BOTT-CHERN CURRENTS AND COMPLEX IMMERSIONSbismut/Bismut/1990.pdf · Vol. 60,No. 1 DUKEMATHEMATICALJOURNAL(C) February 1990 BOTT-CHERNCURRENTSAND COMPLEXIMMERSIONS J.-M. BISMUT, H

Vol. 60, No. 1 DUKE MATHEMATICAL JOURNAL (C) February 1990

BOTT-CHERN CURRENTS ANDCOMPLEX IMMERSIONS

J.-M. BISMUT, H. GILLET, aND C. SOULI

CONTENTS

0. Introduction 255

I. Superconneetion currents and resolutions of vector bundles 257a) A holomorphic chain complex 257b) Assumption (A) on the Hermitian metrics of a chain complex 258c) Wave front sets 259d) Quillen’s superconnections 259e) Double transgression formulas 260f) Convergence of superconnection currents 261

II. A Bott-Chern current 264a) A generalized BottoChern current 264b) The pull-back of the current T(h) by a transversal map 268c) Integration along the fiber of the current T(h) 270

III. The Bott-Chern current as a finite part 271a) The singularity of the Bott-Chern current 271b) The current T(h) as a principal part 277

References 283

0. Introduction. In the whole paper, we will say that i" M’ M is an immersionof smooth manifolds if M’ is a submanifold of M, and if is the correspondinginjection map. In particular the topology of M’ is the topology induced by thetopology of M. In differential geometry, such maps i: M’ M are also calledembeddings.

Let i: M’ M be an immersion of complex manifolds, let r/be a holomorphicvector bundle on M’, let (, v) be a holomorphic complex of vector bundles on Mwhich provides a resolution of the sheaf i. (9M,(/).We assume that the vector bundle r/, the normal bundle N to M’ in M and the

complex are equipped with Hermitian metrics g", gN, h. The purpose of this paperis to construct a current T(h) on M which has three essential properties:

Received January 31, 1989. Revision received September 18, 1989.

255

256 BISMUT, GILLET AND SOUL]

Under certain compatibility assumptions between the various metrics, then thefollowing equation of currents holds

(0.1)2in

T(h) rd-1 (gN) ch(ff,)tM, ch(h)

In (0.1), the forms in the right hand side are obtained in Chern-Weil theory by usingthe holomorphic Hermitian connections associated with the given metrics. Also giu,denotes the current on M given by integration on M’.

The wave front set of the current T(h) is included in the real conormal bundleN to M’ inM.

The current T(h) pulls back naturally by holomorphic maps which are trans-versal to M’.

In the case where M’ if, i.e., if the complex (, v) is acyclic, currents similar toT(h) were already considered by Bott-Chern [BoC-I and Donaldson [D]. In thiscase, the form T(h) was obtained in [BGSl] by using Quillen’s superconnectionsI-QI] and the number operator of the complex (, v), and was called a Bott-Chernform.To construct our current T(h) in full generality, we essentially use the results of

Bismut [B2-1. In [B2-1, it was proved that the Quillen superconnection Cherncharacter forms cou (which here depend on a parameter u which scales the chain mapv) converge as u---} +oo to a current cooo which is explicitly determined. Also thespeed of convergence of c% towards 0900 was studied in [B2-1, and certain keymicrolocal estimates were established. Our construction of T(h) then combinesthe double transgression formulas of [BGSl] for superconnections with the resultsof [B2-1. The current T(h) is here called a Bott-Chern current.The current T(h) is smooth on M\M’, and in general is not locally integrable.

We study the singularity of T(h) near M’, and we also calculate T(h) as a principalpart (in the sense of distributions) of its restriction to M\M’.

In the whole paper, we assume the considered manifolds to be compact. Most ofour results extend to the non compact case in the obvious way.Our main motivation for introducing the current T(h) is related to our previous

work on direct images of Hermitian vector bundles. In fact, in our earlier worki-BGSI, 2, 3], we calculated the curvature of the holomorphic Hermitian connectionon the determinant of a direct image equipped with the Quillen metric I-Q2],and we obtained a differential form version of the Theorem of Riemann-Roch-Grothendieck. In our situation, the determinant lines 2(r/) and 2() (which are thedeterminants of the cohomology of r/and ) are canonically isomorphic. In view ofthe results of [BGSI, 2, 3-1 and of (0.1), one may suspect that the current T(h) willplay a role in comparing the metrics on 2(/) and 2().

In our forthcoming work [BGSS-I, we give a direct verification that the currentsT(h) verify certain functorial properties, which make them natural candidates toappear in a formula comparing the Quillen metrics of 2(r/) and 2(). The wave frontset properties of T(h) play a key role in [BGSS-I, essentially because we can form

BOTT-CHERN CURRENTS AND COMPLEX IMMERSIONS 257

the product of such currents associated with submanifolds M’ and M" intersectingtransversally. Also in [BGSS], we relate our current T(h) to the arithmetic char-acteristic classes of Gillet and Soul6 [GS].Our paper is organised as follows. In Section 1, we recall the main results of [B2].

In Section 2, we construct our current T(h) and we prove it pulls back naturallyunder maps transversal to M’. In Section 3, we describe the singularity of T(h) nearM’ and we calculate T(h) as a principal part.The results contained in this paper were announced in [BGS4].

Acknowledgements. The authors are indebted to the referee of this paper for hiscomments and suggestions.

I. Superconnection currents and resolutions of vector bundles. Let i: M’ M bean immersion of compact complex manifolds, let r/be a holomorphic Hermitianvector bundle on M’, and let (, v) be a holomorphic complex of Hermitian vectorbundles on M which provides a resolution of the sheaf i,6%,(r/).

In this section, we recall the results of Bismut IB2-] on the large parameterbehavior of certain Quillen’s superconnection forms associated with (, v). Inparticular, the microlocal estimates in [B2, Section 3] will play a key role inestablishing the main properties of the Bott-Chern current, which we construct inSection 2.

This section is organized as follows. In a), we describe the main properties of thecomplex (, v). In b), we recall the assumption (A) of [B2] which is a compatibilitycondition between metrics on the complex (, v) and metrics on the normal bundleN to M’ in M and on the vector bundle r/. In c), we introduce the wave front sets ofcurrents on M. In d), we give some properties of Quillen’s superconnections [QI].In e), we recall the double transgression formulas of Bismut-Gillet-Soul6 [BGSI]for superconnection forms. Finally in f), we describe the results established inBismut [B2].

a) A holomorphic chain complex. Let M be a compact connected complexmanifold of complex dimension ’. Let M’ be a finite disjoint union of compactconnected complex submanifolds of M’. Let be the immersion M’ M.

Let N be the complex normal vector bundle to M’ in M, and let N* be its dual.Let

(1.1)

be a chain complex of holomorphic vector bundles on M.Let r/be a holomorphic vector bundle on M’. We assume there is a holomorphic

restriction map r: 01M’ r/ which is such that we have an exact sequence ofsheaves

(1.2) o - CU(m) V (M(m-) "’V M(O) V i,(;M,(n) O.


In particular, the complex (, v) is acyclic on M\M’.For x e M’, 0 < k < m, let Fk,, be the k-th homology group of the complex (, v)x.

Set Fx @’ Fk. x.The following results are consequences of the local uniqueness of resolutions

(see Serre IS, IV, Appendix 1] and Eilenberg [E, Theorem 8]) and are proved in[B2, Section 1].

For k 0, m, x e M’, the dimension of Fk, is locally constant on each Mj,so that Fk is a holomorphie vector bundle on M’.

For x M’, U T,M, let Ovv(x) be the derivative of the chain map v calculatedin any given local holomorphic trivialization of(, v) near x. Then Ovv(x) acts on Fx.When acting on F, Ovv(x) only depends on the image y of U in N,. For x M’,y e N, we now write Ov(x) instead of OvV(X).

For any x M’, y Nx, (Oyv)2(x) 0. If y e N, let iy be the interior multipli-cation operator by y acting on the exterior algebra A(N*). i acts like i (R) 1 onAN* (R) r/. Then, the graded holomorphic complex (F, Ov), on the total space of thevector bundle N, is canonically isomorphic to the Koszul complex (AN* (R) r/, i).

b) Assumption (A) on the Hermitian metrics of a chain complex. We nowassume that o, m are equipped with smooth Hermitian metrics ho, h’-.We equip ()’=o k with the metric h which is the orthogonal sum ofthe metricshe, h’-. Let v* be the adjoint of v with respect to the metric h. Using finitedimensional Hodge theory, we get the identification of smooth vector bundles onM’ for 0 < k < m

(1.3) Fk {f e k Vf O; v*f O}.

As a smooth vector subbundle of Ck, the right hand side of (1.3) inherits aHermitian metric from the metric hCk. Using the identification (1.3), we find that forevery k 0,..., m, Fk is a holomorphic Hermitian vector bundle on M’. Let hdenote the Hermitian metric on Fk. We equip F )’ Fk with the metric hv whichis the orthogonal sum of the metrics hro, hr’-.

Let 0N, O" be Hermitian metrics on the vector bundles N, r/. We equip the vectorbundle AN* (R) r/with the tensor product of the metric induced by 0

N on A(N*) andof the metric

Definition 1.1. Given metrics gN, 9" on N, r/, we will say that the metricshe, h’- on o, m verify assumption (A) if the canonical identification ofholomorphic chain complexes on N

(1.4) (F, Oyv) (AN* (R) r/, i,)

also identifies the metrics.

The following result is proved in [B2, Proposition 1.6].


PROPOSITION 1.2. Given metrics 91, 97 on N, q, there exist metrics h*, h*m ono, m which verify assumption (A).

e) Wave front sets. If 7 is a current on M, we note WF(7) the wave front setof 7. For the definition and properties of wave front sets, we refer to H6rmander[H, Chapter VIII].

Let us just recall that WF() is a closed conic subset of TffM\(O}. Also if pis the projection T*M M, p(WF(7)) is exactly the singular support of ,,whose complement in M is the set of points x such that 7 is C on a neighborhoodofx.

Let N be the real conormal bundle to M’ in M. Let v be the set of currents 7on M which are such that WF(7) c N. In particular, currents in v are smoothon M\M’. By I-H, Definition 8.2.2], , has a natural topology which we nowdescribe.

Let U be a small open set in M, which we identify with an open ball in R2e.Over U, we identify TM with U x R2( Let tp be a smooth current on R2t withcompact support included in U. Let F be a closed conic subset of R2e such thatF cN b on M’ U, and let m be an integer. If 7 is a current, let ’() be theFourier transform of tp7 (which is here considered as a distribution on R2t). If7 v, set

(1.5) Pv, r,,,,(7) sup I11()1.

If 7, is a sequence of currents in v, we will say that 7, converge to 7 ev if

7 7 in the sense of distributions;If U, F, qg, m are taken as before

(1.6) Pv, r,,,,(7, 7) O.

Definition 1.3. P, denotes the vector space of currents 09 on M which havethe following two properties:

09 is a sum of currents of type (p, p);The wave front set WF(to) of o is included in N.

p,,o is the vector space of currents 09 P, which are such that there existcurrents , fl for which

(1.7) WF() N, WF(fl) N, 09 t3 + Off.

We equip P, with the topology induced by v(M). If M’= b, we will writepM, pM, o instead of Pffr, pt; o.

d) Quillen’s superconnections. We now assume that o,-.., m are equipped withHermitian metrics ho h’-. We otherwise use the notations of Section 1.a).

260 BISMUT, GILLET AND SOULI

Set

k k odd

Then +_

is a Z2-graded Hermitian vector bundle. End is naturallyZ2-graded, the even (resp. odd) elements in End commuting (resp. anticommuting)with the operator z + 1 on _+ which defines the Z2 grading.For 0 < k < m, let Vk be the holomorphic Hermitian connection on k" Then

V (’VCk is the holomorphic Hermitian connection on the vector bundle .We now briefly recall the definition of a superconnection in the sense of Quillen

[QI]. The bundle of algebras A(TM) End is naturally Z2-graded. Let S be asmooth section of (A(T*M) End )oad. Then, by definition, V + S is a super-connection on the Z2-graded vector bundle .

In the sequel, V will be considered as a first-order differential operator acting onthe set of smooth sections of A(Tg*M) ( . The curvature (V + S)2 of the super-connection V + S is then a smooth section of (A(T*M) ( End )even.

If A End , we define its supertrace Trs[A] C by

Trs[A] Tr[zA].

We extend Tr as a linear map from A(T*M) End into A(T*M), with theconvention that, if to A(TM), A e End

Tr,[toA] to Try[A].

If B, B’ A(TM) ( End , let [B, B’] be the supercommutator

[B, B’] BB’ (- 1)degBdegB’B’B.

Then by [QI], Tr vanishes on supercommutators.Let tp be the homomorphism from Aeven(Tg*M) into itself which to to A2p(Tg*M)

associates tp(to) (2in)-Pto.Let S be an odd smooth section of A(TM) End . The basic result of Quillen

[QI] asserts that the form qg(Trs[exp(-(V + S)2)]) is closed and represents incohomology the Chern character of o 1 + + (- 1)’m

e) Double transgression formulas.1.d). Let v* be the adjoint of v. Set

We make the same assumptions as in Section

(1.9) V=v+v*.

V is a smooth section of Enddo . For u > 0, let Au be the superconnection

(1.10) A,, V + /V.


Then A is the curvature of the superconnection Au. A is a smooth section of(A(T*M) ) End )eve.

Let Nn be the number operator of the complex . Namely for 0 < k < m, if

hk

Nlth kh.

We now recall a result of [BGSI-I.

THEOREM 1.4. The forms Tr[exp(-A,2)] and Tr[Nn exp(-A,2)] lie in pM anddepend smoothly on u > O. Moreover, for u > O, the followin# identities hold"

(1.11)

Tr[exp(-A2)] -d Tr[ VOu 2w/-

[ ] 1(-_ ) Tr,[Nn exp(-A.2)].TrsV

exp(-A,2)u

In particular,

c3Trs[exp( 2 1

A,)] O Trs[Nn exp( 2-A.)].(1 12) Ou u

Proof. These results are proved in [BGSI, Theorem 1.15]. Note the differenceof signs with respect to [BGSI] since here v decreases the degree in by one, whilein [BGSI], v increases the degree in by one. 121

f) Convergence ofsuperconnection currents.in Sections 1.a) and 1.d). Set

We make the same assumptions as

F+=(R)F,, F_=(R)F,.k even k odd

F F+ F_ is a Hermitian Z2-graded vector bundle. If y N, let fi be theconjugate element of y in N. Then y N represents Y y + Na. In particular,if N is equipped with a metric 9N, [YI2 2 lyl 2.The superconnection formalism of Quillen can also be applied to the Zz-graded

vector bundle F F+ F_. Let (dry)* be the adjoint of Ov with respect to themetric he on F. Then (dry)* is a antiholomorphic function of y. Set

Or V is an odd section of End F. If we use the canonical identification (1.4), then

tgr V r + iy*.


For 0 < k < m, let Vrk be the holomorphic Hermitian connection on the vectorbundle Fk. Then Vr 0)’=o Vrk is the holomorphic Hermitian connection on F.

Let B be the superconnection on F

(1.13) B Vr + cOr V.

The curvature B2 of B is a smooth section of A(T*N) ) End F. The operator NHacts naturally on F. NH is simply multiplication by k on Fk.By [B2, Proposition 3.1], we know that the form on N, Trs[Nn exp(-B2)]

decays as lYl--’ +o faster than exp(- ClYl2) (for one C > 0).6u, denotes the current of integration on the orientable manifold M’.Let C (M) be the set ofcontinuous differential forms onM which have continuous

first derivatives. Let IIcM) be a norm on CI(M) such that IlmllcxM) --’ 0 if andonly if/. tends to 0 uniformly on M together with its first derivatives.We now recall the result of Bismut announced in [BI] and proved in [B2,

Theorems 3.2, 4.1 and 4.3].

THEOREM 1.5. As u -, , we have the following converoence of currents on M

Trs[exp(- Az)] N Tr[exp(- Bz)]6M

(1.14) Tr[x/V exp( 2-A.)]0 in P,

Tr[Nn exp(-A)] fN Tr,[Nu exp(-B2)]bu,

There exists C > 0 such that if la is a smooth differential form on M, then foru>l

fM/2(Trs[exp(-a2u)] --[;NTrs[eXp(--B2)]]tM’} C

(1.15) Trs[/V exp( -A.)]

Ij/ {TraIN,, exp(-A2)]-[fTrs[Nnexp(-B2)]]bM"}If U, F, q, m are taken as in Section 1.c), there exists C > 0 such that for u > 1

( -Au)]-sTrs[exp(-B2)]6M’)<xPv, r,,,m Tr[exp( 2 C


Pv, r,,,,(Trs[x/V exp( 2-A.)]) <

(1.16) ( Au)]-fNTrs[Nnexp(-B2)]bu’)<Pv, r,,,m Trs[Nn exp(- 2

Observe that if the manifolds M and M’ are not assumed to be compact, there isan obvious analogue of Theorem 1.5. The forms # must be taken to have compactsupport, and the various constants C depend explicitly on the compact set containingthe support of the considered forms #.

Recall that the Todd polynomial is an ad-invariant polynomial on matriceswhich is such that if C is a diagonal matrix with diagonal entries x l, xp, then

(1.17) Td(C) I I 1-e-x’

Let (Td-1)’ be the add-invariant polynomial which is such that if C is taken asbefore, then

(1.18) x,+ b =o

If N, r/are equipped with metrics gN, g", we denote by V, V the correspondingholomorphic Hermitian connections, and by (Vs)2, (V")2 their curvatures.

THEOREM 1.6. The form Trs[Nnexp(-A20)] on M is closed. The formTr[Nn exp(-B2)] on M’ is closed. If the metrics ho, h’- verify assumption

(A) with respect to metrics 0, 9 on N, rl, then

(1.19)

Tr[exp(-BZ)] (2i)aim Td-I(-(V)2) Tr[exp(-(V)2)]

Tr[Nn exp(-B2)] -(2in)aim (Td-1)’(_(V)2) Tr[exp(-(V")2)].

Proof. Clearly

(1.20) Tr[Nn exp(-AoZ)] (- 1)k TrE(-(V)2)]o

and so the form (1.20) is closed on M. The form NTr[NH exp(-B2)] is closedby [B2, Theorem 4.3]. The first line of (1.19) is a result of Mathai-Quillen [MQ,Theorem 4.5]. The second line of (1.19) is proved in [B2, Theorem 4.3]. E!


Remark 1.7. Let M be a complex submanifold of M. Assume that M’ M c M’is a complex submanifold ofM and that TM’ TM’ c TM.

Let N be the normal bundle to M’ in M. Then N c N and N N if and only ifM and M’ are transversal. Also (, V)l provides a resolution of r/i, if and only ifMand M’ are transversal.An analogue of the results stated in Theorems 1.5 and 1.6 is proved in [B2,

Theorem 5.1] when replacing M by M. The explicit formulas (1.19) have to beadequately modified.

II. At Bott-Chern current. In this Section, we construct our Bott-Chern singularcurrent T(h) associated with the chain complex (, v) equipped with the metric h.Our construction of T(h) is a straightforward application of results in [BGSl] andin [B2]. We also study the behavior of the current T(h) by pull back, and byintegration along the fiber.

This Section is organized as follows. In a), we construct the current T(h), andwe show that its wave front set is included in the real conormal bundle to M’ in M.In b), we study the pull back of T(h) by a map f transversal to M’. Finally in c), weconsider the integral along the fiber of a submersion of our current T(h).Our assumptions and notations are the same as in Section 1.

a) A generalized Bott-Chern current. Let E be a holomorphic vector bundle ofdimension k on the manifold M, let 9

E be a Hermitian metric on E. Let VE bethe corresponding holomorphic Hermitian connection on E, and let fe (Vn)2 beits curvature. If Q is an ad-invariant polynomial on (k, k) matrices, we use thenotation

In the sequel, ch denotes the function A End(Ck) Tr[exp(A)].We fix once and for all one square root of2in which we note (2it01/2. The formulas

which we will write in the sequel do not depend on this choice.Let q9 be the homomorphism of A(TM) into itself which to A(TR*M) asso-

ciates (2i70-/2. also acts on currents in the same way. More generally, q9 will alsoact on forms or currents on the manifolds M’ and N.We now make the same assumptions as in Section 1.f), and we use the notations

of Section 1.

Definition 2.1. For s C, 0 < Re(s) < 1/2, let (s) be the current on M

1u’-1 Trs[Nn exp( 2Au)] Tr, I-Nn exp(- B2)] 6u, du

(2.1)

By the inequality (1.15) in Theorem 1.5, it is clear that the current ((s) iswell-defined. More precisely, if/ is a smooth form on M, one easily verifies that the


function of s C, 0 < Re(s) < 1/2, M #((s) extends into a meromorphic function ofs C, which is holomorphic at 0.

Definition 2.2. (0) denotes the current on M which is such that if # is anysmooth differential form on M, then

(2.2)

More explicitly, we have the obvious formula

(2.3) (0) f[ {Tr,[Nu(exp(-A2) exp(-- A2o))] }duU

du+ f+, {Tr,[Nnexp(-A2)] (Tr,[Nnexp(-B2)])a,}---F’(1) {Tr,[Nn exp(-Ao2)]-(tTr,[Nnexp(-B2)])6M,}.

By Theorem 1.6, the form Tr[Nn exp(-A)] is closed, and the current

(Tr,[Nn exp(-B)])6,

is also closed.

Remark 2.3. Observe that if the complex (, v) is acyclic, i.e., if M’ q, then thecurrent ((0) is smooth and coincides with the current which was already defined in[BGSl, Section 1.c)].

Recall that he, h’- are the given Hermitian metrics on o, m and that his the metric on which is the orthogonal sum of the metrics he, h’-.

Definition 2.4. Set

(2.4)ch(h) (-1)* ch(h)

0

T(h) t#((0)).

Let y(h) be the current

(2.5) ,(h) (2#0-1/2 q(Tr,[V exp( A,,)])-.2du


Note that by the inequality (1.15) in Theorem 1.5, the current 7(h) is alsowell-defined.

THEOREM 2.5. The current T(h) lies in P,. Thefollowin# identities ofcurrent hold

(2.6)2irr

( O) T(h) 2y(h)

In particular,

The wavefront sets ofthe currents T(h) and (h) are included in Nfft. Ifthe metricsh (he, h’-) verify assumption (A) with respect to metrics 9s, 9 on N, rl, then

(2.8)2iz

T(h) Td-(#) ch(#)6, ch(h).

Proof. By Theorem 1.4, and by formula (2.3), it is clear that the current ((0) isa sum of currents of type (p, p). Let us now prove that the wave front set of ((0) is

included in N. Clearly the form 11 Trs[Nn(exp(-A2) exp(- A))] --du is smootho u

on M. Also by [H, Example 8.2.5], the wave front set of the current

Trs[Nn exp(-- B2)] 6u,

is included in N,. Let pC be the current

Au)] Tr[Nn exp(- B2)] 6M,pc Tr[Nu exp(- 2

We will prove that the wave front set of pc is still included in N,. By usingDuhamel’s formula, and by proceding as in [BGSl, Section 1.c)-I, it is quite easy tosee that/9 is smooth on M\M’. Take now U, F, tp, M as in Section 1.c). By (1.16),we find that

(2.9)+

oo duPv,r,,,m(P) < C < +.


(2.9) exactly says that the wave front set of the current p is included in NI. We havethus proved that WF(’(O))c N. Therefore T(h) P,. Using (1.16) again, thesame argument shows that the wave front set of the current (h) is included in

Set ), (2izr)l/2q)-l((h)). Let # be a smooth even form on M. By definition

Clearly

(2.11) d#y, lim d# Tr.,[x/V exp(- A.2)]2ua-- +oo

a+oolim fffuladTrs[-x//-Vexp( -Au)’12 tdu2--Using Theorems 1.4 and 1.5, we get

(2.12)

We have thus proved the second line in (2.6). Similarly, if # is a smooth odd formon M, by definition

(2.13)

By Theorem 1.6, the currents TrsENn exp(-Ao2)] and (N Tr[Nn exp(-B2)])tMare closed. By Theorem 1.4, the forms Trs[NH exp( 2 pM.Au)] lie in Similarly, by[BGSI, Theorem 1.9-1, the form Tr[NH exp(-B2)] lies in pN. Therefore the cur-rents TrENH exp(-A)] and (N Trs[NH exp(-B2)])tM lie in Puu, and are 8 andclosed. We then get

(2.14)

(8 8)# Tr,ENn exp(--Ag)] 0

fu’ i*(--a)#(fN Trs[Nu exp(-B2)]) 0.

By proceeding as in (2.11) and by using Theorem 1.4 and (2.3), (2.13), (2.14), weimmediately obtain the first line in (2.6). (2.7) follows from (2.6). (2.8) is a consequence


of the Mathai-Quillen formula [MQ, Theorem 4.5] stated in the first line of (1.19)and of (2.7). Theorem 2.5 is proved.

b) The pull-back of the current T(h) by a transversal map. Let/r be a compactconnected complex manifold. Let f be a holomorphic map M --, M.

Definition 2.6. The map f will be said to be transversal to M’ if for anysuch that f() x M’, then

(2.15) {t e T*M; f’r/= 0} c N* {0}.

(2.15) is equivalent to

(2.16) f.TM + TxM’ T,M.

Set/r’ f-l(M’). Then, if f is transver.sal to M’, hr’ is a complex submanifoldof M. Moreover as a subbundle of T*MIM,, the vector bundle N* on M’ pulls backnaturally into a subvector bundle f’N* of T*MI;t2 Iff is transversal to M’, f’N*is exactly the dual )* of the normal bundleFrom now on, we assume that f is transversal to M’. Let be the embedding

M’ M. Let

(f*, f’v): 0

be the holomorphic chain complex of Hermitian vector bundles on r which isthe pull-back by f of the holomorphic chain complex of Hermitian vector bundles(, v).

Let f*(r/) be the holomorphic vector bundle of M’ which is the pull-back by f ofthe holomorphic vector bundle r/on M’.

Since f is transversal to M’, using the local uniqueness of resolutions (IS, ChapterIV, Appendix 1], [E, Theorem 8]), we find that (f*, f’v) provides a projectiveresolution off*r/, i.e., we have the exact sequence of sheaves

(2.17) 0 (gfi(f*,,,) -7,..""y, 7o (9fi(f*o) ,. ’, (gfi,(f*q) ,0.

Also, by [H, Theorem 8.2.4-], since WF((h)) c N, WF(T(h)) N, the currenty(h), T(h) can be unambiguously pulled back into currents f*(y(h*)), f*(T(h*))on M.Of course on 2Q, we can also define currents (f*h*), T(f*h).THEOREM 2.7. Thefollowing identities ofcurrents hold on M’

(2.18)](f*h) f*y(h)

T(f*h) f*T(h).


Proof. We will only prove the second identity in (2.18). For 1 < a < +m, let(a(0) be the current on M

(2.19) (0) f Tr,[Nu(exp(-A2) exp(- A02))]duu

+ f (Trs[Nu exp(-A2)] (Is Trs[Nn exp(-B2)]) 6u,}d-

By Theorems 1.5 and 2.5, we know that as a +

(2.20) (a(0)- ((0) in v(M).

Similarly, we define in the same way currents (j,a.t)(0) on M’ so that, as a

(2.21) ,(0)- ,(0) in (M).

We claim that for any a > 1

(2.22) (,(0) f*(a(o).

In fact, (a(0) is a sum of smooth currents carried by M and smooth currents carriedby M’. Then

By definition, the smooth forms Trs[Nn exp(-A2)] on M pull back intothe corresponding forms on M’ associated with the Hermitian chain complex(f*, f’v).

Since N f,N, we have the equality of forms on M’

(2.23) Tr[Nu exp(-(f*B)Z)] f* u Tr[Nu exp(-BZ)].

In view of (2.19)-(2.23), to obtain (2.22), it remains to prove that

(2.24) f*6u, 6r.

(2.24) can easily be proved by the methods of I-H, Example 8.2.8], i.e., by approxi-mating locally 6u, by a sequence ofsmooth currents, which converge "transversally"to the current 6u, of integration over M’, by pulling back these currents and usingthe transversality of the map f. So we have proved (2.22).


By [H, Theorem 8.2.4], and by (2.20), then as a --, oo

(2.25) f*a(o)--* f*(O)in ,,(

The second equality in (2.11) now follows from (2.21), (2.22), (2.25).

Remark 2.8. Let M be a complex submanifold of M, and let f be the em-bedding M M. Then M is said to intersect M’ transversally if for any x e M c M’,TM + TM’ TM, or equivalently iff: M M is transversal to M’. IfM and Mintersect transversally, the current T(h) has a well defined restriction f*T(h) to2r, which coincides with r(f*h).

c) Integration along the fiber of the current T(h). Let B be a compact complexconnected manifold. Let r be a holomorphic submersion from M into B, withcompact connected fiber Z.We assume that the restriction of r to M’ is also a holomorphic submersion from

M’ on B, whose fibers are denoted Y. Of course Y Z M’. More precisely, thefibers Z intersect M’ transversally.

For k N, let Ck(M) be the set of differential forms on M which are k timescontinuously differentiable. Let IIcM) be a norm on Ck(M) such that if (#n)n isa sequence of elements of Ck(M), then IIIIcM) --’ 0 if and only if #n and its first kderivatives converge to 0 uniformly on M as n +o. We define Ck(B) and IIc)in the same way.

Let # be a smooth differential form on M. In [B2, Theorem 3.2], it was provedthat for any k N, there exists Ck > 0 such that for any u > 1

(2.26)

fzTr[Nnexp(-A2)]-fri’l(fTr[Nnexp(-B2)]) c,(a)

For 0 < Re(s) < 1/2, let r/(h)(#)(s) be the smooth differential form on B

(2.27)1

uS_ A)])(h)(#)(s) #o(Tr[Nn exp(- 2

fri’l(fNtp(Trs[Nnexp(-B2)]))ldu"The fact that for 0 < Re(s) < 1/2, r/(h)(#)(s) is a smooth form on B ofcourse follows

from (2.26). Also r/(h)(/)(s) extends into a meromorphic function of s C which isholomorphic at s 0. In particular r/(h)(#)’(0) is a smooth form on B.On the other hand, let rc,(T(h)) be the current on B which is the image of the

current lT(h) by z, (i.e., the integral along the fiber Z of the current #T(h)). Since


WF(T(h)) c N, then WF(/T(h)) c N.. Also since rqu, is a holomorphic sub-mersion, it follows from [I-I, Theorem 8.2.13] that rc,(#T(h)) is a smooth currenton B, which can therefore be defined pointwise.

THEOREM 2.9. We have the equality ofsmooth differentialforms on B

(2.28) r,(#T(h)) r/(h)(/)’(0).

Proof. We proceed as in the proof of Theorem 2.7. Namely we replace thecurrent T(h) by its approximations Ta(h) qg((’(0)) and r/(h)(#)’(0) by approxi-mating forms r/a(h)(/)’(0) which we define in the obvious way. We claim that

(2.29) r,(#Ta(h)) r/a(h)()’(0).

In fact, T(h) is a sum of currents which are smooth respectively on M and M’ and(2.29) is tautological.Now by Theorem 2.5, as a +

(2.30) T(h) --,/T(h) in ’(M).

Then

(2.31) z,(#Ta(h)) r,(#T(h)) in ’(B).

Theorefore equality (2.28) holds in ’(B). Since both sides of (2.28) are smoothforms, equality (2.28) holds pointwise.

Remark 2.10. Of course a similar result holds for the current (h).

III. The Bott-Chern current as a finite part. In this section, we prove that, ingeneral, the current T(h) is smooth on M\M’ and not locally integrable on M.More precisely, we determine the singularity of T(h) near M’. We show in particularthat if Y is a normal coordinate to M’, then near M’, T(h) - Y1-2 dimN. The integralof T(h) on the complementary of an r/neighborhood ofM’ in M is then equivalentto c Log r/ as q 0. We calculate T(h) as a finite part by substracting off thelogarithmic divergence.Our results mostly rely on the techniques and the estimates of I-B2].This section is organized as follows. In a), we calculate the precise singularity of

the current T(h) near M’. In b), we calculate T(h) as a finite part.Our assumptions and notations are the same as in Sections 1 and 2.

a) The singularity of the Bott-Chern current. Let ON be a smooth Hermitianmetric on the vector bundle N. Let #ru be a smooth Hermitian metric on TM.Identifying N with the orthogonal to TM’ in TM, we assume that 0s is exactly themetric induced by

272 BISMUT, GILLET AND SOUL

if y e N, y represents Y y + y e NR. Let lYl, YI be the norms of y, Y in N, NR.Then YI2 21yl 2.

Let Ss be the unit sphere in N, Ss ( Y e N; YI 1 }. As the boundary of theunit ball in N, Ss is naturally oriented. Let n be the unit vector in NR normal to Ssand pointing outwards.

Let be a smooth differential form on N which is such that there existc, C > 0 for which I1 < c exp(-Clyl2).

Let t be the group of diffeomorphism of N: y N ety. The group is gener-ated by the vector field Y y + ft.Note that on the sphere bundle S, the form +-o ir* ds is well-defined. Indeed

as s +, we use the fact that decays as indicated as lyl--, +c,. As s -oo,irff*0 ff*ir0 decays exponentially fast.

PROPOSITION 3.1. Thefollowin# identity ofdifferentialforms on M’ holds

(3.1) in*o ds.

Proof. Let F be the map: (s, y) R x SN s(Y) e NR. Clearly

(3.2)

Now if j is the embedding SN N,

(3.3) (F*a)(s, y)= j*$*a + dsj*(ird/*a).

Using (3.2) and (3.3), we get (3.1). El

Take Xo e M’. Let U be an open neighborhood of Xo in M, and let z (zl, z)be a holomorphic system of coordinates on U such that V M’ c U is representedby (zk+l 0, z 0). Set

x (z, z)

y (z+,..., z).

Then x is a coordinate system on V. Set e k. For e > 0, let B be the open ballof center 0 and radius e in Ce. We may and we will assume that for e > 0 smallenough, V x B c U.

In the sequel, we always assume that x V, y B. If (x, y) V x B, we considery as an element of Nx.On V x B, A’(TR*M) splits into

A(TM) () A’(TM’) AS(N).i+j=p


If e A,,r)(T*M), let Omax be the component of which has maximal degree2 dim N 2e in the Grassmann variables ofN. Note that at (x, 0), i.e., on M’, maxdoes not depend on the coordinate system (x, y).For u > 0, let Bu be the superconnection on the vector bundle F on N

Bu Vr + x/t3r V.

In the sequel, the differential form on N, Tr[NH exp(-B,2)-I will be consideredas a form on V x Ce.Note that on N

(3.4) Tr[Nn exp(--(vF)2)]max 0,

and so as u 0

(3.5) Trl-Nn exp(-Bu2)-]max 0(U).

Definition 3.2. fir denotes the smooth form on N/{0}

(3.6) fie qg(Tr[Nn exp(-Bg)])maxdu

Because of(3.5) and of the fact that on N\ {0}, the forms TrINn exp(- B,2)] decayexponentially fast as u +, fir is indeed well-defined. Of course fir depends onthe coordinate system (x, y). Clearly

(3.7) TrENn exp(- B,2)] ogu)/2 TrENn exp(- B2)]

and so by Proposition 3.1, we get the equality of differential forms on M’

(3.8) i, flr 2 fv q(Tr,ENn exp(-B2)]).

Let 2 be the volume form on N with respect to the metric 9N. In the coordinatesystem (x, y), 2 extends into a smooth form on 1/x Ce, which is purely vertical (i.e.,only involves the Grassmann variables dy, dy, 1 < < e).

THEOREM 3.3. For any a > O, y N\ {0},

1(3.9) flv(ay) a2 dimN F(Y)"

Let 0(he), o(h) be the smooth forms on M\M’ which are the restrictions of thecurrents 7(h), T(h) to M\M’. Then O(h) is locally integrable on M, and coincidesas a current with 7(h).


If V and e are small enough, there exists C > 0 such that/f(x, y) V x B,

(3.10) [Yl2 dimNloJ(h)(x, Y)- fir(x, Y)I < CIYl.

The current o)(h) fir is integrable on V x B. If the metrics (h, h’) verifyassumption (A) with respect to the metrics gN and g, then

1(3.11) fir’ -(dim N 1)(Td-x)’

"l YI2dimNch’g"

71:dim N

Proof. In the proof, the constants C may vary from line to line.Using (3.6), (3.7), it is clear that if z, is the map (x, y) (x, ay), then z*fle fir.

Also Z*aflr(X, y) a2 dimNfll(X, ay). (3.9) follows.Let z. (resp. 6.) be the form on M (resp. N),

(3.12) au Tr,[Nn exp(-Au2)],

(resp.

(3.13) (di. Tr,[Nn exp(- B2)].)

Set tr. z /v/. By the proofs of [B2, Theorem 3.2] and more specifically by usingequations [B2, (3.89)-(3.104) and (3.107)-(3.109)], we know that if V and e are smallenough, there exist C, C’ > 0 such that for u > 1, x z V, lYl < ex/, then

(3.14)C

Itr*Ou(X, y)- 6 (x, Y)I <- exp(- C’IYl2)

We temporarily decompose . according to the partial degree in the Grassmannvariables of N, so that

2 dim N

(3.15) u z.p=O

In (3.15), (0 < p < 2 dim N) is of partial degree p in the Grassmann variablesdy, d. Then

yt= 2 x,p-O

For 0 < t/< 1, lyl < , we use (3.14) with u l/t/ and y replaced by y/rl. We get

(3.17)2dimN ()0

< Cr/exp ( C’ lY12%


In the right-hand side of (2.3), the first integral f { ) dUu defines a smooth form

on M, and so does not contribute to the singular part of the current ((0) near M’.Also

+oo du ; +oo du(3.18) ,u(x, y)--= ze.2u/lyl2(x, Y)--.

u rl2/2 U

Now if lyl e, then r/=

lYl < e(1 ^ -), then

lyl

ex/u < 1 and so using (3.17), we find that if

(3.19)o e,

a2u/Irl2(X’ y) 61 x, e < Celu exp(-C’e,2u).

From (3.18), (3.19), we deduce that if 6’ denotes the component of 1 with partialvertical degree p, then if Yl < e, 0 < p < 2 dim N

(3.20)

Clyl exp(-C’e2u)(e,x/)p-1y12/e2 U

Observe that

(3.21)

From (3.21), we get for y 4 0

(3.22)

Clearly

(3.23) < C(Log lyl + C’).

Also for p 0, the expression in the right-hand side of (3.20) is bounded as lYl 0.Since 2 dim N > 2, we deduce from (3.20)-(3.23)

(3.24) lyl 2dimN Clyl.


For p > 1, the integrals

(3.25)yl2/e2 U

remain bounded as lYl --’ 0. Also for p > 1, the expressions in the right hand side of(3.20) are bounded. We thus find that for 1 < p < 2 dim N 1,

(3.26) lyl 2dimly auP(X, y) Clyl.

Clearly, since 2 dim N > 2,

(3.27) (/__)2dimN12dimN X, /U

< ClYl2

Using (3.20), (3.22), (3.27), we find that

lyl 2dimN u2dimN(x’ Y)u ’u x, < Clyl.

Equivalently, we get

(3.29) lyl2 dimN (((Zu2 dimN(X, Y))- ffF X, < Clyl,

Using (3.9), (3.24), (3.26), (3.29), we obtain (3.10). Since the function 1/lyl 2 dimN-1

is integrable near 0, we find that the current co(h) fir is integrable.In [B2, eq. (4.9), (4.11)], it was noted that

(3.30) x/ Tr,[c3r V exp(- Bu2)] -ir Tr,[exp(- Bu2)],

so that

(3.31) x/ Tr,[dr V exp(--Bu2)]max O.

Using (3.30), (3.31) and the proof of [B2, Theorem 4.1] (which is in fact identicalto the proof of [B2, Theorem 3.2-1), we find by the same arguments as beforethat

(3.32) lyl2dimvlO(h)(x, y)l < Clyl,

so that 0(h) is integrable near M’. Also by replacing in the analogue of (3.18) the


du f , Oaintegrals { } -- by integrals { } dUu we find that if (h) denotes the density

of the smooth approximating current 7"(h), then we have the uniform estimate fora > 1, x e V, lYl < e,

(3.33) lyl2dimVlOa(h)(x, y)l Clyl.

Using (3.33) and the Dominated Convergence Theorem, we find that as a +oo.Oa(h) converges in the sense of distributions to 0(he). Since by Theorem 1.5, asa +o, ya(h) y(h) and since ya(h) Oa(hg), we deduce that y(h) 0(he).

Also by [MQ, Theorem 4.5-1, [B2, eq. (3.138)-(3.141), (4.22)-!, we find that underassumption (A)

(3.34) Trs[Nn exp(-B2 )]max

_(iu)dimlV(Td-1),(_(Vlv)2 Tr[exp(-(Vn)2)] exp(-ulYI2)22.

From (3.6), (3.34), we get (3.11).

b) The current T(h) as a principal part. The form co(h) is in generalnot integrable on M. Still by using (3.10), it has a well-defined principal part,which defines a current. We now compare the current T(h) with such a principalpart.For r/> 0, let M" denote the set of points ofM whose Riemannian distance to M’

is larger than

THEOREM 3.4. Let # be a smooth evenform on M. Then as l > 0 converges to 0

(3.35) fM,#co(h) + 2 Log r/fM, ,*# f qg(Trs[Nn exp(-B2)])

has a limit which we note #coS(h). Moreover thefollowing identities hold

(3.36)

f#T(h)=f#cY(h’)-fu, i*#fn(2LglYI-r"(1))(Tr[-Nuexp(-B2)l).If the metrics hg, h, verify assumption (A) with respect to the metrics gN andthen thefollowing identities hold

q9 (f Tr[Nn exp(- B2)]) -(Td-1)’(gn) ch(g")


2 Log YI- F’(1))q(Tr[Nn exp(-B2)])

(3.37) dim-I 1-(Td-x)’(gs) ch(g") + Log 2

\ k=l

Proof. Take Xo e M’. Let V be a small open ball in M’ of center Xo. We will usethe same notations as in the proof of Theorem 3.3. In particular, e denotes thedimension of N over V. We also choose geodesic coordinates in the directions ofTRM which are normal to TRM’ with respect to the given Euclidean scalar productof TM. For e > 0, set B { Y R2e; YI < e}. U V x By is then a small realneighborhood of x in M. Again, we identify R2 with the real normal bundle N toM’ in M. Set

o

We may and we will assume that the support of # is included in U. The form i*#on V lifts naturally into a form on V x R2e. Then

(3.38) fM" #c=fu, (#--llYI<’i*#)CO+M" llYI < e(i*#)09"

We know that since # is smooth

I/z(x, y)- (x, 0)l < C lyl.

In the sequel, we lift any differential form on Vinto a differential form on V x R2e.In particular, i*# is considered as a form on V x R2e. Then i*/z has partial verticaldegree 0 and coincides with #(x, 0).

Using Theorem 3.3, we find that

(3.39) limfu (lz-- llrl<i*la)Og= fM(iZ-- llrl<i*la)Og./0

Moreover

(3.40) llrl<(i*#)co- llrl<i*#(9--fl)+ llrl<(i*/z)fl"

Using Theorem 3.3 again, we get

(3.41) lim f llrl<i*#(o9 fl)-- f llrl<i*/z(o- fl).1"-*0 JM JM


Clearly, if we use (3.9), we find that if r/< e

(3.42)

Using (3.8) and (3.39)-(3.42), we see that as u 0, u,#o9 + Log r/u’ i*#(sN i,)has a limit #of given by

(3.43) fM ](’Of M ( llYl<ei*/)(’O + IM llYI<i*/z(O9 13)

Also if we still use the notations in the proof of Theorem 3.3, then

(3.44) fM f dU fM ITlyl2/e’2 du[d Ou [oe2u/iYiZu dlYl2/ez u

From (3.19), (3.44), we deduce easily that for 0 < p < 2N 1, then as T +

where the right-hand side defines a locally integrable current. Set

(3.46) 02dimN 62dimN

Then

(3.47) ;M f IO-udU fM f (ktu

+fuffllrl<#(x,O)(-Su--,,)du


By using (3.19) and (3.44), we find that as T - +o3

(3.48) (po llYl<i,#)-u du"u u

where the right-hand side is of course integrable in all variables. Similarly using(3.19), (3.22), (3.44), we see that as T +o3

(3.49) -U- -’ llYl<#(X, 0)(u u) art.

Also using (3.21), we know that

(3.50) 6(x, y)=Yi- imN 6ulYI2 X,

Therefore

fu ff llrl<#(x, O)-u du fM I T]Y]2

l,/ JIYI1 ]y] <,/O(x, 0)

Now for 0 < u < T32

(3.52) lx/f<lyl<(x/^o#(x, O)lyi2 ,(Log(x//- ^ e) Log());u i’ fs i,,6(x, y).

So, from (3.51) and (3.52), we get

(3.53) ;;; dul ;r fsfrollyl<#o(x, 0) Log T i*# idu

There is C > 0 such that for u > 1

(3.54) < exp(- Cu)).


Also using (3.50) again, we get

(3.55) 1,<lrl< yi2 dimN au X’ 7

From (3.7), (3.8), (3.53)-(3.55), we find that

(3.56)du

rlim+oo {.ft j-,r llrl<#(x’ 0)u-u--Lg Tfr i’# ;r 5,}

Also by (2.3), we know that

(3.57) fu#(’(O) lim {fft/t(ot o)d-Uu f du+ .UOu

T--+ +oo U

-F’(1)(ful.t%-fri*l.tr,)-LogTfri*ls,}Using (3.45), (3.47)-(3.57), we find that

(3.58) ’(0) u o u

u

du+ (#o_ llrl<,i,#) u--u

So from (3.43), (3.58), we find that

(3.59)

llrl> u-U- + (r’(1)- 2 Log e)


Now

(3.60)

Also by (3.7), 6u OLogu)/26x and so

(3.61)

2 llrl<(Log e- Logl Y[)6.

From (3.61), we get

2 llrl>(Logl YI- Log e,)6,.

(3.62) llYI< "-U- llrl’ .,,dUu 2 (Log e- Logl Y[)6.

From (3.59), (3.62), we deduce that

(3.63) #((0) /coy + i*/ (r’(1)- 2 Log] Yl))6x

From (3.63), we obtain (3.36).Assume now that assumption (A) is verified. Then, the first line in (3.37) was

proved in [B2, Theorem 4.3].With the same arguments as in [MQ, Theorem 4.5] and [B2, eq. (3.140), (4.22)],

we get

(3.64) exp2 TrINn exp(- B2)]

1-,, (2ir)aimN(Td-1)’(- (vN)2) Tr[exp(- (V")2)].

t)dimN+


Now for 0 < Re(s) < dim N

(3.65)1 ’-x exp -tlY[2 2

r(s) 2dt

1 +oo s-x dt

r(s) Jo (1 + t)dimNoo 1

udimN-le-U(t +t) du1 - dt r(din N)r(s)

F(dim N s)r(dim N)

By (3.64), (3.65), we find that if 0 < Re(s) < dim N

(3.66) fN (l) Tr[Nu exp(-- B2)]

-(2in)diS(Td -*)’(-(Vs)2) Tr[exp(- (V")2)] F(dimN s)

r(dim N)

Both sides of(3.66) extend into meromorphic functions of s which are holomorphicat s 0. By differentiating (3.66) at s 0, we get

(3.67) 2 Log YI Log 2) Tr[Nn exp(- BZ)]

-(2in)am(Td-x)’(-(VN)2) Tr[exp(-(V")2)] r’(dim N)F(dim N)"

Since F(s + 1) sF(s), we get

dim N-1 1r’(dim N)F’(1) + k

(3.68)F(dim N) =

The second line in (3.37) follows from (3.67), (3.68). 121

Remark 3.5. Clearly the current T(h) does not depend on the metric of TM.Therefore the right hand side of (3.36) does not depend on the metric of TM. Also

M’ i*#v(2 LogIY]- F’(1))Trs[Nn exp(-B2)] only depends on the metric of N.Therefore M/O9 only depends on the metric of N. Equivalently two metrics onTM which induce the same metric on N define the same current o9’(h). This facthas a simple geometric interpretation.

REFERENCES

[BeV1] N. BERLINE AND M. V.RGNE, Z$ros d’un champ de vecteurs et classes caract$ristiques.quivariantes, Duke Math. J. 50 (1983) 539-549.


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I, Comm. Math. Phys. 115 (1988), 49-78.[BGS2] , Analytic torsion and holomorphic determinant bundles, II, Comm. Math. Phys. 115

(1988), 79-126.[BGS3] , Analytic torsion and holomorphic determinant bundles, III, Comm. Math. Phys. 115

(1988), 301-351.[BGS4] , Classes caractristiques secondaires et immersions en 9omdtrie complexe, CRAS 307

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holomorphic sections, Acta Math. 114 (1968), 71 112.[D] S. DONALDSON, Anti-self-dual Yano-Mills connections over complex aloebraic surfaces and

stable vector bundles, Proc. London Math. Soc. 50 (1985), 1-26.[E] S. EmENBERG, Homolooical dimension and local syzyoies, Annals of Math. 64 (1956), 328-336.[GS] H. GmLE’r AND C. SOreLY, Characteristic classes for aloebraic vectors bundles with Hermitian

metrics, to appear in Annals of Math.[HI L. HORMAIDE, The analysis of linear partial differential operators, Vol. I, Grundl. der Math.

Wiss. Band 256, Berlin-Heidelberg-New York, Springer 1983.[MQ] V. MATHAI AND D. QUILLEN, Superconnections, Thorn classes and equivariant differential

forms, Topology 25 (1986), 85-110.[Q1] D. QUILLEN, Superconnections and the Chern character, Topology 24 (1985), 89-95.[Q2] ,Determinants of Cauchy-Riemann operators over a Riemann surface, Funct. Anal. Appl.

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BISMUT; UNIVERSIT PARIS-SUD, MATI-IIMATIQUE, B.M 425, F-91405 ORSAY CEDEX, FRANCE.GILLET: DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS AT CHICAGO, CHICAGO, ILLINOIS,

60638.SOtL: CNRS AND IHES, 35 ROUTE DE CHARTRES, F-91440 BtP.ES/YVErTE, FRANCE.

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BOTT-CHERN CURRENTS AND COMPLEX IMMERSIONSbismut/Bismut/1990.pdf · Vol. 60,No. 1 DUKEMATHEMATICALJOURNAL(C) February 1990 BOTT-CHERNCURRENTSAND COMPLEXIMMERSIONS J.-M. BISMUT, H