L2 Holomorphic Sections of Bundles over Weakly Pseudoconvex Coverings

L2 Holomorphic Sections of Bundles overWeakly Pseudoconvex Coverings

GEORGE MARINESCU1�, RADU TODOR2 and IONUT* CHIOSE31Institut fu« r Mathematik, HumboldtÛniversita« t zu Berlin,Unter den Linden 6, 10099 Berlin,Germany2Faculty of Mathematics, University of Bucharest, Str Academiei 14, Bucharest, Romania3Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794-3651, U.S.A.

(Received: 8 January 2001)

Abstract. We show the existence of L2 holomorphic sections of invariant line bundles over nor-mal coverings of pseudoconvex domains using a L2 generalization of Demailly’s Weyl typeformula.

Mathematics Subject Classi¢cations (2000). 32F32, 32W05.

Key words. pseudoconvex domain, von Neumann dimension, semi-classical estimate,Demailly’s asymptotic formula.

1. Introduction

A well-known theorem of Grauert [11] (his solution of the Levi problem) asserts thata strongly pseudoconvex domain is holomorphically convex. In particular, thedimension of the vector space of holomorphic functions has in¢nite dimension.One can also ask about the existence of L2 holomorphic functions (with respectto a Hermitian metric in the neighbourhood of the closure of the domain). Gromov,Henkin and Shubin [13] actually compute the von Neumann dimension of the spaceof L2 holomorphic functions on coverings of strongly pseudoconvex domains.The von Neumann dimension turns out to be in¢nite. This type of existence resultswere introduced by Atiyah [1].Our goal is to extend this result to situations where weaker pseudoconvexity con-

ditions are required. We consider two classes of (weak) pseudoconvex manifolds.A smooth domain M in a complex manifold X is called pseudoconvex if the Leviform of the de¢ning function is positive semi-de¢nite when restricted to theholomorphic tangent bundle of the boundary. (The domain is called stronglypseudoconvex if we replace ‘semi-de¢nite’ by ‘de¢nite’.) As shown by Grauert [12],pseudoconvex domains exist which possess only constant holomorphic functions,but are domains of meromorphy. Thus, it is natural to study the existence of sectionsin holomorphic line bundles over pseudoconvex domains.

�Partially supported by Sfb 288.

Geometriae Dedicata 91: 23^43, 2002. 23# 2002 Kluwer Academic Publishers. Printed in the Netherlands.

Let E be a semipositive line bundle de¢ned in a neighbourhood ofM and which ispositive near the boundary. We note that the examples constructed by Grauertpossess such line bundles. Then we show that the dimension of the space ofholomorphic L2 sections with values in the kth tensor powers Ek, for large k, growsfaster than a polynomial in kn with leading coef¢cient the integral over M ofthe curvature ðð{=2pÞcðEÞÞn, where n ¼ dimM. If E is positive everywhere, this isthe volume of the domain in the metric given by the curvature of the bundle.Moreover, the same estimate holds if we replace the domain with one of its Galoiscoverings and the usual dimension with the von Neumann dimension. This yieldsthe generalization of the theorem of Gromov^Henkin^Shubin alluded to before.The above-mentioned results are consequences of Theorem 3.8. We also ¢nd there

an upper bound for the von Neumann dimension of the L2 reduced cohomologygroups of type ð0; qÞ, qX 1, with values in Ek, with k very large.We consider further the following class of manifolds. A manifold X is called

weakly 1-complete if it admits a smooth plurisubharmonic (psh) exhaustion functionj (see [20]). The sublevel setsXc ¼ fj < cg for regular values c of j are pseudoconvexdomains. For such domains let us consider a semipositive line bundle which is posi-tive near the boundary. Then the dimension of the space of holomorphic L2 sectionswith values in the high tensor powers Ek grows faster than any polynomial in kn,n ¼ dimM. The existence of an exhaustion function permits us to endow themanifold a complete metric of in¢nite volume and apply the previous estimatefor sublevel sets larger than Mc. Moreover, the estimate also holds for the vonNeumann dimension of the space of L2 sections over a normal covering.Let us note that the estimates for the dimension of the L2 holomorphic sections

for pseudoconvex coverings are new, even in the case of relatively compactpseudoconvex domains. Due to the weak positivity conditions (weakpseudoconvexity and semipositivity), we cannot directly solve the �@@-equation inorder to get holomorphic sections. We argue indirectly, by using the asymptoticMorse inequalities introduced by Siu [22] and Demailly [7] for compact manifolds(see also Berndtsson [3]; for the noncompact case, Nadel and Tsuji [18], Bouche[4], and [17]).In [26] we gave a generalisation of the asymptotic Morse inequalities for coverings

of complete Hermitian manifolds. Here we work with incomplete metrics (and there-fore with the Laplacian with �@@-Neumann boundary conditions), since we cannotapply [26] to obtain the main result of the present paper (see the remarks at theend of Section 3). However, one of the principal ingredients of the proof is takenfrom [26]. It is the generalisation to the covering case of Demailly’s Weyl typeformula for the counting function of the Dirichlet Laplacian D00k acting on high tensorpowers Ek.The plan of the paper is as follows. In Section 2 we recall the necessary background

and results from [26]. In Section 3 we use the Bochner^Kodaira formula in the formgiven by Andreotti and Vesentini [2] and Grif¢ths [10] to reduce the study ofthe Laplacian with �@@-Neumann boundary conditions on @M to the study of the

24 GEORGE MARINESCU ET AL.

Dirichlet Laplacian on a smaller and invariant domain. In Section 4 we apply ourmain result (Theorem 3.8) to the case of weakly 1-complete manifolds and stronglypseudoconvex domains.

2. Estimates of the Spectrum Distribution Function

One of the main ingredients in Demailly’s proof of the asymptoticMorse inequalitiesis aWeyl type formula for the counting function of the Laplacian D00k acting on E

k. Onthe other hand, Shubin gave in [23] a proof of the L2 Morse inequalities for coveringmanifolds in the spirit of Witten’s paper [27]. Our proof uses the technique of [23] togeneralize Demailly’s formula for the Dirichlet Laplacian on a covering manifold.LetM be a complex analytic manifold of complex dimension n on which a discrete

group G acts freely and properly discontinuously. Let ~MM ¼M=G and let p: M�! ~MMbe the canonical projection. We assume that M is paracompact so that G will becountable. Suppose we are given a holomorphic vector bundle ~GG on ~MM and takeits pull-back G ¼ p� ~GG, which is a G-invariant bundle onM. We also ¢x G-invariantHermitian metrics on M and on G.We consider a relatively compact open set ~OO ~MM with smooth boundary and its

preimage O ¼ p�1 ~OO; G acts on ~OO and O=G ¼ ~OO. In general, we will decorate by tildesthe objects living on the quotient. Let U be a fundamental domain of the action of Gon O. This means that (see, e.g., [1]): (a) O is covered by the translations of U , (b)different translations of U have empty intersection and (c) U nU has zero measure(since @ ~OO is smooth). Since ~OO is relatively compact, U has the same property.Let us de¢ne the space of square integrable sections L2ðO;GÞ with respect to a

G-invariant metric onM (and its volume form) and a G-invariant metric on G. ThenL2ðU;GÞ is constructed with respect to the same metrics. There is a unitary action ofG on L2ðO;GÞ. In fact, it is easy to see that

L2ðO;GÞ ffi L2G L2ðU;GÞ ffi L2G L2ð ~OO;GÞ:

We have a unitary action of G on L2G by left translations: g 7 �!lg, wherelgf ðxÞ ¼ f ðg�1xÞ for x 2 G, f 2 L2G. It induces an action on L2ðO;GÞ byg 7 �!Lg ¼ lg Id. Finally, we denote by Ccð� ; �Þ the various spaces of smoothcompactly supported sections.Let us consider a formally self-adjoint, strongly elliptic, positive differential

operator ~PP on ~MM acting on sections of ~GG. Denote by P the G-invariant differentialoperator which is its pull-back toM. From P we construct the following operators:the Friedrichs extension in L2ðO;GÞ of PjCcðO;GÞ and the Friedrichs extension inL2ðU;GÞ of PjCcðU;GÞ. From now on we denote these extensions by P and P0. Theyare closed self-adjoint positive operators and coincide with the Dirichlet Laplacianson O and U .It is known that P is also G-invariant, i.e. it commutes with all Lg. This amounts to

saying that ElðPÞ commutes with Lg, g 2 G, where ðElðPÞÞl is the spectral family of P.

L2 HOLOMORPHIC SECTIONS OF BUNDLES 25

On the other hand, the Rellich lemma tells us that P0 has compact resolvent and,hence, discrete spectrum.

DEFINITION 2.1. For any Hilbert spaceH, we call the Hilbert spaceL2G H a freeHilbert G-module. G-invariant closed spaces of free Hilbert G-modules are calledG-modules.

Since El is G-invariant its image Ran Elis a G-module of the free HilbertG-module L2G L2ðU;GÞ ffi L2ðO;GÞ. The action of G is de¢ned as above byg 7 �!Lg ¼ lg Id.For G-modules one can associate a positive, possibly in¢nite real number, called

the von Neumann or G-dimension, denoted dimG. For notions involving theG-dimension and linear algebra for G-modules we refer the reader to [1, 23] and[16, pp. 75^80] (in the latter proofs from scratch are given).Let us denote by AG the von Neumann algebra which consists of bounded linear

operators in L2G H which commute to the action of G. To describe AG let us con-sider the von Neumann RG algebra of bounded operators on L2G which commutewith all Lg. It is generated by right translations. If we consider the orthonormalbasis ðdgÞg in L

2G where dg is the Dirac delta function at g, then the matrix ofany operatorA 2 RG has the property that all its diagonal elements are equal. There-fore we de¢ne a natural trace onRG as the diagonal element, that is, trGA ¼ ðAde; deÞwhere e is the identity element. AG is the tensor product of RG and the algebra BðHÞof all bounded operators on H. If Tr is the usual trace on BðHÞ then we have a traceon AG by TrG ¼ trG Tr.

DEFINITION 2.2. For any G^module, the projection PL 2 AG and we de¢nedimG L ¼ TrGPL.

We also introduce the notion of G-morphisms. If L1, L2 are two G-modules, then abounded linear operator T : L1�!L2 is called a G-morphism if it commutes with theaction of G. As for the usual dimension, the following statements are true (see [16,pp. 75^80]). If T is injective, dimG L1W dimG L2, and if T has dense image,dimG L1X dimG L2. We denote by rankGT ¼ dimG Ran ðT Þ½ �, where ½V � standsfor the closure of a vector space V .We denote in the sequel NGðl;PÞ ¼ dimG Ran ElðPÞ. Similary, we consider the

spectral distribution (counting) function Nðl;P0Þ ¼ dim Ran ElðP0Þ, whereElðP0Þ is the spectral family of P0; it equals the number of eigenvalues W l.To compare NGðl;PÞ and Nðl;P0Þ we use the analysis of Shubin [23]. However,

there exist a difference in our method. Namely, we take from the beginning the modeloperator P0 to be the operator P itself with Dirichlet boundary conditions on U ,whereas Shubin considers a direct sum of tangent operators to P. So we do nothave to truncate from the outset the eigenfunctions of the model P0. (See also [23,Remark 1.3].) The next result [23, Lemma 2.4] is fundamental.


PROPOSITION 2.3 (Variational principle). Let P be a G-invariant self-adjointpositive operator on a free G-module L2G H where H is Hilbert space. Then

NGðl;PÞ ¼ supn

dimG L j L DomðQÞ; Qð f ; f ÞW lkf k2 ; 8f 2 Lo; ð2:1Þ

where L DomðQÞ runs over G-modules and Q is the quadratic form of P.

With the requisite maximum-minimum result in place, we now begin to examine therelation between the two distribution functions NGðl;PÞ and Nðl;P0Þ.

PROPOSITION 2.4 (Estimate from below). The counting functions of P and P0satisfy the inequality

NGðl;PÞXNðl;P0Þ ; l 2 R: ð2:2Þ

Proof. See [26, Proposition 1.1]. &

In order to get an estimate from above, we have to enlarge a little bit thefundamental domain U and compare the counting function of P to the countingfunction of the Friedrichs extension of P restricted to compactly supported formsin the enlarged domain. For h > 0, the enlarged domain isUh ¼ fx 2 O j dðx;UÞ < hg where d is the distance onM associated to the Riemannmetric on M. We consider the operator P with domain CcðUh;GÞ and take itsFriedrichs extension, denoted PðhÞ0 .

PROPOSITION 2.5 (Estimate from above). There is a constant C > 0 such that

NGðl;PÞWN lþCh2

;PðhÞ0

� �l 2 R; h > 0: ð2:3Þ

Proof. See [26, Proposition 1.4]. &

The estimates from below and above for NGðl;PÞ enable us to study as aby-product the behaviour for l�!1 to obtain the Weyl asymptotics for periodicoperators (due to M. Shubin, see [21] and the references therein).

COROLLARY 2.6. If P is a periodic, positive, second-order elliptic operator asabove, then

liml!1

l�n=2NGðl;PÞ ¼ liml!1

l�n=2Nðl;P0Þ

¼ ð2pÞ�nZU

ZT�xM

Nð1; s0ðPÞðx; xÞÞdx dx:


s0ðPÞðx; xÞ 2 HermðF ;F Þ is the principal symbol of P and Nð1; s0ðPÞðx; xÞÞ is thecounting function for the eigenvalues of this Hermitian matrix.Proof.First let us remark that the last equality is the classical Weyl type formula as

established by Carleman, Ga�rding and others, see [21, p. 72]. It is obvious that

lim l�n=2Nðl;P0ÞW lim l�n=2NGðl;PÞ by the estimate from below. On the other handthe estimate from above gives for l�!1

lim l�n=2NGðl;PÞW lim 1þClh2

� �n=2lþ

Ch2

� ��n=2N lþ

Ch2

;PðhÞ0

� �W limm�n=2Nðm;PðhÞ0 Þ ¼ ð2pÞ

�nðUh

ðT�xM

Nð1; s0ðPÞðx; xÞÞdx dx

for a ¢xed small h. We make h�!0 and obtain the desired formula. &

We are going to apply the above results to the semi-classical asymptotics ask�!1 of the spectral distribution function of the Laplacian 1

kD00k on M. Let ~GG

be a Hermitian holomorphic bundle on ~MM and G ¼ p� ~GG its pull-back.We de¢neCð0;qÞc ð� ; �Þ to be the space of smooth compactly supported ð0; qÞ forms. We denoteby L0; qðM;GÞ the space of ð0; qÞ-forms which are L2 on M with respect to smoothG-invariant metrics on X and G. Let �@@: C0;qc ðM;GÞ�!C0;qþ1c ðM;GÞ be theCauchy^Riemann operator and W: C0;qþ1c ðM;GÞ�!C0;qc ðM;GÞ the formal adjointof �@@ with respect to the given Hermitian metrics on M and G. ThenD00 ¼ �@@Wþ W �@@ is a formally self-adjoint, strongly elliptic, positive and G-invariantdifferential operator.We take E and F two G-invariant holomorphic bundles. Let us form the

Laplace^Beltrami operator D00k on ð0; qÞ forms with values in Ek F . Thus, we will

consider the G-invariant Hermitian bundle G ¼ Lð0;qÞT�M Ek F and applythe previous results for P ¼ ð1=kÞD00kHO where the index O emphasises that theFriedrichs extension gives the operator of the Dirichlet problem on O. Now we haveto make a good choice of the parameter h. We take h ¼ k�1=4. By ð2:2Þ and ð2:3Þ, weobtain the following semi-classical estimate for the Laplacian.

PROPOSITION 2.7. There exists a constant C > 0 such that for l 2 R and k > 0 wehave

N l ;1kD00kHU

� �WNG l ;

1kD00kHO

� �WN lþ

Cffiffiffik

p ;1kD00kHUk�1=4

� �: ð2:4Þ

Demailly has determined the distribution of spectrum for the Dirichlet problemfor ð1=kÞD00k in [7, Theorem 3.14]. For this purpose he introduces ([7, (1.5)]) thefunction nE : M �R�!R depending on the curvature of E and then considers thefunction �nnEðx; lÞ ¼ lime&0 nEðx; lþ eÞ. The function �nnEðx; lÞ is right continuousin l and bounded above on the compacts of M. Denote by a1ðxÞ; . . . ; anðxÞ theeigenvalues of the curvature form icðEÞðxÞ with respect to the metric on M. We


also denote for a multiindex

J f1; . . . ; ng; aJ ¼Xj2J

aj and CðJÞ ¼ f1; . . . ; ng n J:

For VYM, we introduce

IqðV ; mÞ ¼XjJj¼q

ZV

�nnEð2mþ aCðJÞ � aJÞ ds :

PROPOSITION 2.8 (Demailly). Assume that @V has measure zero and that theLaplacian acts on ð0; qÞ forms. Then

limk�!1

k�nN l ;1kD00kHV

� �W IqðV ; lÞ:

Moreover there exists an at most countable set D R such that for l 2 R n D thelimit of the left-hand side expression exists and we have equality.

We return now to the case of a covering manifold and apply Demailly’s formula inð2:4Þ. Let us ¢x e > 0. For suf¢ciently large k, we haveUk�1=4 Ue so the fact that thecounting function is increasing and the variational principle (2.1) yields

N lþCffiffiffik

p ;1kD00kHUk�1=4

� �WN lþ e;

1kD00kHUk�1=4

� �WN lþ e;

1kD00kHUE

� �:

Hence, by (2.4) and Proposition 2.8 (@Ue is negligible for small e),

lim supk

k�nNG l ;1kD00kHO

� �W IqðUe; lþ eÞ:

The use of dominated convergence to make e�!0 in the last integral yields thefollowing asymptotic formula for the Laplacian on a covering manifold:

THEOREM 2.9. The spectral distribution function of ð1=kÞD00kHO on L0;qðO;Ek F Þ

with Dirichlet boundary conditions satis¢es

limk�!1

k�nNG l ;1kD00kHO

� �W IqðU; lÞ: ð7Þ

Moreover, there exists an at most countable set D R such that, for l 2 R n D, thelimit exists and we have equality in (2.11).

We need to know the behaviour of IqðU; lÞ for l�!0 which is given in [7]. First, itis clear that

liml�!0

IqðU; lÞ ¼ IqðU; 0Þ ¼XjJj¼q

ZU

�nnEðaCðJÞ � aJ Þ ds:


By de¢nition ([7, (1.5)]),

�nnEðaCðJÞ � aJÞ ¼2s�2np�n

Gðn� sþ 1Þa1 � � � asj j

Xp2Ns

aCðJÞ � aJ �Xsj¼1

ð2pj þ 1Þjajj

( )n�sþ

;

where the eigenvalues satisfy

ja1jX ja2j � � � X jasj > 0 ¼ jasþ1j ¼ � � � ¼ janj

with the notation ln=2þ (½l�0þ ¼ 0 for l < 0, ½l�0þ ¼ 1 for lX 0). It is clear that, forp 2Ns, the corresponding summand in the above expression vanishes unlesss ¼ n, p1 ¼ � � � ¼ pn ¼ 0, aj < 0 for j 2 J and aj > 0 for j 2 CðJÞ. In particular, if�nnEðaCðJÞ � aJÞ 6¼ 0, {cðEÞ is nondegenerate and has exactly q negative eigenvalues.Let UðqÞ be the set of points x of U such that {cðEÞðxÞ is nondegenerate and has

exactly q negative eigenvalues. For x 2 UðqÞ and jJj ¼ q it follows that

�nnEðaCðJÞ � aJÞ ¼ ð2pÞ�nja1 � � � anj ¼ ð�1Þqð2pÞ�nð{cðEÞÞn

for J ¼ JðxÞ, where

JðxÞ ¼ fj : ajðxÞ < 0g and �nnEðaCðJÞ � aJÞ ¼ 0

for J 6¼ JðxÞ.Therefore

liml�!0

IqðU; lÞ ¼1n!

ZUðqÞð�1Þq

� {

2pcðEÞ

�n: ð2:6Þ

For the formulation of the main result we set UðW 1Þ :¼ Uð0Þ [Uð1Þ.

3. Pseudoconvex Manifolds

Let X be a complex manifold acted upon freely by a discrete group of auto-morphisms G and G�!X be a G-invariant holomorphic vector bundle. We ¢xon X and G G-invariant Hermitian metrics.Let M X be a smooth open set such that:

# M is pseudoconvex,# M is G-invariant,# M=G is compact.

Thus M=G is a pseudoconvex relatively compact domain in the manifold X=G. Weintroduce the functional spaces C 0; qðM;GÞ of smooth ð0; qÞ-forms on M andC 0; qc ðM;GÞ the subspace of forms smooth up to the boundary and with compactsupport inM. We remind that L0; qðM;GÞ is the L2 space of ð0; qÞ-forms, with respectto the ¢xed G-invariant metrics on M and G.The object of our study is the reduced L2 cohomology for �@@. To de¢ne it, we con-

sider the weak maximal extension �@@ : L0; qðM;GÞ�!L0; qþ1ðM;GÞ, which is a closed


densely de¢ned operator with domainDom ð �@@Þ (consisting of elements u such that �@@ucalculated in distributional sense is in L2). The reduced L2 Dolbeault cohomology isde¢ned to be

H0; qð2Þ ðM;GÞ :¼ Ker �@@ \ L0; qðM;GÞ= Ran �@@ \ L0; qðM;GÞ

�where ½V � denotes the closure of the space V . We need to take the closure in order tomake sure thatH0; q

ð2Þ ðM;GÞ is a Hilbert space. The operator �@@ is obviously G-invariantso that H0; q

ð2Þ ðM;GÞ is a G-module.We link the L2 cohomology to the theory of elliptic operators via the Hodge

decomposition for a certain Laplace^Beltrami operator. Let �@@� be the Hilbert spaceadjoint of �@@. We introduce the operator D00 de¢ned by

DomðD00Þ :¼ u 2 Dom ð �@@Þ \Dom ð �@@�Þ : �@@u 2 Dom ð �@@�Þ ; �@@�u 2 Dom ð �@@Þ� �

;

D00u ¼ �@@ �@@�uþ �@@� �@@u for u 2 Dom ðD00Þ :

By general arguments of functional analysis, we know that D00 is selfadjoint ([9,Proposition 1.3.8]). The advantage of using D00 is that it leads to the Hodgedecomposition, since Ker D00 ¼ Ker �@@ \Ker �@@�. It is clear that �@@� and D00 areG-invariant and we have the following lemma (for a proof see, e.g., [13, Proposition1.4]).

LEMMA 3.1. The following weak Hodge decomposition holds:

L0; #ðM;GÞ ¼ Ker D00 % Ran �@@ �

% Ran �@@� �

;

Ker �@@ ¼ Ker D00 % ½Ran �@@ � :

In particular, we have an isomorphism of G-modules:

H0; qð2Þ ðM;GÞ ffi Ker D00 \ L0; qðM;GÞ : ð3:1Þ

We will work with the quadratic form canonically associated to D00, rather thandirectly with D00. Let us recall that the closed quadratic form q ¼ qðAÞ associatedto a positive selfadjoint operator A is de¢ned by

Dom ðqÞ ¼ Dom ðA1=2Þ; qðu; vÞ ¼ ðA1=2u;A1=2vÞ

for u; v 2 Dom ðA1=2Þ.

LEMMA 3.2. The quadratic form associated to D00 is the form Q given byDom ðQÞ :¼ Dom ð �@@Þ \Dom ð �@@�Þ and

Qðu; vÞ ¼ ð �@@u; �@@vÞ þ ð �@@�u; �@@�vÞ ; u; v 2 Dom ðQÞ:

Proof. First remark that Q is a closed form. It is well known (see, e.g., [6,pp. 81^83]) that any closed, positive form is associated to a unique selfadjoint,


positive operator F . The domain of F consists of elements u 2 Dom ðQÞ such thatthere exists w with

Qðu; vÞ ¼ ðw; vÞ for any u 2 Dom ðQÞ : ð3:2Þ

Moreover, for such u, Fu ¼ w. For u 2 Dom ðD00Þ it is clear that u 2 Dom ðQÞ and usatis¢es (3.2) with w ¼ D00u. Therefore D00 F and, since both operators areselfadjoint, F ¼ D00. &

We describe next a core form for Q, i.e. a dense subspace of Dom ðQÞ in the normkukQ ¼

�kuk2 þQðu; uÞ

�1=2. LetM ¼ fx 2 X : rðxÞ < 0g where r is a smooth functionon X which has nonvanishing gradient on @M. We denote

B 0; qðM;GÞ ¼ fu 2 C 0; qc ðM;GÞ : @r ^ �u ¼ 0 on @Mg :

Integration by parts ([9, Propositions 1.3.1^2]) shows that

B 0; qðM;GÞ ¼ C 0; qc ðM;GÞ \Dom ð �@@�Þ ; �@@� ¼ W on B 0; qðM;GÞ : ð3:3Þ

Note that, by ð3:3Þ, the operator �@@Wþ W �@@ on fu 2 B 0; qðM;GÞ : �@@u 2 B 0; qþ1ðM;GÞg ispositive and D00 is one of its selfadjoint extensions.

LEMMA 3.3 (Approximation). B 0; qðM;GÞ is a core form for Q.Proof. In the case the element from Dom ðQÞ ¼ Dom ð �@@Þ \Dom ð �@@�Þ to be

approximated in k � kQ has compact support, the lemma is well known, see [15,Proposition 1.2.4]. Thus, the only thing to do is to approximate elements fromDom ð �@@Þ \Dom ð �@@�Þ with elements with compact support. We can do this thanksto the G-action, which guarantees that the G-invariant metric onM is in some sense‘complete’ . For the details, see [13, Lemma 1.1]. &

Next let us use a form of the Bochner^Kodaira formula introduced by Andreottiand Vesentini [2] and Grif¢ths [10]. Let us assume that there exists a G-invariantHermitian metric on X , Ka« hler near @M, written in local coordinates za as a smoothpositive de¢nite matrix ðgabÞ. We also ¢x a G-invariant Hermitian metric on G,written locally in a holomorphic frame f of G, as a smooth positive function h.There exists a canonical connection on G, compatible with the complex structure

and the metric. The curvature of this connection is denoted cðGÞ. It is a ð1; 1Þ-formon X , cðGÞ ¼

Pyab dza ^ d�zzb, where yab ¼ �@za@�zzb log h. Let ym

b be the curvaturetensor with the ¢rst index raised. Let

u ¼1q !

Xul1 ... lq d�zz l1 ^ � � � ^ d�zz lq f

be a G-valued ð0; qÞ-form on X . We de¢ne the ð0; qÞ-form

cðGÞu ¼1q !

Xyml1um l2 ... lq d�zz l1 ^ � � � ^ d�zz lq f :


We also introduce the Ricci curvature Ric ¼ �cðKX Þ, where KX is the canonicalbundle of X . For a G-valued ð0; qÞ-form u, we set Ric u ¼ �cðKX Þu.We de¢ne next the Levi operator (see [10, p. 418]). First let us remark that we can

choose a G-invariant de¢ning function r forM such that j@rj ¼ 1 in a neighbourhoodof @M, with respect to the Hermitian metric onX . Let us pick, near a boundary pointof M, an orthonormal frame o1; . . . ;on for the bundle of ð1; 0Þ-forms, such thaton ¼ @r. We have

@ �@@r ¼ � �@@on ¼X

lab oa ^ ob ð1W a; bW nÞ :

DEFINITION 3.4. The Levi form of @M is the restriction of @ �@@r to the holomorphictangent bundle of @M; it is given in the dual frame of o1; . . . ;on�1 by the matrixðlabÞ1W a;bW n�1. M is pseudoconvex if the Levi form is everywhere positivesemi-de¢nite.For a ð0; qÞ-form written locally

u ¼1q !

Xua1 ... bq o

a1 ^ � � � ^ oaq f ;

where f is an orthonormal frame in G, we set

Lðu; uÞ ¼1

ðq� 1Þ!

Xlab ua b1 ...bq�1ub b1 ... bq�1 ð1W a; bW nÞ :

The form u 2 B 0; qðM;GÞ if and only if ua1 ... aq ¼ 0 for n 2 fa1; . . . ; aqg. Therefore foru 2 B 0; qðM;GÞ, the summation restricts over 1W a; bW n� 1 and

Lðu; uÞ ¼1

ðq� 1Þ!

Xlab ua b1 ...bq�1ub b1 ... bq�1 X 0 : ð3:4Þ

Finally, let r denote the covariant derivative in the ð0; 1Þ-direction.

LEMMA 3.5. Assume that the G-invariant metric on X is Ka« hler in a G-invariantneighbourhood U of @M. Then for any u 2 B 0; qðM;GÞ with support in U we have

Qðu; uÞ ¼��ru��2 þ �

cðGÞu; u�þ�Ric u; u

�þ

Z@MLðu; uÞ dS: ð3:5Þ

Proof. This formula was given by Grif¢ths [10, p. 429, (7.14)]. &

Next we specialise to the case G ¼ Ek for a G-invariant Hermitian E. Let D00k theLaplacian acting on forms with values in Ek. We shall normalise the quadratic formof the Laplacian:

Qkðu; uÞ ¼1kk �@@uk2 þ k �@@�uk2� �

;


for

u 2 Dom ðQkÞ :¼ Dom ð �@@Þ \Dom ð �@@�Þ \ Lp;qðM;EkÞ:

The bundle E is endowed with a G-invariant Hermitian metric whose curvature ispositive near @M. From now on we ¢x a G-invariant Hermitian metric on X whichcoincides with cðEÞ near @M. Since any two G-invariant metric onM are equivalentthe L2 spaces over M do not change.

LEMMA 3.6. Let E be a G-invariant holomorphic line bundle on X endowedwith a G-invariant Hermitian metric which is positive in a neighbourhood U of@M. Let V U be another G-invariant neighbourhood of @M and 0W RW 1 be asmooth G-invariant function which equals 1 on V and vanishes outside U. Thenfor suf¢ciently large k, any u 2 Dom ðQkÞ \ L0; qðM;EkÞ, qX 1, satis¢es the a-prioriestimate:

kuk2W 12Qkðu; uÞ þ 4ZO

��ð1� r Þu��2 ; ð3:6Þ

where O :¼M n V.Proof. Let u 2 B 0; qðM;EkÞ, qX 1, with support in U . By (3.5) and (3.4) and

cðEkÞ ¼ kcðEÞ,

Qkðu; uÞX�cðEÞu; u

�þ1k

�Ric u; u

�:

Since the metric coincides with cðEÞ near @M, we have cðEÞu ¼ u. Therefore thereexists a constant C0 (depending only of the Ricci curvature of the invariant metricon M) such that

Qkðu; uÞX12kuk2 ; kXC0 : ð3:7Þ

We use now the elementary estimate:

Qkðr u; r uÞW32Qkðu; uÞ þ

6k

sup jdr j2 kuk2 : ð3:8Þ

Obviously C1 ¼ 6 sup jdr j2 <1 . If O ¼M n V estimates (3.7) and (3.8) yield

kuk2W 12Qkðu; uÞ þ 4ZO

��ð1� r Þu��2 ; kX maxfC0; 8C1g ;

for any u 2 B 0; qðM;EkÞ. By the Approximation Lemma 3.3 the estimate holds alsofor u 2 Dom Qk. This proves our contention. &

From relation (3.6) we infer next that the spectral spaces corresponding to thelower part of the spectrum of ð1=kÞD00k on ð0; qÞ-forms, qX 1, can be injected intothe spectral spaces of the G-invariant operator 1kD

00kHO, which correspond to the


Dirichlet problem on O for ð1=kÞD00k. We may apply thus the results from Section 2.This idea appears in Witten’s proof of the Morse inequalities (see [14, 27]) andin [4] in the context of q-convex manifolds in the sense of Andreotti and Grauert.Let us introduce the following spectral spaces: EqkðlÞ ¼ Ran Elðð1=kÞD00kÞ is the

spectral space of ð1=kÞD00k on L0; qðM;EkÞ, Eqk;OðmÞ ¼ Ran Emðð1=kÞD00kHOÞ, thespectral spaces of 1kD

00kHO. As before El stands for the spectral projection. We

denote NqGðl;

1kD

00kÞ ¼ dimG E

qkðlÞ the von Neumann dimension of the spectral spaces

of 1kD00k.

LEMMA 3.7. Assume we are in the hypotheses of Lemma 3.6. Then for suf¢cientlylarge k, for qX 1 and any l < 1=24 the following map is an injective G-morphism:

EqkðlÞ�!E

qk;O 12lþ

8C1k

� �; u 7 �!E12lþ8C1k

1kD00kHO

� �ð1� rÞu : ð3:9Þ

In particular,

NqG l;

1kD00k

� �WNq

G 12lþ8C1k

;1kD00kHO

� �; qX 1 ; l < 1=24 ; ð3:10Þ

The meaning of the map (3.9) is that we cut off the form u and then we project it ontothe spectral space of the Dirichlet Laplacian. We note that (3.10) above shows thatthe spectral spaces EqkðlÞ, for qX 1, l < 1=24, are of ¢nite G-dimension.Proof. To prove the claim let us remark that the map (3.9) is the restriction of an

operatoron L0;qðM;EkÞ of the same form; this is continuous and G-invariant beinga composition of a multiplication with a bounded G-invariant function and aG-invariant projection. To prove the injectivity, we choose u 2 EqkðlÞ, l < 1=24 tothe effect that Qkðu; uÞW lkuk2W 1

24 kuk2. Plugging this relation in (3.6), we get

kuk2W 8ZO

��ð1� r Þu��2 ; u 2 EqkðlÞ ; l < 1=24 : ð3:11Þ

Let us denote by Qk;O the quadratic form of ð1=kÞD00kHO. Then, by (3.8) and (3.11),

Qk;O�ð1� rÞu; ð1� rÞu

�W32Qkðu; uÞ þ

C1kkuk2W 12 lþ

8C1k

� �ZO

��ð1� r Þu��2

which shows that E12lþ8C1kð1=kÞD00kHO� �

ð1� rÞu ¼ 0 implyð1� rÞu ¼ 0 so that u ¼ 0by (3.11). This achieves the proof. &

We now state our main result. We ¢nd a lower bound for the G-dimension of thegroup H0

ð2ÞðM;EkÞ of L2 holomorphic sections in Ek and give an upper boundfor the growth of the G-dimension of the higher L2 cohomology groups. The L2

condition is understood with respect to G-invariant metrics on the base X and alongthe ¢bers of the bundle.


THEOREM 3.8. Let M be a smooth pseudoconvex domain which admits a freeholomorphic action of a discrete group G such that the action extends in a complexneighbourhood X of M. Assume that M=G is compact and there exists a Hermitianholomorphic G-invariant line bundle E on X which is positive near @M. Then, fork�!1,

dimGH0ð2ÞðM;EkÞX

kn

n!

ZðM=GÞðW 1Þ

� {

2pcðEÞ

�nþ oðknÞ ; ð3:12Þ

where n ¼ dimM.The higher L2 reduced cohomology groups (qX 1) have ¢nite G-dimension and

satisfy the inequalities for k�!1:

dimGH0;qð2Þ ðM;EkÞW

kn

n!

ZðM=GÞðqÞ

ð�1Þq� {

2pcðEÞ

�nþ oðknÞ ; ð3:13Þ

Proof. Since Dom Qk;O can be embedded in Dom Qk, an easy consequence of thevariational principle (2.1) is that N0Gðl; ð1=kÞD

00kÞXN0Gðl; ð1=kÞD

00kHOÞ. By Theorem

2.9 for q ¼ 0

limkk�nN0G l;

1kD00k

� �X I0ðU; lÞ ; l < 1=24 ; l 2 R n D: ð3:14Þ

(Note that O can be chosen smooth.)We ¢nd now an upper bound for N1Gðl; ð1=kÞD

00kÞ. Fix an arbitrary d > 0. For

k > 8C1=d we have

N1 l;1kD00k

� �WN1G 12lþ

8C1k

;1kD00kHO

� �WN1G 12lþ d;

1kD00kHO

� �;

hence, by (2.5), limkk�nN1Gðl; ð1=kÞD00kÞW I1ðU; 12lþ dÞ. We can let d�!0 so that

limkk�nN1G l;

1kD00k

� �W I1ðU; 12lÞ ; l < 1=24 ; l 2 R n D : ð3:15Þ

We consider next �@@:L0; 0ðM;EkÞ�!L0; 1ðM;EkÞ. Since D00k commutes with �@@, it followsthat the spectral projections of D00k commute with �@@ too, showing thus �@@E0kðlÞ E

1kðlÞ.

Therefore, we have the G-morphism �@@: E0kðlÞ�!E1kðlÞ, where �@@l denotes the restriction

of �@@ (by the de¢nition of E0kðlÞ, �@@l is bounded by kl). Since for any G-morphism Awe have dimG Ran ðAÞ½ � ¼ dimG Ker ðAÞ?, we see that

dimG Ker �@@l þ dimG Ran ð �@@lÞ �

¼ dimG E0kðlÞ:

Moreover, dimG Ran ð �@@lÞ �

W dimG E1kðlÞ and they are ¢nite. Therefore by (3.14) and

(3.15),

dimGH0ð2ÞðM;EkÞX dimG Ker �@@l X kn

hI0ðU; lÞ � I1ðU; 12lÞ

iþ oðknÞ ;


for l < 1=24 and l 2 R n D. We can now let l go to zero through these values. Thelimits I0ðU; 0Þ and I1ðU; 0Þ are calculated in [7] (see (2.6)) and if we identify thefundamental domain U with O=G, we get

dimGH0ð2ÞðM;EkÞX

kn

n!

ZðO=GÞðW 1Þ

� {

2pcðEÞ

�nþ oðknÞ :

Now we can let O exhaust M and the desired formula (3.12) follows.To prove (3.13) we remark that, for lX 0,

dimG Ker ðð1=kÞD00kÞ \ L0; qðM;EkÞWNq

Gðl; ð1=kÞD00kÞ ;

and as in formula (3.15),

limkk�n dimG Ker ðð1=kÞD00kÞ \ L

0; qðM;EkÞW IqðU; 12lÞ;

l < 1=24; l 2 R n D :

We invoke now the Hodge isomorphism (3.1)

Ker ðð1=kÞD00kÞ \ L0; qðM;EkÞ ffi H0; q

ð2Þ ðM;EkÞ ;

and let l�!0 through values l 2 R n D. By (2.6) we get the inequality (3.13) for thedomain of integration O=G so we can let O�!M. The proof is thus complete.&

COROLLARY 3.9. In the conditions of Theorem 3.8 assume thatZðM=GÞðW 1Þ

� {

2pcðEÞ

�n> 0 : ð3:16Þ

Then for large k the space H0ð2ÞðM;EkÞ is nontrivial. More precisely,

dimGH0ð2ÞðM;EkÞekn :

Remarks. (i) Condition (3.16) may be seen as a very weak positivity condition. Itmeans that the integral of the curvature over its positivity points exceeds the integralof the curvature over the points where the curvature is nondegenerate and has exactlyone negative eigenvalue. So the curvature may have negative eigenvalues. Theinequality (3.16) is certainly satis¢ed if the curvature is everywhere semipositive,to the effect that ðM=GÞð1Þ ¼ ;. Therefore the integral in (3.12) extends over allM=G. If E is everywhere positive, this integal expresses the volume of M=G inthe metric given by cðEÞ.(ii) Assume that E is positive everywhere on M. Then (3.6) becomes

kuk2W 12Qkðu; uÞ ; k, 1 ; qX 1 : ð3:17Þ

Thus, Ker D00k ¼ 0 and by Hodge isomorphism we obtain the asymptotic vanishing


result

H0;qð2Þ ðM;EkÞ ¼ 0 ; k, 1 ; qX 1 :

For G ¼ fIdg, this is implicit in Takegoshi [25]. Moreover, (3.17) shows thatRan ð �@@Þ \ L0; qðM;EkÞ is closed for very large k and qX 1. Therefore, for anyf 2 L0; qðM;EkÞ, qX 1, k, 1, with �@@f ¼ 0, there exists u 2 L0; q�1ðM;EkÞ with�@@u ¼ f . By solving the �@@-equation with singular logarithmic weights, we can showthat the L2 holomorphic sections of Ek, k, 1, separate points ofM and give localcoordinates on M (see [13, Theorem 0.7] for the case when M=G is Stein).(iii) The inequalities (3.12), (3.13) remain true if @M ¼ ;. This implies thatM=G is

a compact manifold and we have obtained estimates for the growth of the vonNeumann dimension of the L2 cohomology analogous to the Morse inequalitiesof Demailly [7]. Note that since @M ¼ ; we need no positivity condition for E nearthe boundary. Corollary 3.9 holds true in this situation too. It can be seen as ageneralisation of a theorem of Napier [19] which shows that if E is positive, thenthere are suf¢ciently many sections in Ek (more precisely, M is Ek-convex).(iv) Let us explain brie£y why it is necessary to work directly with incomplete

metrics. Let M be a relatively compact pseudoconvex domain. If there exists aKa« hler metric near @M, there exists a complete metric on M which is Ka« hler near@M. Indeed, if we denote with d a function which, near @M, equals the distanceto @M, it is known [8, Principal Lemma], that the complex Hessian @ �@@w ofw ¼ � logðdÞ is uniformly bounded from below near @M. By adding e@ �@@w, (e- 1),to a Ka« hler metric we obtain the desired complete metric.In order to apply the method from [26] we need to have an analogue of estimate

(3.6). But, on the one hand, the boundedness from below of the eigenvalues ofcðEÞ with respect to the complete metric just introduced cannot be veri¢ed. Onthe other hand, if we want to get around this dif¢culty by introducing the weightexpð�wÞ ¼ d along the ¢bers of E, we obtain, in degree 0, an L2 cohomology strictlylarger as the one considered in this paper. Therefore, the results in [26] do not implyTheorem 3.8.

4. Weakly 1-Complete Manifolds

In this section we apply the main Theorem 3.8 to the case of weakly 1-completemanifolds and strongly pseudoconvex domains. We consider coverings ofpseudoconvex domains which appear as sublevel sets of a weakly 1-completemanifold ~XX . We denote by j: ~XX�!R the smooth psh exhaustion function and~XXc ¼ fj < cg for c 2 R. Let n ¼ dimX .

THEOREM 4.1. Let X be a complexmanifold which admits a free holomorphic actionof a discrete groupG such that X=G ¼ ~XX is a weakly 1-complete noncompact manifold.Let Xc ¼ p�1ð ~XXcÞ, where p is the covering map. Assume that E is a G-invariant


Hermitian holomorphic line bundle in a neighbourhood of Xc and is positive near @Xc.Then,

limk!1

k�n dimGH0ð2ÞðXc;E

kÞ ¼ 1;

where the L2 condition is considered with respect to any G-invariant metrics on Xc andEXc

.

A similar result for the trivial covering and E an everywhere positive line bundlehas been obtained by S. Takayama [24].

Proof. Consider d > c such that E is de¢ned on Xd ¼ p�1ð ~XXdÞ. We construct ametric of in¢nite volume on ~XXd in the following way. Let ehh be a metric on~EE ¼ E=G which has positive curvature on ~XXd n ~XXc�e where e > 0 is small enough(ehh is the push down of the given metric on E). Leteoo be a metric on ~XXd which coincideswith {cð ~EEÞ on ~XXd n ~XXc�e . Let w: ð�1; dÞ�!R be a convex increasing function suchthat limt!c wðtÞ ¼ 1 (e.g. wðtÞ ¼ ðc� tÞ�2). Then eoo0 ¼ eooþ {@ �@@wðjÞ is a metric of in¢-nite volume if w increases very fast. We consider the new metricehh0 ¼ehh expð�wðjÞÞwith curvature {cð ~EE;ehh0Þ ¼ {cð ~EEÞ þ {@ �@@wðjÞ ¼ eoo0. We take the pull-backs of themetrics eoo0 and ehh0 on Xd and denote them with o0 and h0. We consider apseudoconvex domain Xe ¼ p�1 ~XXe (e regular value of j, c < e < d) and applyTheorem 3.8 for Xe and the metric h0. In particular (3.12) becomes

limkk�n dimGH0

ð2ÞðXe;EkÞX

1n!

Zð ~XXeÞðW 1Þ

� {

2pcð ~EE;ehh0Þ�n :

The restriction morphism

Fk: H0ð2ÞðXe;E

kÞ�!H0ð2ÞðXc;E

kÞ

is clearly an injective G-morphism. Moreover, the L2 condition in the latter spacemay be taken with respect to any G-invariant metrics in the neighbourhood ofXc. Hence,

limkk�n dimGH0

ð2ÞðXc;EkÞX

1n!

Zð ~XXeÞðW 1Þ

� {

2pcð ~EE;ehh0Þ�n : ð4:1Þ

We split the integral in (4.1) into two parts:Zð ~XXeÞðW 1Þ

� {

2pcð ~EE;ehh0�n ¼ Z

ð ~XXcÞðW 1Þ

� {

2pcð ~EE;ehh0Þ�n þ Z

~XXen ~XXc

�eoo02p

�n:

The ¢rst integral is ¢nite and may be negative. The second represents the volume of~XXe n ~XXc in the metric eoo0. Since eoo0 has in¢nite volume on ~XXd the second term goesto in¢nity as e�!d. Therefore, by letting e�!d in (4.1), we obtain the desiredresult. &


Remarks. (i) We can actually prove more. Namely, for each k there exists asequence of G-invariant spaces Lk H0

ð2ÞðXc;EkÞ \ C0;0ðXc;EkÞ such that

k�n dimG Lk½ ��!1: This follows from a variant of the main Theorem 3.8 forcomplete Hermitian manifolds proved in [26]. Indeed, the metric o0 is completeon Xd and cðE; h0Þ ¼ o0 at least outside Xc. Then, by [26], for the L2 cohomologywith respect to o0 and h0 we have limk k�n dimGH0

ð2ÞðXd ;EkÞ ¼ 1 since the

volume of ~XXd is in¢nite. The range Lk of the injective G-morphismH0ð2ÞðXd ;E

kÞ�!H0ð2ÞðXc;E

kÞ is contained in C0;0ðXc;EkÞ and Lk½ � has the samevon Neumann dimension as H0

ð2ÞðXd ;EkÞ.

(ii) Using (3.13) we obtain for qX 1 and k, 1:

dimGH0;qð2Þ ðXc;E

kÞWkn

n!

ZðXc=GÞðqÞ

ð�1Þq� {

2pcðEÞ

�nþ oðknÞ : ð4:2Þ

Let us examine (4.2) for the case G ¼ fIdg. Then by the representation theorem ofTakegoshi ([25], Theorem 6.2), H0;q

ð2Þ ðXc;EkÞ ffi HqðXc;OðEkÞÞ for qX 1 and

k, 1. We thus obtain that the growth of the dimension of the analytic sheafcohomology dimHqðXc;OðEkÞÞ is at most polynomial in kn. This was proved in [17]using the �@@ operator theory without boundary conditions as an answer to a questionof Ohsawa [20].We show next how we can recover some results fromGromov et al. [13]. As before,

we consider a complex manifold X with a free action of a discrete group G and aG-invariant domain M.

PROPOSITION 4.2 ([13]). Assume that M is a strongly pseudoconvex domainsuch that M=G is compact. Then dimG

H0ð2ÞðMÞ \ C

0;0ðMÞ�¼ 1. Moreover,

dimGH0;qð2Þ ðMÞ <1 for qX 1.

Proof. Since ~MM ¼M=G is a compact strongly pseudoconvex domain in X=Gwe may take a de¢ning function ejj for ~MM such that ~MM ¼ fejj < 0g and @ �@@ejj ispositive de¢nite near @ ~MM. This follows immediately from the de¢nition by com-position of a de¢ning function as in de¢nition with an increasing convex function(e.g. t 7 �! expðAtÞ � 1 for large A > 0). Consider e > 0 suf¢ciently small such that~MMe ¼ fejj < eg is also strongly pseudoconvex. Let w: R�!R be a function suchthat wðtÞ ¼ 0 for tW 0 and w0ðtÞ > 0, w00ðtÞ > 0 for t > 0. Then by replacing ejjwith wðejjÞ we can assume that @ �@@ejjX 0 everywhere and @ �@@ejj > 0 near @ ~MMe.Let j be the pull^back of ejj to X and let Me the pre-image of ~MMe by the cover-ing map. Me is a pseudoconvex domain and the trivial bundle E ¼ X �C

endowed with the metric expð�jÞ is semipositive and positive near @Me. ByTheorem 3.8,

dimGH0ð2ÞðMe;EkÞX

kn

n!

Z~MMe

� {

2p@ �@@ejj�n þ oðknÞ ; ð4:3Þ


for k�!1. Since

H0ð2ÞðMe;EkÞ ¼ f : Me�!C :

ZMe

j f j2 expð�kjÞ <1� �

and j < e on Me we see that H0ð2ÞðMe;EkÞ H0

ð2ÞðMeÞ for any k. By (4.3),

dimGH0ð2ÞðMeÞXC kn þ oðknÞ

for some C > 0 : By letting k�!1 we obtain dimGH0ð2ÞðMeÞ ¼ 1. We take L to be

the range of the injective G-morphism of restriction H0ð2ÞðMeÞ�!H0

ð2ÞðMÞ. ThenL C0;0ðMÞ and dimG L½ � ¼ 1 as claimed.To prove the ¢niteness of the G-dimension for qX 1 we use the ¢niteness statement

in Theorem 3.8. By this result we know that dimGH0;qð2Þ ðM;EkÞ <1 for suf¢ciently

large k. We ¢x such a k. The sections from L0; 2ðM;EkÞ are forms with coef¢cientsin the weighted L2 space L2ðM; expð�kjÞÞ. Since jjj is bounded on M we see thatL0; 2ðM;EkÞ ¼ L0;qðMÞ as sets and the two norms are equivalent. HenceH0;qð2Þ ðM;EkÞ ffi H0;q

ð2Þ ðMÞ. &

Proposition 4.2 recovers a part of [13, Theorems 0.1^2], where it is, moreover,proved that any point of @M is a peak point for H0

ð2ÞðMÞ \ C0;0ðMÞ. In [13, Theorem

0.3], it is proved that for any integer N > 0 there exists a G-invariant subspaceL H0

ð2ÞðMÞ \ C0;0ðMÞ such that dimG½L� ¼ N. See also Remark (i) above.

PROPOSITION 4.3 ([13]). LetM be a pseudoconvex domain with holomorphic actionof a discrete groupG on a complex neighbourhood X ofM such thatM is invariant andM=G is compact. Suppose that in a G-invariant neighbourhood of @M there exists aG-invariant smooth strictly psh function j. Then dimGH0

ð2ÞðMÞ ¼ 1 anddimGH

0;qð2Þ ðMÞ <1 for qX 1.

Proof. Suppose thatM ¼ fr < 0g, for a smooth invariant function r. Choose e > 0such that j is strictly psh on f�e < j < eg. Let w: R�!R be a function such thatwðtÞ ¼ 0 for tW � e and w0ðtÞ > 0, w00ðtÞ > 0 for t > �e. Then wðjÞ is a Gînvariantpsh function which is strictly psh near @M. Therefore the trivial bundleE ¼ X �C endowed with the metric expð�wðjÞÞ is semipositive and positive near@M. The conclusion of the Proposition follows now from Theorem 3.8 as in thepreceding proof. &Proposition 4.3 recovers a part of [13, Theorems 0.5^6], where j is allowed to be

non^smooth and it is moreover proved that any point of strong pseudoconvexityin @M is a peak point for H0

ð2ÞðMÞ \ C0;0ðMÞ.

Remark. The results in Sections 3 and 4 are still valid if we replace Ek with Ek Fwhere F is a G-invariant Hermitian holomorphic vector bundle de¢ned on the neigh-bourhood of M. Therefore we can replace in these sections the L2-cohomology forð0; qÞ-forms with the L2 cohomology for ðp; qÞ-forms.


Acknowledgements

We express our hearty thanks to our Professor V. Iftimie and Professors M. Shubinand G. Henkin for sending us the preprint of [13] and for very useful discussions.Special thanks are due to the referee, whose careful reading and suggestions leadto the improvement of the paper.

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L2 Holomorphic Sections of Bundles over Weakly Pseudoconvex Coverings