w/ri) = (l/n,«j) = P2,/2,*(0) -

(l/n,w) = p25/2,n(—2/n) = (0, it;). The Kobayashi distances be-tween the points 0,1/n and -2/n on the unit disc approaches zero asn approaches oo.

Intuitively speaking, for non-compact manifolds, the points at"infinity" plays a very important role. In fact Green [Gn4] showedthat

Theorem 17 (Green) Let D be a union of (possibly singular) hyper-surfaces DI , . . . , Dm hypersurfaces in a complex manifold M. ThenM — D is Kobayashi-hyperbolic if

(0 There is no non-constant holomorphic curve f : C —> M - D;(ii) There is no non-constant holomorphic curve

f : C -> A! H ... H Dik - (Dh U ... U Dh)

for all possible choices of distinct indices so that {ii,.. .,*fc}U{j'll"-!Jl} = { l , » - 1 ™ } -

An important special case of this is the theorem:

Corollary (Green) The complement of q hyperplanes in generalposition in CPn is Kobayashi-hyperbolic for q > In + 1. The num-ber 2n + l is sharp.


Let {PJ(ZQ, . . . ,2n) | 1 < j < k} be a set of homogeneouspolynomials with coefficients in an algebraic number field k. Let Vbe the common zeros of Pj (i < j < fc) in CPn. We assume that Vis irreducible and smooth. Lang conjectured that if V is hyperbolicthen for any finite extension K of the field k, the set of rational pointsV(K] in V over K (i.e., V(K] = {(ZQ, . . . , zn) \ if there exists i so thatZi ^ 0 and ^/^ G 1£ for all j} is finite. If V is a hyperbolic affinealgebraic manifold in kN defined by polynomials {Qj(zi, . . . , ZN) =0, 1 < j < k} then the set of integral points of V over K is finite. Thisconjecture is verified for the case of curves of genus g > 2 (Falting);for V = Pn — {2n + 1 hyperplanes in general position} (Ru-Wong)and for V = complement of an ample divisor of an abelian variety(Fairings).

Green ([Gn3]) also proved that if the hyperplanes are distinct butnot in general position, one can still conclude that the complement isErody -hyperbolic, namely it contains no non-trivial holomorphic curvefrom C. The corresponding statement in number theory:

"Pn(k) — {2n + 1 distinct hyperplanes} contains only finitely manyintegral points "

is still open. The proof of Ru-Wong for the case of hyperplanes ingeneral position involves an extension of the Siegel-Roth-Schmidt typeestimate for which the general position assumption is necessary.

Returning to the discussion of hyperbolic manifolds, the follow-ing Theorem of Brody ([B]) is very important in constructing exam-ples.

Theorem 18 (Brody) Small smooth deformations of a compact hy-perbolic manifold are hyperbolic.

Thus the set of compact hyperbolic manifolds is open. The fol-lowing example of Brody and Green shows that it is not a closed setin general.

Example (Brody-Green) The hypersurface in CP3 defined by

V£ = z$ + (ezoZl)d/2 + (ezQz2)

d/2 = OJ

Pit-Mann Wong 135

is hyperbolic for any e ^ 0 and where d > 50 is an even integer. Fore = 0, Vb is a Fermat variety which is clearly not hyperbolic. Notethat by the Lefschetz theorem, V£ is simply connected. Since VQ isnon-singular (it is the Fermat surface of degree d) it follows that Ve

is non-singular for small e. A Fermat surface of any degree admitscomplex lines, for instance take \i, and 77 be any d-th roots of —1, thenZQ = fjiZi and 22 = r)Z3 is a complex line in the Fermat surface ofdegree d.

For the non-compact case, the problem of deformation of hy-perbolic manifolds is much more complicated, additional assumptionsare needed. The concept of "hyperbolic embeddedness" is needed, weshall not get into this here. The readers are encouraged to look intothe very interesting paper of Zaidenberg [Z].

Classically, hyperbolicity is studied via the behavior of curvature.The most well-known Theorem is the Schwarz-Pick-Ahlfors lemma:

Theorem 19 Let M be a complex hermitian manifold with holomor-phic sectional curvature bounded above by a (strictly) negative con-stant. Then M is Kobayashi-hyperbolic.

For a Riemann surface, Milnor [Mi] (see also Yang [Ya]) ob-served that the classical condition on the holomorphic curvature canbe relaxed to the condition that the curvature satisfies K(r) < —(1 + e)/(r2logr) asymptotically where r is the geodesic distancefrom a point. The following higher dimensional analogue of Milnor'sresult is due to Greene and Wu ([G-W] p. 113 Theorem G'). A pointO of a Kahler manifold M is called a pole if the exponential map atO is a diffeomorphism of the tangent space at O onto M. Let r be thegeodesic distance from O. Let Sr be the geodesic sphere and X bethe outward normal. Then Z = X — \/^lJX is called the (complex)radial direction. The radial curvature is defined to be the sectionalcurvature of the plane determined by the radial direction. With theseterminologies we can now state the Theorem of Greene and Wu:

Theorem 20 Let M be a complex Kahler manifold with a polesuch that (i)the radial curvature is everywhere non-positive and< — l/(r2logr) asymptotically, (ii)the holomorphic sectional curva-ture < -1/r2 asymptotically. Then M is Kobayashi-hyperbolic.


The next Theorem ([K-W]) gives a criterion of hyperbolicitywithout requiring information on the precise rate of decay of the cur-vature. A complex manifold M of complex dimension n is said tobe strongly q-concave if there exists a continuous function <p on Msuch that (z)for all real numbers c the set {z e M | <p(z) < c} iscompact, (iz)the Levi form idd(p is semi-negative and has at leastn — q negative eigenvalues (in the sense of distributions) everywhereoutside a compact set. Alternatively, M is strongly ^-concave if thereexists a continuous function (p on M such that (z')for all real numbersc the set {z G M \ (p(z) > c} is compact, (ii)fhe Levi form idd(p issemi-positive and has at least n — q positive eigenvalues (in the senseof distributions) everywhere outside a compact set.

Theorem 21 (Kreuzman-Wong) Let M be a complete Kdhler mani-fold of complex dimension m such that both the holomorphic sectionalcurvature and the Ricci curvature are (strictly) negative. Assume thatM is strongly 0-concave and that the universal cover is Stein then Mis Kobayashi hyperbolic.

A complete simply connected Riemannian manifold M of non-positive Riemannian sectional curvature is said to be a visibility man-ifold if any two points at infinity (denoted M(oo)) can be joinedby a unique geodesic in M. A complete simply-connected Rieman-nian manifold with sectional curvature bounded above by a negativenumber (i.e., K < —62) is a visibility manifold. More generally acomplete simply-connected Riemannian manifold with strictly nega-tive sectional curvature (i.e., K < 0) and radial curvature — K(r), fromsome fixed point, satisfying the condition

oo/ rK(r) dr = oo

is a visibility manifold.

Corollary (Kreuzman-Wong) Let M be a complete Kdhler manifoldsuch that its universal cover satisfies the visibility axioms and that theRiemannian sectional curvature satisfies -a2 < K < 0. Assume thatM admits a finite volume quotient. Then M is Kobayashi hyperbolic.

Pit-Mann Wong 137

The proof of Theorem 21 (and the corollary) relies on the corn-pacification theorem (again, we see that the "infinity" plays a crucialrole) of Nadel and Tsuji [N-T] extending the result of Siu and Yau[S-T] on compacification of Kahler manifolds of finite volume andnegative pinching (both above and below) of the Riemannian sectionalcurvature. The theorem of Siu and Yau gives more precise informa-tion on the compactification, in the case of Kahler manifolds, of thecorresponding theorem of Gromov ([B-G-S]) in the Riemannian case.Both the theorem of Nadel-Tsuji and that of Siu- Yau have the originin the work of Andreotti-Tomassini [A-T] on pseudoconcave mani-folds. These theorems are natural generalization of the well-knowncompacification theorem for finite volume quotients of bounded sym-metric domains.

Let D be an irreducible algebraic curve in CP2. At a point pin D let A\, . . . , A& be local irreducible components of D containingp. Let L be a projective line through p and denote by rrij = min^{intersection multiplicity of L n Aj}. Then (mi — 1, . . . , m^ — 1) arethe orders of irreducible singularities at p. Let

b = V (m, - 1)^ 3 '

be the total order of singularities of D. Denote by D* the dual curveof D. The curve D is birationally equivalent to its dual .D*. The nor-malization of D and D* are isomorphic. Denote by 6* the total orderof singularities of D*.

Theorem 22 (Grauert-Peternell [G-P]) Let D be an irreduciblealgebraic curve in CP2 of genus g > 2. Assume that b* + x(D] < 0(where b* is the total order of irreduible singularities of D) and thatevery tangent ofD* intersects D* in at least two points. Then CP2 — Dis hyperbolic.

At present one of the major open problems in the theory ofhyperbolic manifolds is the following conjecture.

Conjecture : For a generic algebraic curve D of degree d > 5 inCP2, the complement CP2 — D is Kobayashi-hyperbolic.


The space of algebraic curves of degree in d in CP2 is a projectivemanifold, denoted $d. By a generic curve of degree d, we mean anelement of $d — Sd where Sd is a subvariety of lower dimension. Theconjecture is of course false without the "generic" condition. One ofthe difficulty of the conjecture is to describe the exceptional subvarietySd- The interested readers are refer to the paper of Grauert [G] formany interesting ideas.

We now turn to the Second Main Theorem of Value DistributionTheory. It all begins from the fundamental work of Nevanlinna inone complex variable. Let / : C —> CP1 be a holomorphic map. Thecharacteristic function T(/, r) is defined by

At

where 0 < s < r and u is the Fubini-Study metric on CP1. For apoint a G CP\ denote by ra(/, a, r) the number of preimages (countingmultiplicities) of a inside the disk of radius r. The counting function

J aj r) is defined by

XV.I.T) -

The proximity function ra(/, a, r) is defined by

dAr

where ||#;a|| = |<rc,a>[/||ic|| ||a|| is the projective distance between£ and a. Here ||z||2 = |zo|2 + |£i|2, ||a||2 = |a0|

2 + |ai|2 and <x,a> =XQCLQ + xiai. The characteristic function, counting function and prox-imity function are related by the First Main Theorem of Nevanlinna

Theorem 23 (FMT) Letf:C-^ CP1 be a non-constant holomorphicmap and let a be a point ofCP1. Then

Pit-Mann Wong 139

The FMT plays a similar role in Nevanlinna Theory as the roleplayed by the Artin-Whaple product formula in Number Theory. Thecounterpart of Roth's Theorem in Number Theory is The Second MainTheorem of Nevanlinna:

Theorem 24 (SMT) Let f : C — > CP1 be a non-constant holomorphicmap and {ai, . . . , aq} be a finite subset of CP1. Then for any e > 0there exists a set A of finite Lebesgue measure such that the followingestimate holds for all r E [s, oo) - A:

Note that, by the FMT, the left hand side of the SMT can bereplaced by <?T(/, r) - £i<,-<g #(/ , a,-, r).

For a point a G CP1, the defect S/(a) is defined to be

r) 1== "Corollary Lef / : C — >• CP1 fee a non-constant holomorphic map.Then for a finite set {ai, . . . , ag} o/ CP1, the sum of defects satisfiesthe estimate

V S < 2.~

The factor 2+e in the SMT correspondes to the exponent 2+£ inRoth's Theorem. The fact that CP1 — {3 distinct point} is hyperbolicis a consequence of Nevanlinna' s SMT. This corresponds to the factin number theory that Thue-Siegel Theorem (the integral points ofP1^)— {3 distinct point} is finite) follows from Roth's Theorem. Theproofs of these two statements are completely analogous (cf . Ru-Wong[R-W]).


Nevanlinna's Theorem can be extended from CP1 to arbitraryRiemann surface in the following form. Let M be a Riemann surfacewith hermitian metric

ds2 = h^- — dz A dz.2?r

Denote by R the Gaussian curvature of h\ i.e.,

fl2

-I

Theorem 25 Let M be a Riemann surface with hermitian metricds2 and let f : C — > M be a non-constant holomorphic map. Let{ai, . . . , aq} be a finite set ofM. Then for any e > 0 there exists a setA of finite Lebesgue measure such that

3 \Z\<t

holds for all r E [s, oo) - A. Consequently, the sum of defects satisfiesthe estimate

f (

3 \Z\<t

Here the characteristic function T(/, ds2, r) is given by

3 \Z\<t

If the Gaussian curvature R is constant (= c) then

3 \Z\<t

Pit-Mann Wong 141

and Theorem 25 takes a simpler form:

Corollary Same assumption as in Theorem 25 and assume that theGaussian curvature R = c of the Riemann surface M is constant. Thenfor any e > 0 there exists a set A of finite Lebesgue measure such that

qT(f, ds\ r] - N(f, *i> r) < (2c + e)T(/, ds\ r)

holds for all r £ [5, oo) - A Consequently, the sum of defects satisfiesthe estimate

8(f,aj)<2c.Vt" Jl —

For the Riemann sphere with the Fubini-Study metric, the cur-vature R = 1 (parabolic); for the torus (elliptic), the canonical metricis a flat metric, i.e. R = 0; and for surface of genus > 2 (hyperbolic)with the canonical metric the curvature R = — 1. Thus

Corollary (0 IfM=CP1then^1^qS(f,aj)<2;(ii) I f M = T = torus then Ei<y</(/» flj) ^ °> in

particular every non-constant holomorphic map fromC into T is surjective;

(Hi) If genus M > 2 then there is no non-constant holo-morphic map from C into M, i.e., M is hyperbolic.

The corresponding Theorems in number theory assert that thefollowing spaces contain only finitely many integral points over anynumber field k:

(i) (Thue, Roth, Schmidt) P1 - {3 distinct point};(/i) (Siegel) T1 - {one point};(Hi) (Mordell conjecture) compact Riemann surfaces of

genus > 1.

Nevanlinna's Theorem can also be generalized to higher dimen-sion for holomorphic maps between equidimensional manifolds. Thisextension is due to Carlson-Griffiths [C-G] and Griffiths-King [G-K].Lei / : C" — > M71 be a holomorphic map into a projective manifold.Let D be an ample divisor on M represented as the zero set of a


holomorphic section s of a holomorphic line bundle I over M. Theproximity function is defined by

dBr

where Br is the ball of radius r in C"1 and da is the rotationallyinvariant measure of the boundary, normalized so that the volume ofthe boundary dBr is 1. Specifically,

Let I be a holomorphic line bundle over M and let h be a her-mitian metric on I with Chern form p. The characteristic function of/ is defined by

where u =

We state the SMT of Carlson-Griffiths-King in the sharper formof Wong [W2] (see also Goldberg-Grinshtein [G-G], Lang [L4] andCherry [Ch]):

Theorem 26 Let I be a positive holomorphic line bundle over a pro-jective manifold of dimension n and Z>i, . . . , Dq £ |I| be divisors suchthat D = DI + . . . + Dq is of simple normal crossing. Let T be the dualof the canonical bundle ofM. Let f : C71 — > M71 be a non-degenerate(Jacobian not identically zero or equivalently, the image contains anon-empty open set). Then for any s > 0, there exists a set A of finiteLebesgue measure such that the estimate

Pit-Mann Wong 143

n(l + e)loglogT(/,r,r)

^n(l + e)logloglogT(/>IJr)

on(! + e)loglogr

holds for all r £ [s, oo) — A.

Corollary With the same assumptions as in Theorem 25 and assumein addition that f is transcendental Then

Theorem 26 holds if one replaces C™ by an affine algebraic man-ifold N of dimension m > n = dimM and under the same non-degeneracy assumption; namely, the image of the map / contains anon-empty open set. Stoll extended Theorem 26 to the case wherethe domain is a parabolic manifold. In this more general case, theright hand side of the estimate of Theorem 26 is more complicated;terms involving the Ricci curvature of the parabolic manifold alsoappears. We refer the readers to Stoll [Sto2] for details. The corre-sponding statement in number theory of the estimate in Theorem 26is conjectured by Lang. This sharper form of the Roth's Theorem isstill open.

Nevanlinna's Theory can also be extended to the non-equidimen-sional case under a much weaker non-degeneracy assumption. Thiscase is much harder and much deeper; so far the only satisfactoryresult is the case of hyperplanes in CPn even though there are someprogress in the more general case. The main ideas of handling linearlynon-degenerate holomorphic curves are contained already in Ahlfors[A] (also H. Weyl and J. Weyl [W-W]; for a different approach seeCartan [Ca]). Unlike the case of Nevanlinna and also the case ofCarlson-Griffiths-King where the first derivative of the holomorphic


map contains all the necessary informations needed; the linearly non-degenerate condition involves, for curves in CP71, derivatives of / oforder up to n. The informations contained in the derivatives are relatedby the Pliicker Formula. Ahlfor's Theory was extended by Stoll to thecase of linearly non-degenerate meromorphic maps from C™ into CPn.Stoll realized that the associated maps in the higher dimensional case,unlike the case of curves, are in general only meromorphic rather thanholomorphic. This is so even if the original map is assumed to beholomorphic. Thus it is necessary to develop the whole theory formeromorphic maps. The Theory of Ahlfors and Stoll was extended byMurray where the domain is assumed to be Stein, and Wong where thedomain is assumed to be affine algebraic or parabolic. The followingsharper form of the SMT is due to Stoll- Wong [S-W].

Theorem 26 Let M be an affine algebraic manifolds of dimension mand f : M — > CPn be a linearly non-degenerate meromorphic map.Let ai,...,aqbe hyperplanes ofCPn in general position. Then for anye > 0 there exists a set A of finite Lebesgue measure such that

+ (2 + e) log log T(/, r) + — log+

+ (log+log+log+T(/}r)

+ O(log+log+log+r)

for all r G [s, oo) — A and where &M is the degree ofM.

The SMT for linearly non-degenerate curves in CPn correspondesto the subspace Theorem of Schmidt in number theory. The SMT canalso be extended to the case of hyperplanes of CPk in n-subgeneralposition. The result is first conjectured by H. Cartan and is known asCartan conjecture. The conjecture is first resolved in the affirmative byNochka and also in the Thesis of Chen. The corresponding result wasdue to Ru and Wong using ideas from the works of Nochka and Chen.

Pit-Mann Wong 145

Theorem 27 Let M be an affine algebraic manifolds of dimension mand f : M — >• CPk be a linearly non-degenerate meromorphic map. Letai, . . . , aq be a hyperplanes ofCP* in n-subgeneral position (k < n).Then for any e > 0 there exists a set A of finite Lebesgue measuresuch that

(f> %', 0 < (« + 1 +

for all r G [s, oo) — A

Actually it is possible to obtain a more precise estimate as inTheorem 26.

For holomorphic curves from C into Abelian varieties, a SMTwas obtained by Ochiai [Oc] and also by Noguchi [Nog] using jetmetrics. However these results are not in the best possible form. In aforth coming paper we shall treat the case of holomorphic curves inspaces of constant sectional curvature. A sharp form of the SMT canbe obtained via the use of Pliicker's formula and also the technique ofSiu described below. Recently R. Kobayashi, using a rather differentmethod seems to obtain a fairly sharp SMT in the case of holomorphiccurves in Abelian varieties.

The main ingredients of the proof are: Green- Jensen's Formula,Nevanlinna's lemma and Pliicker's Formula.

Green- Jensen Formula Let (p be a function of class C2 or a plurisub-harmonic function or a plurisuperharmonic function on C71. Then

M<*l-

=r

l- j

where ddc[ip] denotes differentiation in the sense of distribution.


In particular the lemma applies to (p = log|/| where / is ameromorphic function.

To state the lemma of Nevanlinna we need a definition. A non-negative, non-decreasing function g defined on [0, oo) is called agrowth function if for any to > 0

00

100.

Jto

A typical growth function is g(t) = (log(l +1))1+£ where e > 0.

Nevanlinna's Calculus Lemma Let T be a non-negative, non-decreasing, absolutely continuous function defined on the interval[s, oo) where s > 0. Let g be a growth function. Then there existsa measurable subset [s, oo) with finite Lebesgue measure such that

T'(r] < T(r) g(T(r)}

holds for all r G [s, oo) — A.

This technical lemma is fundamental in all the estimates en-countered in Nevanlinna theory. Another lemma which is of techni-cal as well as theoretical importance is Pliicker's Formula. Let S bea Riemann surface with hermitian metric h and (M, g) be a com-plex manifold of dimension n with constant sectional curvature c. Let/ : S —> M be a holomorphic curve and fk be the fc-th associatecurves. Assume that fk ^ 0 for 1 < k < n. Define differential forms

We may now state the Pliicker Formula (cf. [W3]):

Pliicker's Formula for Spaces of Constant Curvature Let (M, g)be a hermitian manifold of constant curvature c and S a Riemannsurface with hermitian metric h. Let f : S -* M be a holomorphic

Pit-Mann Wong 147

curve which is non-degenerate of order k; i.e., the associate curvefk £ 0. Then

fRic 61 = 92-281+0êfc_i + 9fc+i - 29fc; 2 < k < n - 1

on S - {C e S|Afc(C) = 0}. Afote ito G0 = &n = 0.

One of the main reasons that the case of curves in CPn workswell is that the associate (osculating) curves are holomorphic (to seethis one can either follow the method of Ahlfors or Wong [W3]). If themetric connection of the target space is holomorphic then of course allhigher derivatives of the curve are also holomorphic and £ priori so arethe associate curves (which are the wedge product of the derivatives).However, connections are usually not holomophic (almost never is,for details see Wong [W3]); for instance the connection of the Fubini-Study metric is not holomorphic. On the other hand, meromorphicconnections do exist on projective varieties, hence osculating curvesdefined via these connections are also meromorphic. This is the mainidea of Siu's SMT.

Theorem 28 Let M be a projective surface (i.e., complex dimension2) with a meromorphic connection D. Let the a holomorphic sectionof a holomorphic line bundle $ over M such thatt®D is holomorphic.Let f : C —> M be a holomorphic curve which is non-degenerate inthe sense that the image of f is not contained entirely in the pole setof D and that f A Df ^ 0. Let I be a holomorphic line bundle overM with a non-trivial holomorphic section s such that D = [s = 0] isnon-singular. Then for any e > 0 and A > 1 there exists a set A offinite Lebesgue measure such that

(I - e)T(/, r) + N ( f , D, r) < AT(/, f ® I* , r) + o(T(/, I, r)

for all r E [s, oo) — A.

Siu's Theorem provides some very interesting new exampleseven though this approach does not yet produce the "right" estimatein many of the important cases. The problem lies in the difficulty


of controlling the pole order of the meromorphic connection, makingoptimal estimate in the SMT unattainable.

Another long standing problem which is solved only in the lastfew years is the problem of moving target. A hyperplane in CPn =PtC71*1) may be identified with a point in the dual P((C7l+1)*). Butinstead of considering fixed hyperplanes 01, . . . , aq G P((Cn+1)*) oneconsiders holomorphic curves P((Cn+1)*). In theone dimensional case, Nevanlinna conjectured that the deficit estimateof a holomorphic curve / : C — > CP1 remains valid if the growth ofcharacteristic functions T(gj,r) of the moving hyperplanes is slowerthan the growth of the characteristic function T(/,r). Chuang [Chu]made significant progress on this problem. The conjecture is finallysolved by Steinmetz in 1986 for curves into CP1. The case of curves inCPn is solved by Ru-Stoll [R-S1,2] recently. Bardis [Ba] and O'Shea[OS] extended the Theory of moving targets to the case where thedomain is also of higher dimension; deficit estimates are obtained un-der additional assumptions. We shall only state the SMT of Steinmetzand Ru-Stoll here. Given a family of holomorphic maps {31, . . . , gq}from C into P((C7l+1)*), the field of meromorphic functions gener-ated by {0i, . . . , gq} is the smallest subfield S3 of the field of mero-morphic functions on C containing all the coordinate functions offljj 1 < j < 9- A holomorphic curve / : C — > CPn is said to belinearly non-degenerate over S3 if the coordinate functions of / doesnot satisfies any non-trivial linear equation with coefficients in S3.

Theorem 29 Let f : C — > CPn be a holomorphic curve and g\ , . . . , gq

: C — » P((C™+1)*) be q holomorphic maps considered as moving hy-perplanes ofCP71 in general position. Assume that T(QJ, r)/T(/, r) — >•0 as r — » oo and that f is linearly non-degenerate over S3. Then forany e > 0 there exists a set A of finite Lebesgue measure such that theestimate

(/,&,r) < (n + l + e)T(/,r)

holds for all r £ [s, oo) - A. Consequently

Pit-Mann Wong 149

In fact Ru-Stoll obtained a version of the SMT for moving tar-gets corresponding to the Cartan conjecture. For this they introduce aconcept called fc-flat (we refer the readers to [R-S2] for details.

Theorem 30 Let f : C —> CPn be a holomorphic curve and g\,..., gq

: C —» P((C7l+1)*) be q holomorphic maps considered as moving hy-perplanes ofCPn in general position. Assume that T(QJ} r)/T(/, r) —>0 as r —> oo and that the dimension of the map f is k-flat over S3. Thenfor any e > 0 there exists a set A of finite Lebesgue measure such thatthe estimate

'mtf' »> 0 < (2n - fc + 1 + e) T(/, r)

holds for all r E [s, oo) — A and where the Uj 's are the Nochka weightsassociated to the QJ 's. Consequently

(Ill) Remarks

From the results listed in the two previous sections, the simi-larities between the two theories seem quite striking. The results inthe Theory of curves are more complete due to the fact that thereare more tools available. The analytic machineries are more power-ful; the idea of Nevanlinna using invariants defined by integrals (e.g.,characteristic functions, proximity functions) makes estimates easierto obtain (pointwise estimates are replaced by integral estimates). Fur-thermore, the proofs of the various Theorems in Nevanlinna Theoryare quite uniform. The basic approach and the basic steps are essen-tially the same. The key ingredients are the Jensen formulas (corre-sponds to the Artin-Whaple's Product Formula), Ahlfors' Theory ofassociate (osculating) curves (corresponds to the successive minimain the geometry of numbers) and Nevanlinna's calculus lemma esti-mating the derivative of a positive convex increasing function by thefunction itself.

Even though Nevanlinna's lemma is elementary in nature, it hasthe effect of making many estimates routine. Without this technical


lemma Nevanlinna Theory would be much more complicated. Unfor-tunately, there is as yet no good analog of Nevanlinna's lemma inNumber Theory. This perhaps is the main reason that the proofs indiophantine approximation are not as uniform; many estimates are ob-tained via ingenious process which are perhaps not so "natural". Thesearch of a good analog of Nevanlinna's lemma should be one of themain technical goal in the theory of diophantine approximations.

If one compares the theory of successive minima to the the-ory of associate curves one notices that the later is much more well-developed. The center-piece of the theory of associate curves is theFormula of Pliicker, relating the invariants of higher order osculatingcurves to that of the lower order osculating curves. The counterpart ofPliicker's Formula in the theory of successive minima has yet to bedeveloped. The precise relations among the successive minima seemrather complicated at this point. A better understanding of these fun-damental relationships would go a long way in developing the theoryof diophantine approximations.

References

[A] Ahlfors, L. The Theory of Meromorphic Curves, Acta Soc.Sci. Fenn. Ser. A, vol. 3, no. 4 (1941).

[A-G] Andreotti A. and H. Grauert, Algebraische Korper von auto-morphen Funktionen, Nachr. Akad. Wiss. Gottingen,Math-.-Phys. Kl. H (1961), 39^8.

[Ar] Armitage, The Thue-Siegel-Roth Theorem in Characteristicp, J. Alegbra 9, 1968).

[A-T] A. Andreotti and G. Tomassini, Some Remarks on Pseudo-concave Manifolds, Essays on Topology and Related Topics(dedicated to G. de Rham), Springer-Verlag, 1970.

[B] R. Brody, Compact manifolds and hyperbolicity, Trans. Am.Math. Soc., 235 (1978), 213-219.

[Ba] Bardis, M., Defect relation for meromorphic maps definedon covering parabolic manifolds, Notre Name Thesis 1990.

Pit-Mann Wong 151

[B-G-S] W. Ballman, M. Gromov and V. Schroeder, Manifolds ofNonpositive Curvature, Progress in Math. 61 (1985),Birkhauser.

[Ca] Cartan, H., Sur les zeros des combinaisons lintaires de pfonctions holomorphes donnees, Mathematica (cluj) 7, 80-103 (1933).

[C-G] Carlson, J. and Ph. Griffiths, Defect relation for equidimen-sional holomorphic mappings between algebraic varieties,Ann. of Math. 95 (1972), 557-584.

[Ch] Chen, W. Cartan's Conjecture: Defect Relations for Mero-morphic Maps from Parabolic Manifold to Projective Space,University of Notre Dame Thesis 1987.

[Chu] Chuang, C.-T., Une generalization d'une inegalite deNevanlinna. Sci. Sinica 13 (1964), 887-895.

[Cy] Cherry, W., The Nevanlinna error term for coverings gener-ically surjective case, preprint (1991).

[E-N] P. Eberlein and B. O'Neill, Visibility Manifolds, Pacific J.of Math. 46 (1973), 45-109.

[Fl] Faltings, G. Endlichkeitssatze fur abelsche Varietdten uberZahlkorpern, Invent. Math. 73, 349-366 (1983).

[F2] , Diophantine Approximations on Abelian Varieties,Preprint (1989),

[G] Grauert, H. Jetmetriken undhyperbolische Geometric, Math.Zeitschr. 200, 149-168 (1989).

[Ge] Gelfond, Transcendental and Algebraic Numbers, DoverPublications, Inc. (1960).

[G-G] Goldberg, A. A. and V. A. Grinshtein, The LogarithmicDerivative of a Meromorphic Function, Math. Notes 191976, AMS translation 320-323.

[G-K] Griffiths, Ph. and J. King, Nevanlinna theory and holomor-phic mappings between algebraic varieties, Acta Math. 130(1973), 145-220.

[G-U] Grauert, H. and U. Peternell, Hyperbolicity of the comple-ment of plane curves. Manuscripta Math. 50 (1985), 429-441.

[Gnl] Green, M., The complement of the dual of a plane curve andsome new hyperbolic manifolds. Value Distribution Theory,


Part A edited by Kujala and Vitter, Marcel Dekker, NewYork (1974), 119-131.

[GN2] , Some examples and counter-examples in value dis-tribution theory in several variables. Compositio Math.

[Gn3] , Some Picard theorems for holomorphic maps toalegbraic varieties. Amer. J. Math. 97 (1975), 43-75.

[Gn4] , Holomorphic maps into complex projective spaceomitting hyperplanes, Trans. Am. Math. Soc. 169, 89-103(1972).

[G-W] Green R. and H. Wu, Function Theory on Manifolds WhichPossess a Pole, Springer Lecture Notes 669 (1979).

[Kl] Kobayashi, S. Hyperbolic Manifolds and Holomorphic Map-pings, Marcel Dekker, New York, 1970.

[K2] , Intrinsic distances, measures and geometric func-tion theory, Bull. Amer. Math. Soc., 82 (1976), 357-416.

[Kh] Khinchin, Continued Fractions, The University of ChicagoPress, 1964.

[K-K1] Kieraan P. and S. Kobayashi, Satake Compactification andExtension of Holomorphic Mappings, Invent. Math. 16(1972), 237-248.

[K-K2] , Comments on Satake Compactification and theGreat Picard Theorem, J. Math. Soc. Japan 28 (1976), 577-580.

[K-W] Kreuzman, M. J. and P. M. Wong, Hyperbolicity of nega-tively curved Kdhler manifolds, Math. Annalen. 287 (1990),47-62.

[LI] Lang. S. Report on diophantine approximations, Bull. Soc.Math. France 93 (1965), 177-192.

[L2] , Fundamentals of Diophantine Geometry, Springer-Verlag, New York (1983).

[L3] , Hyperbolic and Diophantine Analysis, Bull. AMS,14, 159-205 (1986).

[L4] , The error terms in Nevanlinna Theory, Duke Math.J. 56. (1988), 193-218.

[Mai] Mahler, K. On a Theorem ofLiouville in Fields of PositiveCharacteristic, Canadian J. Math. 1, 1949.

Pit-Mann Wong 153

[Ma2] , Lectures on Diophantine Approximations, Univer-sity of Notre Dame Press (1961).

[M] Milnor, J., On deciding whether a surface is parabolic orhyperbolic, Amer. Math. Monthly, 84 (1977), 43-46.

[Nel] Nevanlinna, R., Einige Eindeutigkeitssatze in de Theorie dermeromorphen Funktionen, Acta Math. 48 (1926), 367-391.

[Ne2] , Le Theoreme de Picard-Borel et la Theorie desFonctions Meromorphes, Chelesa Publ. Co. New York(1974).

[Ne3] , Eindeutige analytische Functioned 2nd ed. DieGrundle. Math. Wiss. 46 (1953), Springer-Verlag.

[No] Nochka, E. I., On the theory of meromorphic functions, So-viet Math. Dokl,. 27 (2) (1983).

[N-T] Nadel, A. and H. Tsuji, Compactification of complete Kdhlermanifolds of negative Ricci curvature, J. Diff. Geom. 28(1988), 503-512.

[Ol] Osgood, C. F., A number theoretic-differential equationsapproach to generalizing Nevanlinna theory, Indiana J. ofMath. 23, 1-15 (1981).

[O2] , Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds, or better, J. Number Theory 21, 347-389 (1985).

[Oc] Ochiai, T., On holomorphic curves in algebraic varietieswith ample irregularity, Invent. Math. 43, 83-96 (1977).

[OS] O'Shea, A. Notre Dame Thesis, 1991.[Ri] Ridout, The p-adic generalization of the Thue-Siegel-Roth

theorem, Mathematika 5, 1958.[Ro] Roth, K. F., Rational approximations to algebraic numbers,

Mathematika 2, 1-20 (1955).[Roy] Royden, H. Remarks on the Kobayashi metric, Proc. Mary-

land Conference on Several Complex Variables, SpringerLecture Notes 183 (1971).

[R-S1] Ru, M. and W. Stall. Courbes holomorphes evitant des hy-perplans mobiles. C. R. Acad. Sci. Paris t. 310 Serie I.45^8 (1990).

[R-S2] , The Cartan Conjecture for Moving Targets.Proc. Symp. Pure Math. 52 (1991) 477-507.


[R-W] Ru, M. and P.M. Wong, Integral Points of !*"(£) - {In +1hyperplanes in general position}, Invent. Math. 106 (1991),195-216.

[Sa] Satake, I., On compactifications of the quotient spaces forarithmetically defined discontinuous groups, Ann. of Math.72 (1960), 555-580.

[Schl] Schlickewei, H. P., The p-adic Thue-Siegel-Roth-Schmidttheorem, Archiv der Math. 29, 267-270 (1977).

[Schml] Schmidt, W. M., Simultaneous Approximation to Algebraicnumbers by Elements of a Number Field, Monatshefte furMath. 79, 55-66 (1975).

[Schm2] , Diophantine Approximation, Springer LectureNotes 785, Springer-Verlag, New York (1980).

[Si] Siegel, C. L., Approximation algebraischer Zahlen, Math.Zeitschr. 10, 173-213 (1921).

[Sil] Silverman, J. H., The Arithmetic of Elliptic Curves, Grad.Texts in Math. 106, Springer-Verlag (1986).

[Siu] Siu, Y.T., Defect relation for holomorphic maps betweenspaces of different dimensions, Duke Math. J. 55 (1987),213-251.

[Shi] Shiffman, B., Introduction to Carlson-Griffiths equidistri-bution theory, Springer Lecture Notes in Math. 981 (1983),44-89.

[Sh2] , A general second main theorem for meromorphicfunctions on C1, Amer. J. of Math. 106 (1984), 509-531.

[S-T] Siu, Y.T. and S.-T. Yau, Compactification of negativelycurved complete Kahler manifolds of finite volume. Seminaron Diff. Geom. edited by S. T. Yau, Ann. of Math Studies102 (1982), 363-380.

[Stm] Steinmetz, N., Eine Verallgemeinerung des zweiten Nevan-linnaschen Hauptsatz.es, J. Reine Angew. Math. 368 (1986),134-141.

[Stol] Stoll, W. Die beiden Hauptsatze der Wertverteilungstheoriebei Funktionen mehrerer komplexen Verlanderlichen I, ActaMath. 90 (1953), 1-115; II, Acta Math. 92 (1954), 55-169.

[Sto2] , Value distribution on parabolic spaces, SpringerLecture Notes in Math. 600 (1977).

Pit-Mann Wong 155

[Sto3] , Deficit and Bezout Estimates, Valuation Distribu-tion Theory, Part B, edited by R. O. Kujala and A. VitterIII, Marcel Dekker (1973).

[Sto4] , Introduction to value distribution theory ofmero-morphic maps, Springer Lecture Notes in Math. 950 (1982),210-359.

[StoS] , The Ahlfors-Weyl theory of meromorphic maps onparabolic manifolds, Springer Lecture Notes in Math. 981(1983), 101-219.

[S-W] Stoll, W. and P. M. Wong, Second Main Theorem ofNevan-linna Theory for Non-equidimensional Meromorphic Maps,preprint 1991.

[T] Thue, A., Uber Annaherungswerte algebraischer Zahlen, J.reine ang. Math. 135, 284-305 (1909).

[U] Uchiyama, Rational Approximations to Algebraic Functions,J. Fac. Sci. Hokkaido Univ. 15, 1961 p 173-192).

[VI] Vojta, P. Diophantine Approximations and Value Distribu-tion Theory, Springer Lecture Notes 1239, Springer-Verlag(1987).

[V2] , Mordell's conjecture over function fields, Invent.Math. 98, 115-138 (1989).

[V3] , Siegers Theorem in the compact case, preprint(1989).

[V4] , A Refinement of Schmidt's Subspace Theorem,Amer. J. Math. Ill, 489-518 (1989).

[Vi] Vitter, A., The lemma of the logarithmic derivative in severalcomplex variables, Duke Math. J., 44 (1977), 89-104.

[Wl] Wong, P. M., Defect Relations for Maps on ParabolicSpaces and Kobayashi Metric on Projective Spaces Omit-ting Hyperplanes, University of Notre Dame Thesis (1976).

[W2] , On The Second Main Theorem ofNevanlinna The-ory, Amer. J. Math. Ill, 549-583 (1989).

[W3] , On Holomorphic Curves in Spaces of ConstantHolomorphic Sectional Curvature, preprint 1990, to appearin Proc. Conf. in Complex Geometry, Osaka, Japan (1991).


[W4] , Holomorphic mappings into Abelian Varieties,Amer. J. Math. 102, 493-501 (1980).

[Wu] Wu, H., The equidistribution theory of holomorphic curves,Ann. of Math Studies 64 (1970), Princeton University Press.

[W-W] Weyl, H. and J. Weyl, Meromorphic functions and analyticcurves, Ann. of Math. Studies #12, Princeton UniversityPress (1943).

[Y] P. Yang, Curvature of complex submanifolds ofC71, J. Diif.Geom. 12(1977), 499-511.

[Ye] Ye, Z., On Nevanlinna's Erroe Terms, preprint 1990.[Z] Zaidenberg, M. G., Stability of Hyperbolic Imbeddedness

and Construction of Examples, Math. USSR Sbornik 62(1989), 331-361.

Some Recent Results and Problems inthe Theory of Value-Distribution

Lo Yang

Dedicated to Professor Wilhelm Stoll on the occasion of hisinauguration as the Duncan Professor of Mathematics.

For meromorphic functions of one complex variable, the theoryof value-distribution has tremendously developed already since thetwenties of this century. Although it has a long history, there are stillsome interesting and remarkable results during the recent years. For in-stance, Drasin [6] proved that the F. Nevanlinna conjecture is correct;Lewis and Wu [13] made a significant step in proving the Arakelyan'sconjecture [1]; Osgood [18] and Steinmetz [20] independently provedthe defect relation for small functions, and so on.

In this lecture, I would like to mention some recent results andproblems which are based on my own interests.

1. Precise estimate of total deficiency of meromorphicfunctions and their derivatives

Let f ( z ) be a transcendental meromorphic function in the fi-nite plane and a be a complex value (finite or infinite). According toR. Nevanlinna

ra (r, TTT-

It is clear that 0 < £(a, /) < 1. If 5(a, /) is positive, then ais named a deficient value with respect to f ( z ) and <5(a, /) is itsdeficiency. The most fundamental result of Nevanlinna theory can bestated as follows [12, 17, 23].

157

158 Some Recent Results and Problems

Any transcendental meromorphic function f ( z ) in the finite planehas countable deficient values at most and the total deficiency doesnot exceed 2. i.e.

£%,/)< 2.aeC

It is the famous Defect Relation (or Deficient Relation). In thegeneral case, the upper bound 2 is sharp. If the order A or the lowerorder p, of f ( z ) is assigned, then the following deficient problem canbe introduced. (Edrei [8])

Problem 1. Let T^, be the set of all the meromorphic functionsof finite lower order p,. Can we determine

= sup

Do extremal functions exist? If so, what other properties characterizeextremal functions?

When /^ is less than 1, Edrei [8] obtained a precise estimate forthe total deficiency by using the spread relation proved by Baernstein[2]. In fact, he proved

2-aiii AWT, | < A i < l .

The Problem 1, however, is still open for meromorphic functions oflower order bigger than 1, although a suitable bound has been sug-gested by Drasin and Weitsman [7] as follows:

where

2sinf(2M-

and

Lo Yang 159

A2(/i) = 2 -2cos|(2/i-[2//])

[2/i] +

Now we consider the derivative f^ (z) of order k, where k is apositive integer. Hayman [11] pointed out that

£%,/<*>) <£±f.In 1971, Mues [15] improved this result to

fc* + 5fc + 4

Recently I proved [26]

Theorem 1. Let f ( z ) be a transcendental meromorphic functionin the finite plane and k be a positive integer. Then we have

It is clear that for any positive integer fc, we always have

2fc + 2 fc2 + 5/c + 4 fc + 22/c + l < fc2 + 4fe + 2 "^ fc + 1

and

fc2 + 5fc + 4 2fc + 2 fc + 2 fc2

fe2 + 4fc + 2 2fc + 1 k + 1 *2 + 4fc + 2"

Although Theorem 1 gives a much better estimate for^2 8(o>i f ^ ) > it does not include fl(oo, /^). For this reason, we haveaGCanother estimate [26].

Theorem 2. Let /(#) be a transcendental meromorphic functionof finite order in the finite plane and k be a positive integer. Then wehave


2fc(l-e(oo,/))2 - - 6(00, /))'

where 6(00, /) is the ramification index of oo with respect to /,defined by

6(00, /) = l-lim sup ^1r-K» T(r, /)

In particular, if 6(00, /) < 1, then we have

k—*oo

It is natural to discuss the precise estimate of total deficiency ofboth the function itself and its derivative. For this subject, Drasin [5]posed the following questions.

Problem 2. Let f(z) be meromorphic and of finite order in thefinite plane. If X) ̂ (a? /) = 2 and £(oo, /) = 0, do we must have

Problem 3. Let f(z) be meromorphic in the finite plane with, /) = 0. Can we have

aec bee

If not, what is best bound?Quite recently, I proved the following theorem.

Theorem 3. Let f(z) be a transcendental meromorphic functionof finite order in the finite plane and k be a positive integer. Then wehave

Lo Yang 161

3.

The equality holds if and only if either

(i) e(oo,/) = l,aec

or

( i i )fc = l, 6(cx),/) = 0J

Theorem 3 gives a positive answer to the Problem 2 by compar-ing Theorem 3 and the known fact

aec

Theorem 3 answers also the Problem 3, when / is of finite order.

2. Conjectures of Frank, Goldberg and Mues

Mues [15] posed the following conjecture, when he improvedthe Hayman's estimate.

Problem 4. Let f(z) be a transcendental meromorphic functionin the finite plane and k be a positive integer. Then the followingrelation should be true.

£«(«, /<«)<!.aec

If f(z) satisfies one of the following conditions, then the Muesconjecture can be easily verified.

(i) The order of / is finite and 6(00, /) < 1 - |;(ii) / has only poles with multiplicity < fc;(iii) The order of / is finite and

, / )> 2.aec


It seems that the Mues conjecture is true, when / has finite order.Connecting the Problem 4, G. Frank and A. Goldberg recently

raised two similar conjectures respectively.

Problem 5. Let f(z) be a transcendental meromorphic functionin the finite plane and k be a positive integer. If e is an arbitrary smallpositive number, then the following estimate seems true.

kN(r, /)<(! + e)N(r, j^) + S(r, /),

where

S(r>/) = 0{log(rT(rl/))}l

except for r in a set with finite linear measure.

Problem 6. Let f ( z ) be a transcendental meromorphic functionin the finite plane. Then the following inequality should be correct.

When k > 2, the Frank conjecture (Problem 5) is much strongerthan the Goldberg conjecture (Problem 6). The Mues conjecture is adirect consequence of any one of them, when the order of f(z) isfinite.

3. Best coefficients of Hayman inequality

In 1959, Hayman [1 1] obtained a very interesting and remarkableinequality in which the characterstic function T(r, /) can be bonndedby two counting functions. It is impossible without introducing thederivatives.

T(r, /) < (2 + l)

(3.1)

Lo Yang 163

Later on, Hayman [12] adopted this inequality as the principalresult of Chapter 3 in his book. He also posed the following question.

Problem 7. What are the best coefficients of the inequality (3.1).Concerning this problem, the following inequality was obtained

by the author [25] about two years ago.

Theorem 4. Let f ( z ) be a transcendental meromorphic functionin the finite plane and k be a positive integer. If e is an arbitrary smallpositive number, then we have

The proof of Theorem 4 is based on the following lemma.

Lemma. Under the conditions of Theorem 4, we have

(r, / ) < ( + e)N(r, ) + ( + ^(r, -- + flf(r, /).

If the Goldberg conjecture is true, then we have

T(r, /) < N(r, ) + N(r, —) + S(r, /). (3.2)

If the Frank conjecture is true, then we also have the inequality(3.2) in the case of k > 2 and

T(r, /)<(! + e)N(r, j) + (l+ e}N(r, ^-) + S(r, /)

in the ease of k = 1.

4. Normal families and fix-points of meromorphic functions

A theorem which makes the connection between the normalityof a given family of meromorphic functions a.nd the lack of fix-pointsof both these functions and their derivatives, was proved by the author[24] in 1986.


Theorem 5. Let T be a family of meromorphic functions in aregion D and k be a positive integer. If, for every function f(z) of F,both f(z) and f^k\z) (the derivative of order fc) have no fix-pointsin JD, then .T7 is normal there.

Since the iteration of f ( z ) is very important and grows muchfaster than f ( z ) and /(fe) (z), it is natural to pose the following problem.

Problem 8. Let F be a family of entire functions, D be a regionand k be a fixed positive integer. If, for every function f(z) ofF, bothf ( z ) and fk(z) (the iteration of order k of /(z)) have no fix-pointsin jD, is .F normal there?

Schwick [19] proved several criteria for normality. Among oth-ers, he proved

Theorem 6. Let F be a family of meromorphic functions in aregion JD and n and fc be two positive integers. If for every function

(/»)(*) ^ i

and

n > f c + 3, (4.1)

then F is noinial in £>. Moreover, if f is a family of holomorphicfunctions, then the condition (4.1) can be replaced as

n > f c + l. (4.2)

It seems to me that the following assertion should be true.

Problem 9. Let f be a family of meromorphic functions in aregion D and n and k be two positive integers with (4.1). If, for everyfunction j(z) of T, (fn)^ has no fix-points, then F is normal in D.When F is a fajnily of holomorphic functions, then (4.1) can also bereplaced by (4,2).

Lo Yang 165

5. Common Borel directions of ameromorphic function and its derivatives

Let f ( z ) be meromorphic and of order A in the finite plane,where 0 < A < oo. Valiron [21] proved there exists a direction B :arg z = #o(0 < 0o < 2?r) such that, for any positive number e andany complex value a, we always have

Um logn(r,00,£,/ = a)

r->oo log T

except two values of a at most, where n(r, 0o> e? / = &) denotes thenumber of zeros of / - a in the region (\z\ < r) D (| arg z — OQ\ < e).Such direction is named a Borel direction of order A of f ( z ) .

Since /(*) (z) is also meromorphic and of order A in the finiteplane, it has a Borel direction too. Valiron [21] asked the followingquestion.

Problem 10. Let f ( z ) be meromorphic and of order A in thefinite plane, where 0 < A < oo. Is there a common Borel direction off ( z ) and all its derivatives?

Concerning this problem, the known result is

Theorem 7. Let f ( z ) be meromorphic and of order A in thefinite plane, where 0 < A < oo. If f ( z ) has a Borel exceptional value(which is either a finite complex value or the infinity), then there existsa common Borel direction of f ( z ) and all its derivatives.

The papers concerning Theorem 7 are due to Milloux, ZhangK. H., Zhang Q. D. and myself [14, 23, 27].

On the other hand, the following fact is also knorwn.There exists a meromorphic function fo(z) of order one in the

finite plane such that its derivative /Q(Z) has more Borel directionsthan fa(z).

For instance, the function


which was pointed out by Steinmetz, is such an example. fo(z) hasthe Borel directions arg z = |, TT and ̂ , but /o(^) has these and inaddition the direction arg z = |.

6. Optimum condition to ensurethe existence of Hayman direction

Corresponding to the Hayman's inequality, we may ask if thereare some similar results in the angular distribution. My followingtheorem [22] aims at answer of this question.

Theorem 8. Let f ( z ) be a meromorphic function in the finiteplane. If

TV«. f\

DO, (6.1),(log r)3

then there exists a direction (H) : arg z = 0o such that, for anypositive number £, an arbitrary integer k and any two finite complexvalues a and b(b ̂ 0), we have

lim {n(r, 00, ej = a) + n(r, 00, e, / = &)} = oo.r— >oo

It is convenient to name such kind of direction as Hayman di-rection.

For a meromorphic function, since the condition ensuring a Juliadirection is

<6-2)

Drasin [3] raised the following question in 1984.

Problem 11. Is Theorem 8 still true, if the condition (6.1) isreplaced by (6.2)?

It seems that the answer of Problem 1 1 is positive, since ChenH. H. [4] proved recently the following fact.

Theorem 9. Let f ( z ) be a meromorphic function in the finiteplane. If the condition (6.2) is satisfied, then there exists a direction

Lo Yang 167

(H) : arg z = OQ such that for any positive number e, an arbitrarypositive integer k and any two finite complex values a and b(b ̂ 0),we have

linxsup . . .r->oo log T

7. Picard type theorems and theexistence of singular directions

Bloch principle says a family of holomorphic (or meromorphic)functions in a region satisfying a condition (or a set of conditions)uniformly which can only be possessed by the constant functions inthe finite plane, must be normal there. In simple words, there usually isa criterion for normality to correspond a Picard type theorem. Similarlywe have the following question.

Problem 12. Corresponding to every Picard type theorem, isthere a singular direction? To be precise, let P be a property (or a setof properties) such that any entire function (or a meromorphic functionin the finite plane) satisfying P, must be a constant. Then for anytranscendental entire function (or a meromorphic function with somesuitable condition of growth), is there a ray arg z = 0o(0 < 90 < 2?r)such that f ( z ) does not satisfy P in the angle | arg z — 0$\ < e, forany small positive number e.

The direction in the Problem 12 is a direction of Julia type. Wecan also pose a problem for a direction of Borel type.

For instance, the following fact is well known.Let f ( z ) be meromorphic in the finite plane. If (fn)^ ^ 1 for

two positive integers n and k with n > fc+3, then f ( z ) must reduce toa constant. When f ( z ) is entire, the condition n > k + 1 is sufficient.

Problem 13. Let f ( z ) be meromorphic and of order A in thefinite plane, where 0 < A < oo. Is there a direction arg z = 00(0 <#o < 2?r) such that for any positive number e two arbitrary positiveintegers n and k with n > k + 3 and any finite, non-zero complexvalue a, we have


tog n(r. ft. e. (/•)<*) = a) =A?

r-K» log r

When f ( z ) is entire, the condition n > k + 1 seems sufficient.

8. Growth, number of deficientvalues and Picard type theorem

Picard type theorems are not only involved in criteria for nor-mality and singular directions, but also connect with the growth andnumber of deficient values.

Problem 14. Let P be a property (or a set of properties) givenby the Problem 12. Suppose that f ( z ) is entire (or meromorphic) andof finite lower order ^ in the finite plane and that Lj : arg z = 9j(j =1, 2, • • •, J; 0 < 0i < 0% < • - - < 0j < 2?r) are finite number of raysissued from the origin. If f(z) satisfies P in C\(U^=1I/j) then theorder A of,f(z) seems to have the estimate

and the number of finite non-zero deficient values does not exceed J.The property P can be chosen asf\\ T ( "y\ —r- C\ QTl/^ T\ /I y 1 —r- 1 ( If (~ fi I *I Ay I \/&} -f— \J dllvi. / \Ai] -f— J. I n/ t " 15

(ii) f ( z ) — af(z)n 7^ 6, where a and 6 are two finite complexvalues with a ^ 0 and n > 5 is a positive integer;

(iii) (f(z)nyk) ^ 1, where n and fc are two positive integerswith n > k + 3, when / is meromorphic and n > k + 1, when / isentire.

9. Value-distribution with respect to small functions

Let f ( z ) be meromorphic in the finite plane and a be a complexvalue. The theory of value-distribntion investigates the distribution ofzeros of f ( z ) — a. It is natural to instead of the complex value a byanother meromorphic function a(z) with the condition

(9.1)

Lo Yang 169

Therefore, corresponding to results in the theory of value-dis-tribution, we can ask similar questions with respect to small func-tions. Some of them are very difficult and significant. For instance,R. Nevanlinna himself asked if his defect relation can be extended tosmall functions. Up to few years ago, Osgood [18] and Steinmetz [20]independently settled this problem with a positive answer.

Theorem 10. Let f ( z ) be a transcendental meromorphic functionin the finite plane and A be the set of all meromorphic functions a(z)with the condition (9.1). Then we have

All the complex values including the infinity are contained in A.There are still some problem. We mention only one of them here.Recently, Lewis and Wu [13] proved

Theorem 11. If f ( z ) is an entire function of finite lower order,then there exists a positive number TO not depending on / such thatthe series ^P(<$(a, /))5~r is convergent for any r with 0 < r < TQ.

Problem 15. Is Lewis and Wu's result still true for small func-tions?

Acknowledgement. The author is very grateful to the Depart-ment of Mathematics, University of Notre Dame, Prof. W. Stoll andProf. P. M. Wong for their hospitality.

REFERENCES

[1] Arakelyan, N. U., Entire function of finite order with a setof infinite deficient values (in Russian). Dokal. USSR, 170(1966), 999-1002.

[2] Baernstein, A., Proof of Edrei's spread conjecture, Proc.London Math. Soc., 26 (1973), 418-434.


[3] Earth, K. R, Brannan, D. A. and Hayman, W. K., ResearchProblems in Complex Analysis, Bull. London Math. Soc., 16(1984), 490-517.

[4] Chen, H. H., Singular directions corresponding to Hayman'sinequality (Chinese), Adv. in Math. (Beijing), 16 (1987),73-80.

[5] Drasin, D., An introduction to potential theory and mero-morphic functions, Complex analysis and its applications,Vol. 1, IAEA, Vienna, 1976, 1-93.

[6] Drasin, D., Proof of a conjecture ofF. Nevanlinna concern-ing functions which have deficiency sum two, Acta Math.,158 (1987), 1-94.

[7] Drasin, D. and Weitsman, A., Meromorphic functions withlarge sums of deficiencies, Adv. in Math., 15 (1974), 93-126.

[8] Edrei, A., Solution of the deficiency problem for functionsof small lower order, Proc. London Math. Soc., 26 (1973),435-445.

[9] Edrei, A. and Fuchs, W. H. J., On meromorphic functionswith regions free of poles and zeros, Acta Math., 108 (1962),113-145.

[10] Frank, G. and Weissenborn, G., Rational deficient functionsof meromorphic functions, Bull. London Math. Soc., 18(1986), 29-33.

[11] Hayman, W. K., Picard values of meromorphic functionsand their derivatives, Ann. of Math., 70 (1959), 9-42.

[12] Hayman, W. K., Meromorphic functions, Oxford, 1964.[13] Lewis, J. L. and Wu, J. M., On conjectures of Arakelyan

and Littlewood, J. d'Analyse Math., 50 (1988), 259-283.[14] Milloux, H., Sur les directions de Borel desfonctions entiere,

de leurs derivees et de leurs integrales, J. d'Analyse Math.,1 (1951), 244-330.

[15] Mues, E., Uber eine Defekt und Verzweigungsrelationfiir dieAbleitung Meromorpher Funktionen, Manuscripta Math., 5(1971), 275-297.

[16] Nevanlinna, R., Le Theoreme de Picard-Borel et la theoriedesfonctions meromorphes, Coll Borel, 1929.

Lo Yang 171

[17] Nevanlinna, R., Analytic functions, Springer-Verlag, Berlin,1970.

[18] Osgood, C. F., Sometimes effective Thue-Siegel-Schmidt-Ne-vanlinna bounds or better, J. Number theory, 21 (1985),347-389.

[19] Schwick, W., Normality Criteria for families of meromor-phic functions, J. D'Analyse Math., 52 (1989), 241-289.

[20] Steinmetz, N., Eine Verallgemeinerung des zweiten Nevan-linnaschen Hauptsatzes, J. fur Math., 368 (1986), 134-141.

[21] Valiron, G., Recherches sur le theoreme de M. Borel dans latheorie desfonctions meromorphes, Acta Math., 52 (1928),67-92.

[22] Yang, Lo, Meromorphic functions and their derivatives, J.London Math. Soc., (2) 25 (1982), 288-296.

[23] Yang, Lo, Theory of value-distribution and its new research(in Chinese), Science Press, Beijing, 1982.

[24] Yang, Lo, Normal families and fix-points of meromorphicfunctions, Indiana Univ. Math. J., 35 (1986), 179-191.

[25] Yang, Lo, Precise fundamental inequalities and sum of de-ficiencies, Sci. Sinica, 34 (1991), 157-165.

[26] Yang, Lo, Precise estimate of total deficiency of meromor-phic derivatives, J. d'Analyse Math., 55 (1990), 287-296.

[27] Yang, Lo and Zhang Qingde, New singular directions ofmeromorphic functions, Sci. Sinica, 27 (1984), 352-366.

Documents

Proceedings Symposium on Value Distribution Theory in Several Complex Variables