Holomorphic Morse Inequalities and Bergman Kernelsgmarines/PREPRINTS/mmbook.pdf · holomorphic Morse inequalities of Demailly. They give asymptotic bounds on the Morse sums of the

Holomorphic Morse Inequalities and Bergman Kernels

Xiaonan Ma and George Marinescu

December 2, 2005

2

Contents

1 Introduction 1

2 Demailly’s holomorphic Morse inequalities 72.1 Connections on the tangent bundle . . . . . . . . . . . . . . . . 7

2.1.1 Levi-Civita connection . . . . . . . . . . . . . . . . . . . 72.1.2 Holomorphic Hermitian connection . . . . . . . . . . . . 122.1.3 Bismut connection . . . . . . . . . . . . . . . . . . . . . 15

2.2 Spinc Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . 152.2.1 Clifford connection . . . . . . . . . . . . . . . . . . . . . 152.2.2 Dirac operator and Lichnerowicz formula . . . . . . . . . 182.2.3 Modified Dirac operator . . . . . . . . . . . . . . . . . . 20

2.3 Bismut’s Lichnerowicz formula for (∂E

+ ∂E,∗

)2 . . . . . . . . . 22

2.3.1 The operator ∂E

+ ∂E,∗

. . . . . . . . . . . . . . . . . . 22

2.3.2 Bismut’s Lichnerowicz formula for (∂E

+ ∂E,∗

)2 . . . . . 252.4 Asymptotic of the heat kernel . . . . . . . . . . . . . . . . . . . 25

2.4.1 Statement of the result . . . . . . . . . . . . . . . . . . . 262.4.2 Localization of the problem . . . . . . . . . . . . . . . . 272.4.3 Rescaling of the operator D2

p . . . . . . . . . . . . . . . . 302.4.4 Uniform estimate on the heat kernel . . . . . . . . . . . 322.4.5 Proof of Theorem 2.4.1 . . . . . . . . . . . . . . . . . . . 37

2.5 Demailly’s holomorphic Morse Inequalities . . . . . . . . . . . . 372.5.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . . 382.5.2 A heat kernel proof of Theorem 2.5.1 . . . . . . . . . . . 38

2.6 Bibliographic notes . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Applications and Extensions of the Holomorphic Morse Inequalities 413.1 The Siu–Demailly Criterion . . . . . . . . . . . . . . . . . . . . 41

3.1.1 Big line bundles . . . . . . . . . . . . . . . . . . . . . . . 423.1.2 Moishezon manifolds . . . . . . . . . . . . . . . . . . . . 45

3.2 Abstract Morse inequalities for the L2–cohomology . . . . . . . 483.2.1 L2–cohomology and Hodge theory . . . . . . . . . . . . . 483.2.2 The Fundamental Estimate . . . . . . . . . . . . . . . . 513.2.3 Asymptotic distribution of eigenvalues . . . . . . . . . . 543.2.4 Morse Inequalities for the L2 cohomology . . . . . . . . . 60

ii Contents

3.3 Uniformly positive line bundles . . . . . . . . . . . . . . . . . . 633.3.1 A general cohomology estimate . . . . . . . . . . . . . . 64

3.4 Siu-Demailly criterion for isolated singularities . . . . . . . . . . 663.4.1 Hyperconcave manifolds . . . . . . . . . . . . . . . . . . 663.4.2 Proof of Theorem 3.4.1 . . . . . . . . . . . . . . . . . . . 70

3.5 Holomorphic Morse Inequalities for q-convex manifolds . . . . . 733.6 Holomorphic Morse Inequalities for Coverings . . . . . . . . . . 76

3.6.1 Covering manifolds, von Neumann dimension . . . . . . 763.6.2 Holomorphic Morse inequalities . . . . . . . . . . . . . . 78


4 Asymptotic Expansion of the Bergman Kernel 814.1 Spectral gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.1.1 Spectral gap of modified Dirac operators . . . . . . . . . 824.1.2 Spectral gap of the Kodaira-Laplacian 2p . . . . . . . . 83

4.2 Near diagonal expansion of Bergman kernel . . . . . . . . . . . 844.2.1 Diagonal asymptotic expansion of Bergman kernel . . . . 844.2.2 Localization of the problem . . . . . . . . . . . . . . . . 864.2.3 Rescaling and Taylor expansion of the operator D2

p . . . 87

4.2.4 Sobolev estimate on the resolvent (λ− Lt2)−1 . . . . . . . 90

4.2.5 Uniform estimate on the Bergman kernel . . . . . . . . . 944.2.6 Bergman kernel of L . . . . . . . . . . . . . . . . . . . . 974.2.7 Proof of Theorem 4.2.1 . . . . . . . . . . . . . . . . . . . 994.2.8 The coefficient b1: a proof of Theorem 4.2.2 . . . . . . . 1014.2.9 Proof of Theorem 4.2.3 . . . . . . . . . . . . . . . . . . . 103

4.3 Off-diagonal expansion of the Bergman kernel . . . . . . . . . . 1034.3.1 From heat kernel to Bergman kernel . . . . . . . . . . . 1044.3.2 Uniform estimate on the heat kernel and the Bergman

kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.3.3 Proof of Theorems 4.3.1 . . . . . . . . . . . . . . . . . . 1094.3.4 Proof of Theorems 4.3.2 . . . . . . . . . . . . . . . . . . 110


5 Several Applications of the Asymptotic Expansion 1135.1 The Kodaira Embedding Theorem . . . . . . . . . . . . . . . . . 1135.2 Generalizations to non-compact manifolds . . . . . . . . . . . . 117

5.2.1 Complete Hermitian mnaifolds . . . . . . . . . . . . . . . 1175.2.2 Covering Manifolds . . . . . . . . . . . . . . . . . . . . . 120

5.3 Singular polarizations and the Shiffman - Ji - Bonavero - TakayamaCriterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.3.1 Singular Hermitian metrics on line bundles . . . . . . . . 1215.3.2 Bergman kernel and the generalized Poincare metric . . . 1245.3.3 The singular holomorphic Morse inequalities of Bonavero 1295.3.4 Some examples of Moishezon manifolds . . . . . . . . . . 133

5.4 Compactification of manifolds with pinched negative curvature . 133

Contents iii

5.5 Toeplitz Operators . . . . . . . . . . . . . . . . . . . . . . . . . 1375.6 Weak Lefschetz theorems . . . . . . . . . . . . . . . . . . . . . . 1415.7 Bibliographic notes . . . . . . . . . . . . . . . . . . . . . . . . . 143

6 Bergman Kernels on symplectic manifolds 1456.1 Bergman Kernels of modified Dirac operators . . . . . . . . . . 145

A Elements of analytic and hermitian geometry 149A.1 Line bundles and divisors . . . . . . . . . . . . . . . . . . . . . . 149A.2 Currents on complex manifolds . . . . . . . . . . . . . . . . . . 150A.3 Hermitian geometry . . . . . . . . . . . . . . . . . . . . . . . . . 153A.4 Pseudoconvex and pseudoconcave manifolds . . . . . . . . . . . 158

B Spectral analysis of self-adjoint operators 163

C Heat kernel and finite propagation speed 169C.1 Heat kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169C.2 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

D Harmonic oscillator 175D.1 Harmonic oscillator on R . . . . . . . . . . . . . . . . . . . . . . 175D.2 Harmonic oscillator on vector spaces . . . . . . . . . . . . . . . 178

iv Contents

Chapter 1

Introduction

Let X be a compact complex manifold and let L be a holomorphic line bundleon X, and let Hq(X, O(L)) be the q-th cohomology group of the sheaf ofholomorphic sections of L on X.

Many important results in algebraic and complex geometry are derived bycombining a vanishing with an index theorem, or from the asymptotic resultson the tensor power Lp when p →∞. One of the most famous examples is Ko-daira vanishing theorem which asserts that if L is positive, then Hq(X, O(Lp))vanish for q > 1 and large p. The key remark is that the spectrum of theKodaira-Laplace operator 2p acting on (0, q)-forms, q > 1, with values inthe tensor powers Lp, shifts to the right linearly in the tensor power p. As aconsequence the kernel of 2p is trivial on forms of higher bidigree and the van-ishing theorem follows by Hodge theory and Dolbeault isomorphism. Moreover,Riemann-Roch-Hirzebruch Theorem implies that Lp has a lot of holomorphicsections on X for large p, which indeed embed the manifold X in a projectivespace.

An important generalization which we will emphasize are the asymptoticholomorphic Morse inequalities of Demailly. They give asymptotic bounds onthe Morse sums of the ∂ Betti numbers dim Hq(X, O(Lp)) in terms of certainintegrals of the curvature form of L. The holomorphic Morse inequalities pro-vide a useful tool in complex geometry. They are again based on the asymptoticspectral behavior of the Kodaira-Laplace operator 2p for large p.

The applications of vanishing theorems and holomorphic Morse inequali-ties are numerous. Let us mention here only the Kodaira embedding theorem,the classical Lefschetz hyperplane theorem for projective manifolds, the com-putation of the asymptotics of the Ray-Singer analytic torsion by Bismut andVasserot, as well as the solution of the Grauert-Riemenschneider conjectureby Siu and Demailly or the compactification of complete Kahler manifolds ofnegative Ricci curvature by Nadel and Tsuji. Donaldson’s work on symplecticsub-manifolds is also inspired from here.

The holomorphic Morse inequalities are global statements which can bededuced from local informations such as the behavior of the heat or Bergmankernels. In this refined form we find the asymptotic expansion of the Bergman

2 Chapter 1. Introduction

kernel of Lp as p → ∞ which had a tremendous impact on the research inthe last years. Especially, let’s single out its applications in Donaldson’s pro-gram on the existence of Kahler metrics with constant scalar curvature in rela-tion to Mumford-Chow stability. Other applications include convergence of theinduced Fubini-Study metric, distribution of zeroes of random and quantumpolynomials or sections, Berezin-Toeplitz quantization and sampling problems.

An other important operator which we will study, also in view of thegeneralization to symplectic manifolds, is the Dirac operator acting on hightensor powers of L on symplectic manifolds. For a Kahler manifold the twiceof the Kodaira-Laplacian is the square of the Dirac operator.

In the present book we will give a self-contained and unified approachof the holomorphic Morse inequalities and the asymptotic expansion of theBergman kernel by using the heat kernel, and we present also various appli-cations. Our point of view is from the local index theory, especially from theanalytic localization techniques developed by Bismut-Lebeau. Basically, theholomorphic Morse inequalities are a consequence of the small time asym-ptotic expansion of the heat kernel. The Bergman kernel corresponds to thelimit of the heat kernel when time goes to infinity, and is more sophisticate.A simple principle in this book is that the existence of the spectral gap of theoperators implies the existence of the asymptotic expansion of the correspond-ing Bergman kernel if the manifold X is compact or not, or singular, or withboundary. Moreover, we will present a general and algorithmic way to computethe coefficients in the expansion.

Let us now give a rapid account of the main results discussed in our book.The first chapter is dedicated to Demailly’s holomorphic Morse inequali-

ties which originally arose in connection with the generalization of the Kodairavanishing theorem for Moishezon manifolds proposed by Grauert and Riemen-schneider. They conjectured that a compact complex manifold X possessinga semi-positive line bundle L which is positive at one point is Moishezon.The conjecture was solved by Siu and Demailly. The solution of Demailly [41]involves the following strong Morse inequalities,

q∑j=0

(−1)q−j dim Hj(X, O(Lp)) 6pn

n!

∫X(6q)

(−1)q(√

−12π

RL)n

+ o(pn) (1.1)

as p −→ ∞. Here X(6 q) is the set of points where RL the curvature of Lis non-degenerate and RL ∈ End(T (1,0)X), defined by RL(u, v) = gTX(RLu, v)for u, v ∈ T (1,0)X (cf. (2.4.2)), has at most q negative eigenvalues. For q = n wehave equality, so we obtain an asymptotic Riemann-Roch-Hirzebruch formula.

Demailly’s discovery was triggered by Witten’s influential analytic proofof the standard Morse inequalities [133]. Witten analyzes the spectrum of theSchrodinger operator ∆t = ∆ + t2|df |2 + tV , where t > 0 is a real parameter,∆ is the Bochner-Laplacian acting on forms on X, f is a Morse function on Xand V is a 0-order operator. For t −→∞, the spectrum of ∆t approaches thespectrum of a sum of harmonic oscillators attached to the critical points of f .

3

In Demailly’s holomorphic Morse inequalities the role of the Morse functionis played by the Hermitian metric on the line bundle and the Hessian of theMorse function becomes the curvature of the bundle. The original proof wasbased on the study of the semi-classical behavior as p → ∞ of the spectralcounting functions of the Bochner-Laplacians 2p. Heat equation proofs weresubsequently given by Bismut [13], Demailly [43] and Bouche [26].

We present here a new approach based on the asymptotic of the heat kernelof the square of the Dirac operator exp(−u

pD2

p). For this purpose we give in

Sections 2.1-2.3 a self-contained account of the connections on the tangentbundle, Dirac operator and Lichnerowicz formula. The analytic core follows inSection 2.4 where, inspired by the work of Bismut-Lebeau [17, §11], we presenta new proof for the asymptotic of the heat kernel. In Section 2.5 we apply theseresult to obtain a heat equation proof of the holomorphic Morse inequalitiesfollowing Bismut [13].

Chapter 3 starts with the proof of the Siu-Demailly criterion which answersthe Grauert-Riemenschneider conjecture. For q = 1 the Morse inequalities (1.1)give

dim H0(X, O(Lp)) >pn

n!

∫X(61)

(√−12π

RL)n

+ o(pn) , p −→∞ . (1.2)

Therefore if L satisfies ∫X(61)

(√−12π

RL)n

> 0 , (1.3)

(in particular, if L is semi-positive and positive at one point), there are a lotof sections in H0(X, O(Lp)), which by taking quotients deliver n independentmeromorphic functions, i.e. X is Moishezon. In the sequel we prove the Morseinequalities for the Dolbeault L2–cohomology spaces for a non-compact ma-nifold satisfying the fundamental estimate (Poincare inequality) at infinity.Using this more abstract formulation of the Morse inequalities we can find alower bound for the growth of holomorphic section space for uniformly positiveline bundles (Theorem 3.3.1) and an extension of the Siu-Demailly criterionfor compact complex spaces with isolated singularities.

We end the chapter with a study of a class of manifolds satisfying pseudo-convexity conditions in the sense of Andreotti-Grauert, namely q-convex andweakly 1-complete manifolds and also covering manifolds. Pseudoconvex man-ifolds are very important in complex geometry and analysis.

In Chapter 4 we study the asymptotic expansion of the Bergman kernel.We assume now L is positive, equivalently, let hL be a Hermitian metric onL and let RL be the curvature of the holomorphic Hermitian connection ∇L

on (L, hL), we assume that ω =√−12π

RL defines a Kahler form on X. LetgTX be the associated Kahler metric on TX. Let E be a holomorphic vectorbundle on X with a Hermitian metric hE. By Kodaira vanishing theorem,Hq(X, O(Lp ⊗ E)) = 0 for p large enough and q ≥ 1. The Bergman kernel Pp

of Lp⊗E for p large enough, is the smooth kernel of the orthogonal projection


from C∞(X,Lp ⊗ E), the space of smooth sections of tensor powers Lp ⊗ E,on the space of holomorphic sections of Lp⊗E, or, equivalently, on the kernel

of the Kodaira-Laplacian 2p on Lp ⊗ E. More precisely, let {Spi }

dp

i=1 be anyorthonormal basis of H0(X, O(Lp⊗E)) with respect to the global inner productinduced by gTX , hL and hE. Then for p large enough,

Pp(x, x′) =

dp∑i=1

Spi (x)⊗ (Sp

i (x))∗ ∈ (Lp ⊗ E)x ⊗ (Lp ⊗ E)∗x′ , (1.4)

Pp(x, x) =

dp∑i=1

|Spi (x)|2, if E = C. (1.5)

The Bergman kernel is studied in Tian [129], Ruan [108], Zelditch [134], Catlin[34], Bleher-Shiffman-Zelditch [19], Z. Lu [80] in various generalities, establish-ing the asymptotic expansion for high powers of L. Moreover, the coefficients inthe asymptotic expansion encode geometric information about the underlyingcomplex projective manifolds.

Our approach continues the method applied in Chapter 2. Moreover, wetreat both the Dirac operator and the Kodaira-Laplacian in the same time bymeans of the modified Dirac operator. The key point in our approach is thatfor Spec 2p, the spectrum of 2p (or the square of the modified Dirac operator),there exist µ0, CL > 0 such that for p ≥ 1

Spec 2p ⊂ {0}∪ ]2µ0p− CL, +∞[. (1.6)

Now the spectral gap property (1.6) and the finite propagation speed of solu-tions of hyperbolic equations allows first to localize our problem on the asym-ptotic of Pp(x0, x

′) to a problem in the neighborhood of x0, thus to work onR2n, and the extended operator on R2n herites also the spectral gap property.Now by combining the spectral gap property, the rescaling on the coordinatesand the functional analysis techniques, we can conclude our final result. Thuswe obtain the following asymptotic expansion (cf. Theorem 4.2.2):

Pp(x, x) ∼∞∑

r=0

br(x)pn−r, (1.7)

where br(x) ∈ End(E)x are smooth coefficients, which are polynomials in RTX ,RE and their derivatives with order ≤ 2r − 1. Moreover

b0 = IdE, b1 =1

4π

[2RE(wj, wj) +

1

2rX IdE

]. (1.8)

where rX is the scalar curvature of (TX, gTX) and {wj}nj=1 is an orthonormal

base of T (1,0)X. In the case of trivial bundle E the term b1 was calculated by Lu[80] and used by Donaldson [50] in his work on the existence of Kahler metricswith constant scalar curvature in relation to the Hilbert-Chow stability.

5

We go on in Sections 4.2 and 4.3 and find the full off-diagonal expansion ofthe Bergman kernel Pp(x, x′). We use the methods from Dai-Liu-Ma [37] wherethe asymptotic of the Bergman kernel of the Dirac operators was related tothe asymptotic of the heat kernel (cf.Theorem 4.3.2), and also from [84] wherewe worked on the generalized Bergman kernel associated to the renormalizedBochner-Laplacian.

In Chapter 5 we present some applications of the asymptotic expansion.First, it provides an analytic proof of the Kodaira embedding theorem initi-ated by Bouche and Shiffman-Zelditch. Secondly, we can establish the existenceof the expansion on compact sets of a non-compact manifold, as long as thespectral gap exists. One interesting situation is the case of Zariski open setsin compact complex spaces endowed with the generalized Poincare metric.The expansion of the Bergman metric implies a new proof of the Shiffman-Ji-Bonavero criterion for a Moishezon manifold. Thirdly, we obtain again Morseinequalities which are suitable for the study of the compactification of mani-folds with pinched negative curvature. Finally, using the full off-diagonal ex-pansion of the Bergman kernel we study the properties of the Toeplitz operatorsand Toeplitz-Berezin quantization.

In Chapter 6, we find the asymptotic expansion of the Bergman kernelassociated to the modified Dirac operator and the renormalized Bochner-Laplacian, as well as their applications.

We hope the material of this book can also be used by gratuated students.To help the readers, we add four appendices. In the appendix A, we presentsome basic facts on Hermitian geometry. In appendix B, we collect some factson the self-adjointness of an operator. In appendix C, we explain in detailthe relation of the heat kernel and the finite propagation speed of solutionsof hyperbolic equations. In appendix D, we find the basic facts about theharmonic oscilator.

The book should also serve as analytic introduction to the applicationsto algebraic geometry of the holomorphic Morse inequalities as developed byDemailly and his school, as well as to Donaldson’s program about the existenceof Kahler metrics of constant scalar curvature. We refer the reader to therecent book of Gauduchon [54] for an introduction to the geometric side ofthis program.

The present manuscript is just the first draft of our book. We will continu-ously complete and polish our presentation. We trust the reader’s benevolence.

Notation : In the whole book, if there is no other specific notification,when in a formula a subscript index appears two times, then we sum up withthis index.


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