Holomorphic Morse Inequalities And Bergman Kernels

Progress in Mathematics Volume 254

Series Editors H. Bass J. Oesterlé A. Weinstein

Xiaonan Ma George Marinescu

Holomorphic Morse Inequalities and Bergman Kernels

Birkhäuser Basel xBoston xBerlin

Authors: Xiaonan Ma Centre de Mathématiques Laurent Schwartz (C.M.L.S.) École Polytechnique 91128 Palaiseau Cedex France e-mail: [email protected]

George Marinescu Mathematisches Institut Universität zu Köln :H\HUWDO± .|OQ Germany e-mail: [email protected]

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Library of Congress Control Number : 2007922259

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Ferran Sunyer i Balaguer (1912–1967) was a selftaught Catalan mathematician who, in spite of a serious physical disability, was very active in research in classical mathematical analysis, an area in which he acquired international recognition. His heirs created the Fundaci´ o Ferran Sunyer i Balaguer inside the Institut d’Estudis Catalans to honor the memory of Ferran Sunyer i Balaguer and to promote mathematical research. Each year, the Fundaci´ o Ferran Sunyer i Balaguer and the Institut d’Estudis Catalans award an international research prize for a mathematical monograph of expository nature. The prize-winning monographs are published in this series. Details about the prize and the Fundaci´ o Ferran Sunyer i Balaguer can be found at http://ffsb.iec.cat This book has been awarded the Ferran Sunyer i Balaguer 2006 prize. The members of the scientific commitee of the 2006 prize were: Antonio C´ ordoba Universidad Aut´ onoma de Madrid Paul Malliavin Universit´e de Paris VI Joseph Oesterl´e Universit´e de Paris VI Oriol Serra Universitat Polit`ecnica de Catalunya, Barcelona Alan Weinstein University of California at Berkeley

Ferran Sunyer i Balaguer Prize winners since 1997: 1997

Albrecht B¨ottcher and Yuri I. Karlovich Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators, PM 154

1998

Juan J. Morales-Ruiz Differential Galois Theory and Non-integrability of Hamiltonian Systems, PM 179

1999

Patrick Dehornoy Braids and Self-Distributivity, PM 192

2000

Juan-Pablo Ortega and Tudor Ratiu Hamiltonian Singular Reduction, PM 222

2001

Martin Golubitsky and Ian Stewart The Symmetry Perspective, PM 200

2002

Andr´e Unterberger Automorphic Pseudodifferential Analysis and Higher Level Weyl Calculi, PM 209 Alexander Lubotzky and Dan Segal Subgroup Growth, PM 212

2003

Fuensanta Andreu-Vaillo, Vincent Caselles and Jos´e M. Maz´on Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, PM 223

2004

Guy David Singular Sets of Minimizers for the Mumford-Shah Functional, PM 233

2005

Antonio Ambrosetti and Andrea Malchiodi Perturbation Methods and Semilinear Elliptic Problems on Rn , PM 240 Jos´e Seade On the Topology of Isolated Singularities in Analytic Spaces, PM 241

2006

Xiaonan Ma and George Marinescu Holomorphic Morse Inequalities and Bergman Kernels, PM 254

To Ling and Cristina

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Demailly’s Holomorphic Morse Inequalities 1.1 Connections on vector bundles . . . . . . . . . . . . . . . . . . 1.1.1 Hermitian connection . . . . . . . . . . . . . . . . . . . 1.1.2 Chern connection . . . . . . . . . . . . . . . . . . . . . . 1.2 Connections on the tangent bundle . . . . . . . . . . . . . . . . 1.2.1 Levi–Civita connection . . . . . . . . . . . . . . . . . . . 1.2.2 Chern connection . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Bismut connection . . . . . . . . . . . . . . . . . . . . . 1.3 Spinc Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Clifford connection . . . . . . . . . . . . . . . . . . . . . 1.3.2 Dirac operator and Lichnerowicz formula . . . . . . . . 1.3.3 Modified Dirac operator . . . . . . . . . . . . . . . . . . 1.3.4 Atiyah–Singer index theorem . . . . . . . . . . . . . . . 1.4 Lichnerowicz formula for
E . . . . . . . . . . . . . . . . . . . E E,∗ . . . . . . . . . . . . . . . . . . 1.4.1 The operator ∂ + ∂ 1.4.2 Bismut’s Lichnerowicz formula for
E . . . . . . . . . . 1.4.3 Bochner–Kodaira–Nakano formula . . . . . . . . . . . . 1.4.4 Bochner–Kodaira–Nakano formula with boundary term 1.5 Spectral gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Vanishing theorem and spectral gap . . . . . . . . . . . 1.5.2 Spectral gap of modified Dirac operators . . . . . . . . . 1.6 Asymptotic of the heat kernel . . . . . . . . . . . . . . . . . . . 1.6.1 Statement of the result . . . . . . . . . . . . . . . . . . 1.6.2 Localization of the problem . . . . . . . . . . . . . . . . 1.6.3 Rescaling of the operator Dp2 . . . . . . . . . . . . . . . 1.6.4 Uniform estimate on the heat kernel . . . . . . . . . . . 1.6.5 Proof of Theorem 1.6.1 . . . . . . . . . . . . . . . . . .

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1

9 9 11 12 13 17 21 21 22 24 26 28 29 30 34 35 40 43 43 47 49 49 50 53 55 60

x

Contents

1.7 1.8

Demailly’s holomorphic Morse inequalities . . . . . . . . . . . . . . Bibliographic notes . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 66

2 Characterization of Moishezon Manifolds 2.1 Line bundles, divisors and blowing-up . . . . . . . . . . . 2.2 The Siu–Demailly criterion . . . . . . . . . . . . . . . . . 2.2.1 Big line bundles . . . . . . . . . . . . . . . . . . . 2.2.2 Moishezon manifolds . . . . . . . . . . . . . . . . . 2.3 The Shiffman–Ji–Bonavero–Takayama criterion . . . . . . 2.3.1 Singular Hermitian metrics on line bundles . . . . 2.3.2 Bonavero’s singular holomorphic Morse inequalities 2.3.3 Volume of big line bundles . . . . . . . . . . . . . . 2.3.4 Some examples of Moishezon manifolds . . . . . . 2.4 Algebraic Morse inequalities . . . . . . . . . . . . . . . . . 2.5 Bibliographic notes . . . . . . . . . . . . . . . . . . . . . .

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69 80 80 84 97 97 101 111 117 121 124

3 Holomorphic Morse Inequalities on Non-compact Manifolds 3.1 L2 -cohomology and Hodge theory . . . . . . . . . . . . 3.2 Abstract Morse inequalities for the L2 -cohomology . . 3.2.1 The fundamental estimate . . . . . . . . . . . . 3.2.2 Asymptotic distribution of eigenvalues . . . . . 3.2.3 Morse inequalities for the L2 cohomology . . . 3.3 Uniformly positive line bundles . . . . . . . . . . . . . 3.4 Siu–Demailly criterion for isolated singularities . . . . 3.5 Morse inequalities for q-convex manifolds . . . . . . . 3.6 Morse inequalities for coverings . . . . . . . . . . . . . 3.6.1 Covering manifolds, von Neumann dimension . 3.6.2 Holomorphic Morse inequalities . . . . . . . . . 3.7 Bibliographic notes . . . . . . . . . . . . . . . . . . . .

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127 134 134 137 144 148 152 160 167 167 169 173

4 Asymptotic Expansion of the Bergman Kernel 4.1 Near diagonal expansion of the Bergman kernel . . . . . . . . 4.1.1 Diagonal asymptotic expansion of the Bergman kernel 4.1.2 Localization of the problem . . . . . . . . . . . . . . . 4.1.3 Rescaling and Taylor expansion of the operator Dp2 . . 4.1.4 Sobolev estimate on the resolvent (λ − L2t )−1 . . . . . 4.1.5 Uniform estimate on the Bergman kernel . . . . . . . 4.1.6 Bergman kernel of L . . . . . . . . . . . . . . . . . . 4.1.7 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . 4.1.8 The coefficient b1 : a proof of Theorem 4.1.2 . . . . . . 4.1.9 Proof of Theorem 4.1.3 . . . . . . . . . . . . . . . . .

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175 176 178 179 183 187 189 191 194 196

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Contents

4.2

xi

Off-diagonal expansion of the Bergman kernel 4.2.1 From heat kernel to Bergman kernel . 4.2.2 Uniform estimate on the heat kernel and the Bergman kernel . . . . . . . . 4.2.3 Proof of Theorem 4.2.1 . . . . . . . . 4.2.4 Proof of Theorem 4.2.3 . . . . . . . . Bibliographic notes . . . . . . . . . . . . . . .

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199 204 204 209

5 Kodaira Map 5.1 The Kodaira embedding theorem . . . . . . . . . . . . . . 5.1.1 Universal bundles . . . . . . . . . . . . . . . . . . . 5.1.2 Convergence of the induced Fubini–Study metrics . 5.1.3 Classical proof of the Kodaira embedding theorem 5.1.4 Grassmannian embedding . . . . . . . . . . . . . . 5.2 Stability and Bergman kernel . . . . . . . . . . . . . . . . 5.2.1 Extremal K¨ ahler metrics . . . . . . . . . . . . . . . 5.2.2 Scalar curvature and projective embeddings . . . . 5.2.3 Gieseker stability and Grassmannian embeddings . 5.3 Distribution of zeros of random sections . . . . . . . . . . 5.4 Orbifold projective embedding theorem . . . . . . . . . . 5.4.1 Basic definitions on orbifolds . . . . . . . . . . . . 5.4.2 Complex orbifolds . . . . . . . . . . . . . . . . . . 5.4.3 Asymptotic expansion of the Bergman kernel . . . 5.4.4 Projective embedding theorem . . . . . . . . . . . 5.5 The asymptotic of the analytic torsion . . . . . . . . . . . 5.5.1 Mellin transformation . . . . . . . . . . . . . . . . 5.5.2 Definition of the analytic torsion . . . . . . . . . . 5.5.3 Anomaly formula . . . . . . . . . . . . . . . . . . . 5.5.4 The asymptotics of the analytic torsion . . . . . . 5.5.5 Asymptotic anomaly formula for the L2 -metric . . 5.5.6 Uniform asymptotic of the heat kernel . . . . . . . 5.6 Bibliographic notes . . . . . . . . . . . . . . . . . . . . . .

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211 212 213 219 221 224 224 226 229 231 235 235 238 239 246 250 250 251 253 256 260 262 269

6 Bergman Kernel on Non-compact Manifolds 6.1 Expansion on non-compact manifolds . . . . . . . . . . . 6.1.1 Complete Hermitian manifolds . . . . . . . . . . 6.1.2 Covering manifolds . . . . . . . . . . . . . . . . . 6.2 The Shiffman–Ji–Bonavero–Takayama criterion revisited 6.3 Compactification of manifolds . . . . . . . . . . . . . . . 6.3.1 Filling strongly pseudoconcave ends . . . . . . . 6.3.2 The compactification theorem . . . . . . . . . . .

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271 271 275 276 281 281 286

4.3

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xii

Contents

6.4 6.5

Weak Lefschetz theorems . . . . . . . . . . . . . . . . . . . . . . . 290 Bibliographic notes . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

7 Toeplitz Operators 7.1 Kernel calculus on Cn . . . . . . . . . . . . . 7.2 Asymptotic expansion of Toeplitz operators . 7.3 A criterion for Toeplitz operators . . . . . . . 7.4 Algebra of Toeplitz operators . . . . . . . . . 7.5 Toeplitz operators on non-compact manifolds 7.6 Bibliographic notes . . . . . . . . . . . . . . .

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295 298 302 310 312 314

8 Bergman Kernels on Symplectic Manifolds 8.1 Bergman kernels of modified Dirac operators . . . . 8.1.1 Asymptotic expansion of the Bergman kernel 8.1.2 Toeplitz operators on symplectic manifolds . 8.2 Bergman kernel: mixed curvature case . . . . . . . . 8.2.1 Spectral gap . . . . . . . . . . . . . . . . . . 8.2.2 Asymptotic expansion of the Bergman kernel 8.3 Generalized Bergman kernel . . . . . . . . . . . . . . 8.3.1 Spectral gap . . . . . . . . . . . . . . . . . . 8.3.2 Generalized Bergman kernel . . . . . . . . . . 8.3.3 Near diagonal asymptotic expansion . . . . . 8.3.4 The second coefficient b1 . . . . . . . . . . . 8.3.5 Symplectic Kodaira embedding theorem . . . 8.4 Bibliographic notes . . . . . . . . . . . . . . . . . . .

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315 316 318 320 320 322 324 324 327 328 333 338 343

A Sobolev Spaces A.1 Sobolev spaces on Rn . . . . . . . . . . . . . . . . . . . . . . . . . 345 A.2 Sobolev spaces on Rn+ . . . . . . . . . . . . . . . . . . . . . . . . . 348 A.3 Sobolev spaces on manifolds . . . . . . . . . . . . . . . . . . . . . . 348 B Elements of Analytic and Hermitian Geometry B.1 Analytic sets and complex spaces . . . . . B.2 Currents on complex manifolds . . . . . . B.3 q-convex and q-concave manifolds . . . . . B.4 L2 estimates for ∂ . . . . . . . . . . . . . B.5 Chern-Weil theory . . . . . . . . . . . . .

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351 354 362 365 369

C Spectral Analysis of Self-adjoint Operators C.1 Quadratic forms and Friedrichs extension . . . . . . . . . . . . . . 375 C.2 Spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 C.3 Variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . 381

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xiii

D Heat Kernel and Finite Propagation Speed D.1 Heat kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 D.2 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 E Harmonic Oscillator E.1 Harmonic oscillator on R . . . . . . . . . . . . . . . . . . . . . . . 393 E.2 Harmonic oscillator on vector spaces . . . . . . . . . . . . . . . . . 397 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

Introduction Let X be a compact complex manifold and L be a holomorphic line bundle on X. We denote by H q (X, L) the qth cohomology group of the sheaf of holomorphic sections of L on X. Many important results in algebraic and complex geometry are derived by combining a vanishing property with an index theorem, or from the asymptotic results on the tensor powers Lp when p → ∞. One of the most famous examples is the Kodaira–Serre vanishing theorem which asserts that if L is positive, then H q (X, Lp ) vanish for q  1 and large p. The key remark is that the spectrum of the Kodaira–Laplace operator
p acting on (0, q)-forms, q  1, with values in the tensor powers Lp , shifts to the right linearly in the tensor power p. As a consequence the kernel of
p is trivial on forms of higher degree and the vanishing theorem follows by the Hodge theory and the Dolbeault isomorphism. Moreover, the Riemann–Roch–Hirzebruch theorem implies that Lp has a lot of holomorphic sections on X for large p, which indeed embed the manifold X in a projective space. An important generalization which we will emphasize is the asymptotic holomorphic Morse inequalities of Demailly. They give asymptotic bounds on the Morse sums of the ∂-Betti numbers dim H q (X, Lp ) in terms of certain integrals of the curvature form of L. The holomorphic Morse inequalities provide a useful tool in complex geometry. They are again based on the asymptotic spectral behavior of the Kodaira–Laplace operator
p for large p. The applications of these vanishing theorems and holomorphic Morse inequalities are numerous. Let us mention here only the Kodaira embedding theorem, the classical Lefschetz hyperplane theorem for projective manifolds, the computation of the asymptotics of the Ray-Singer analytic torsion by Bismut and Vasserot, as well as the solution of the Grauert–Riemenschneider conjecture by Siu and Demailly or the compactification of complete K¨ahler manifolds of negative Ricci curvature by Nadel and Tsuji. Donaldson’s work on the existence of symplectic submanifolds was inspired by the same circle of ideas. The holomorphic Morse inequalities are global statements which can be deduced from local information such as the behavior of the heat or Bergman kernels. In this refined form we can establish the asymptotic expansion of the Bergman kernel associated to Lp as p → ∞, which have had a tremendous impact on research in

2

Introduction

the last years. Especially, let’s single out its applications in Donaldson’s approach to the existence of K¨ahler metrics with constant scalar curvature in relation to the Mumford–Chow stability which was mainly motivated by a conjecture of Yau. Other applications include the convergence of the induced Fubini–Study metrics, the distribution of zeroes of random sections, the Berezin–Toeplitz quantization and sampling problems. Another important operator which we will study, also in view of the generalization to symplectic manifolds, is the Dirac operator acting on high tensor powers of L on symplectic manifolds. For a K¨ ahler manifold the square of the Dirac operator is twice the Kodaira Laplacian. In the present book we will give for the first time a self-contained and unified treatment to the holomorphic Morse inequalities and the asymptotic expansion of the Bergman kernel by using heat kernels, and we present also various applications. Our point of view comes from the local index theory, especially from the analytic localization techniques developed by Bismut–Lebeau. Basically, the holomorphic Morse inequalities are a consequence of the small time asymptotic expansion of the heat kernel. The Bergman kernel corresponds to the limit of the heat kernel when the time parameter goes to infinity, and the asymptotic is more sophisticated. A simple principle in this book is that the existence of the spectral gap of the operators implies the existence of the asymptotic expansion of the corresponding Bergman kernel, no matter if the manifold X is compact or not, or singular, or with boundary. Moreover, we will present a general and algorithmic way to compute the coefficients of the expansion. Let us now give a rapid account of the main results discussed in this book. In the first chapter we introduce the basic material. After giving a selfcontained presentation of the connections on the tangent bundle, Dirac operator and Lichnerowicz formula, we specify them for the Kodaira Laplacian, especially we study in detail the Bochner–Kodaira–Nakano formula without and with boundary term. These various formulas are fundamental and have a lot of applications. We will use them repeatedly throughout the text. As a direct application, we establish immediately classical vanishing results and the spectral gap property for Kodaira Laplacians and modified Dirac operators. The latter will play an essential role in our approach to the asymptotic expansion of Bergman kernel. The last two sections of this chapter are dedicated to Demailly’s holomorphic Morse inequalities. They originally arose in connection with the generalization of the Kodaira vanishing theorem for Moishezon manifolds proposed by Grauert and Riemenschneider, who conjectured that a compact connected complex manifold X possessing a semi-positive line bundle L, which is positive at at least one point, is Moishezon. The conjecture was solved by Siu and Demailly. The solution of Demailly involves the following strong Morse inequalities:  q √ n
pn −1 L (−1)q−j dim H j (X, Lp )  (−1)q 2π R + o(pn ) (1) n! X(
q) j=0

Introduction

3

as p −→ ∞, where RL is the curvature of L (cf. (1.5.15)), and X( q) is the set of points where R˙ L ∈ End(T (1,0) X), defined by RL (u, v) = g T X (R˙ L u, v) for u, v ∈ T (1,0) X and a Riemannian metric g T X on T X, is non-degenerate and has at most q negative eigenvalues. For q = n we have equality, so we obtain an asymptotic Riemann–Roch–Hirzebruch formula. Demailly’s discovery was triggered by Witten’s influential analytic proof of the standard Morse inequalities. Witten analyzes the spectrum of the Schr¨ odinger 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 X and V is a 0-order operator. For t −→ ∞, the spectrum of ∆t approaches the spectrum of a sum of harmonic oscillators attached to the critical points of f . In Demailly’s holomorphic Morse inequalities, the role of the Morse function is played by the Hermitian metric on the line bundle and the Hessian of the Morse function becomes the curvature of the bundle. The original proof was based on the study of the semi-classical behavior as p → ∞ of the spectral counting functions of the Kodaira Laplacians
p on Lp . Subsequently, Bismut gave a heat kernel proof which involves probability theory, and then Demailly and Bouche were able to replace the probability technique by a classical heat kernel argument. We present here a new approach based on the asymptotic of the heat kernel of the Kodaira Laplacian, exp(− up
p ). The analytic core follows in Section 1.6 where, inspired by the work of Bismut–Lebeau, we present a new proof for the asymptotic of the heat kernel. In Section 1.7 we apply these results to obtain a heat equation proof of the holomorphic Morse inequalities following Bismut. In Chapter 2 we study the properties of the field of meromorphic functions. We establish further two fundamental results about Moishezon manifolds. Then we give the proof of the Siu–Demailly criterion which answers the Grauert– Riemenschneider conjecture. For q = 1, the Morse inequalities (1) give  √ n pn −1 L R + o(pn ) , p −→ ∞ . (2) dim H 0 (X, Lp )  n! X(
1) 2π Therefore if L satisfies

 X(
1)

√

−1 L 2π R

n > 0,

(3)

(in particular, if L is semi-positive and positive at at least one point), there are a lot of sections in H 0 (X, Lp ), which by taking quotients deliver n independent meromorphic functions, i.e., X is Moishezon. In Section 2.4 we present an algebraic reformulation of the holomorphic Morse inequalities. In Chapter 3 we prove the Morse inequalities for the Dolbeault L2 -cohomology spaces for a non-compact manifold satisfying the fundamental estimate (Poincar´e inequality) at infinity. Using this more abstract formulation of the Morse inequalities, we can find a lower bound for the growth of the holomorphic section

4

Introduction

space for uniformly positive line bundles (Theorem 3.3.5) and an extension of the Siu–Demailly criterion for compact complex spaces with isolated singularities. We end the chapter with a study of a class of manifolds satisfying pseudoconvexity conditions in the sense of Andreotti–Grauert, namely q-convex and weakly 1-complete manifolds and also covering manifolds. Pseudoconvex manifolds are very important in complex geometry and analysis. In Chapter 4, we study the asymptotic expansion of the Bergman kernel. We assume now that L is √positive, equivalently, there exists a Hermitian metric −1 L R defines a K¨ ahler form on X, where RL is the hL on L, such that ω = 2π curvature of the holomorphic Hermitian connection ∇L on (L, hL ). In the rest of the Introduction we denote by g T X the associated K¨ ahler metric to ω on T X. We also let E be a holomorphic vector bundle on X with a Hermitian metric hE . Since L is positive, the Kodaira–Serre vanishing theorem shows that H q (X, Lp ⊗ E) = 0

(4)

for p large enough and q  1. Thus the whole cohomology of Lp ⊗ E concentrates in degree zero. The Bergman kernel Pp (x, x ) associated to Lp ⊗ E for p large enough, is the smooth kernel of the orthogonal projection Pp 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
p on Lp ⊗E. More dp be any orthonormal basis of H 0 (X, Lp ⊗ E) with respect to precisely, let {Sip }i=1 the global inner product induced by g T X , hL and hE (cf. (1.3.14)). Then for p large enough, 

Pp (x, x ) =

dp


Sip (x) ⊗ (Sip (x ))∗ ∈ (Lp ⊗ E)x ⊗ (Lp ⊗ E)∗x
.

(5)

i=1

Especially, Pp (x, x) =

dp


|Sip (x)|2 ,

if E = C.

(6)

i=1

The Bergman kernel has been studied by Tian, Yau, Bouche, Ruan, Catlin, Zelditch, Lu, Wang, and many others, in various generalities, establishing the asymptotic expansion for high powers of L. Moreover, it was discovered that the coefficients in the asymptotic expansion encode geometric information about the underlying complex projective manifolds. Our approach to the study of the asymptotic expansion continues the method applied in Chapter 1. We treat both the Dirac operator and the Kodaira Laplacian in the same time by means of the modified Dirac operator. The key point of our method is that the spectrum Spec(
p ) of
p (or of the half of the square of the

Introduction

5

Dirac operator) has a spectral gap, cf. Section 1.5. This means that there exists C > 0 such that for p  1, Spec(
p ) ⊂ {0}∪ ]2π p − C, +∞[.

(7)

We can divide our approach in three steps. The first step is to establish the spectral gap property (7). The second is the localization: the spectral gap property (7) and the finite propagation speed of solutions of hyperbolic equations allow us first to localize the asymptotic of Pp (x0 , x ) in the neighborhood of x0 . We pullback and extend the operator to Tx0 X ∼ = R2n , and verify that it inherits also the spectral gap property. The third step is to work on R2n . Here we combine the spectral gap property, the rescaling of the coordinates and functional analysis techniques, to conclude the proof of our final result. Moreover, by using a formal power series trick, we get a general and algorithmic way to compute the coefficients in the expansion. Certainly, for the last two steps it makes no difference whether the manifold X is compact or not. Thus in various new situations, we only need to verify the spectral gap property (cf. Chapters 5, 6, 8). We obtain finally the following asymptotic expansion (cf. Theorem 4.1.2): Pp (x, x) ∼

∞


br (x)pn−r ,

(8)

r=0

where br (x) ∈ End(E)x are smooth coefficients, which are polynomials in RT X , RE and their derivatives with order  2r − 2. Moreover  1  E 1 (9) b0 = IdE , b1 = 2R (wj , w j ) + rX IdE , 4π 2 where rX is the scalar curvature of (T X, g T X ) and {wj }nj=1 is an orthonormal basis of T (1,0) X. In the case of trivial bundle E the term b1 was calculated by Lu and used by Donaldson in his work on the existence of K¨ ahler metrics with constant scalar curvature. We also find the full off-diagonal expansion of the Bergman kernel Pp (x, x ) with the help of the heat kernel. In Chapter 5, we study in detail the metric aspect of the Kodaira map as an application of the asymptotic expansion of the Bergman kernel. First, we present an analytic proof of the Kodaira embedding theorem following an original idea of Bouche, and we study the convergence of the induced Fubini–Study metric. Then the Kodaira map Φp : X −→ P(H 0 (X, Lp )∗ ), defined by Φp (x) = {s ∈ H 0 (X, Lp ) : s(x) = 0} for x ∈ X, is an embedding for p large enough and for any l ∈ N, there exists Cl > 0 such that 1  Cl    2 , (10)  Φ∗p (ωF S ) − ω  l p p C (X) where ωF S is the Fubini–Study form on P(H 0 (X, Lp )∗ ). By using the Kodaira embedding, we also discuss briefly the relation of the Bergman kernel and the existence of K¨ ahler metrics with constant scalar curva-

6

Introduction

ture. Then, as an easy consequence of our approach, we describe the asymptotic expansion of the Bergman kernel on complex orbifolds, and the metric aspect of the Kodaira map. Finally, we give an introduction to the Ray-Singer analytic torsion and study its asymptotic behavior. The analytic torsions have a lot of applications, especially in Arakelov geometry. This seems to be quite independent of our subject, but in fact, Donaldson has used the analytic torsion in his study of the existence of K¨ ahler metrics with constant scalar curvature. In Chapter 6 we establish the existence of the expansion on compact sets of a non-compact manifold, as long as the spectral gap exists. One interesting situation is the case of Zariski open sets in compact complex spaces endowed with the generalized Poincar´e metric. The expansion of the Bergman kernel implies a new proof of the Shiffman–Ji–Bonavero–Takayama criterion for a Moishezon manifold. Then we obtain again Morse inequalities which are suitable for the study of the compactification of complete K¨ ahler manifolds with pinched negative curvature. In Chapter 7, using the full off-diagonal expansion of the Bergman kernel, we study the properties of Toeplitz operators and the Berezin–Toeplitz quantization. For f ∈ C ∞ (X, End(E)), we define the Toeplitz operator {Tf,p } as the family of linear operators Tf,p : L2 (X, Lp ⊗ E) −→ L2 (X, Lp ⊗ E) ,

Tf,p = Pp f Pp .

(11)

One of our main goals is to show that the set of Toeplitz operators is closed under the composition of operators, so they form an algebra. More precisely, let f, g ∈ C ∞ (X, End(E)), then there exist Cr (f, g) ∈ C ∞ (X, End(E)) with Tf,p Tg,p =

∞


p−r TCr (f,g),p + O(p−∞ ),

(12)

r=0

where Cr are differential operators. In particular C0 (f, g) = f g. If f, g ∈ C ∞ (X), then √ −1 T{f,g},p + O(p−2 ), [Tf,p , Tg,p ] = p

(13)

here {f, g} is the Poisson bracket of f, g on (X, 2πω). In Chapter 8, we find the asymptotic expansion of the Bergman kernel associated 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 graduate students. To help the readers, we add five appendices. In Appendix A, we recall the Sobolev embedding theorems and basic elliptic estimates. In Appendix B, we present useful material from Hermitian geometry. We also introduce the basics of Chern–Weil and Chern–Simons theories. In Appendix C, we collect some facts about self-adjoint

Introduction

7

operators. In Appendix D, we explain in detail the relation of the heat kernel and the finite propagation speed of solutions of hyperbolic equations. Finally, in Appendix E, we explain the basic facts about the harmonic oscillator. The book should also serve as an analytic introduction to the applications to algebraic geometry of the holomorphic Morse inequalities as developed by Demailly and his school, as well as to Donaldson’s approach to the existence of K¨ ahler metrics of constant scalar curvature. To keep the book within reasonable size, we list several classical results without proofs, and we indicate the corresponding references in the bibliographic notes of each chapter. The literature concerning the various themes we treat is quite vast and contains many important contributions. We could not include them all in the Bibliography, and restrained to the references which directly influenced our work. Prerequisites for this book are a course on differentiable manifolds and vector bundles. This book is not necessarily meant to be read sequentially. The reader is encouraged to go directly to the chapter of interest. Basically, Chapters 1 and 4 introduce the main technical ideas, and other chapters are various generalizations and applications. Here is a roadmap for our book. Chap. 2N Chap. @ qq8 N q  q N q  q N qq qqq  N&  / Chap. 3 _ __ _/ Chap. Chap.< 1N  8 <
5

6

7

8

Notation We denote by C, N, Q, R, Z the complex, natural, rational, real, integer numbers, and C∗ = C \ {0}, N∗ = N \ {0}, R∗ = R \ {0}, R+ = [0, ∞[, R∗+ =]0, ∞[, Q+ = Q ∩ R+ . For u ∈ R, we denote by u the integer part of u. For α = (α1 , . . . , αm ) ∈ Nm , B = (B1 , . . . , Bm ) ∈ Cm , we write by |α| =

m
j=1

αj ,

α! =

 (αj !), j

Bα =



α

Bj j .

j

SL(n, C) is the space of C-valued n × n matrices with determinant 1. O(n) is the orthogonal group of degree n over R. U (n) is the unitary group of degree n over C. We denote by dim or dimC the complex dimension of a complex (vector) space. We denote also by dimR the real dimension of a space.

8

Introduction

For a complex vector bundle E on a manifold X, rk(E) denotes its rank, and IdE the identity morphism. Also, det(E) := Λrk(E) (E) is its determinant line bundle, E ∗ its dual bundle and End(E) := E ⊗ E ∗ . The space of smooth sections of E over X is denoted by C ∞ (X, E). If Q is an operator, we denote by Ker(Q) its kernel, Im(Q) its image set. If U is a subset of V , we write U ⊂ V . If U is a relatively compact subset of V , we write U  V . The characteristic function 1U of U is defined as 1 on U and 0 on the complement of U . 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 with this index.

Acknowledgments We are happy to acknowledge our debt to Professor Jean-Michel Bismut. His impact can be felt throughout the book. The first-named author in particular – having been a student of Professor Jean-Michel Bismut – wishes to express his hearty thanks for discussions on various subjects and his enlightened support since 1994. Over the years, the second-named author benefited from many inspirational discussions with Jean-Pierre Demailly about holomorphic Morse inequalities. The subject of Bergman kernels was tremendously influenced by the work of Louis Boutet de Monvel and Johannes Sj¨ ostrand, with whom we had the privilege of discussing the topics. We thank Xianzhe Dai, Kefeng Liu and Weiping Zhang who helped us understand better the Bergman kernel. We thank also Christophe Margerin, Pierre Milman, Richard Thomas and Xiaowei Wang for several useful conversations and the staff of Birkh¨auser for its patience and dedication. We express our gratitude to the institutions which welcomed us, Humboldt ´ Universit¨ at zu Berlin, Centre de Math´ematiques Laurent Schwartz de l’Ecole Polytechnique (Palaiseau), Johann Wolfgang Goethe Universit¨ at (Frankfurt am Main) and Universit¨ at zu K¨ oln. In particular, our collaboration started at the Humboldt Universit¨ at zu Berlin, where, in his Thesis from 1922, Stefan Bergman discovered the kernel function in response to a question of Erhard Schmidt. We reproduce here the account of Menahem Max Schiffer from 19811: “Bergman participated in Schmidt’s seminar and was charged to give a lecture on the development of arbitrary functions with finite square integrals in terms of an orthogonal set. As he told me, he misunderstood the task and instead of dealing with real functions over a real interval, he attacked the problem for analytic functions over a complex domain. He found the task hard but attacked it courageously and carried it through. This was the genesis of his famous theory of the kernel function.” 1 M.M.

Schiffer, Stefan Bergman (1885–1977) in memoriam, Ann. Pol. Math. 39 (1981), 5–9.

Chapter 1

Demailly’s Holomorphic Morse Inequalities The first aim of this chapter is to provide the background material on differential geometry for the whole book. Then, in the last two sections, we present a heat kernel proof of Demailly’s holomorphic Morse inequalities, Theorem 1.7.1. This chapter is organized as follows. In Section 1.1 we review the theory of connections on vector bundles. In Section 1.2, we explain different connections on the tangent bundle and their relations. In Section 1.3, we define the modified Dirac operator for an almost complex manifold and prove the related Lichnerowicz formula. We explain also the Atiyah–Singer index theorem for the modified Dirac E E,∗ is a modified Dirac operator. In Section 1.4, we show that the operator ∂ + ∂ operator, and we establish the Lichnerowicz and Bochner–Kodaira–Nakano formulas for the Kodaira Laplacian. In Section 1.5, we deal with vanishing theorems for positive line bundles and the spectral gap property for the modified Dirac operator and the Kodaira Laplacian. In Section 1.6, we establish the asymptotic of the heat kernel which is the analytic core result of this chapter. Finally, in Section 1.7, we prove Demailly’s holomorphic Morse inequalities.

1.1 Connections on vector bundles In this section, we review the definition on connections and the associated curvatures. Section 1.1.1 reviews some general facts on connections on vector bundles, and we specify them to the holomorphic case in Section 1.1.2.

1.1.1 Hermitian connection Let E be a complex vector bundle over a smooth manifold X. Let T X be the tangent bundle and T ∗ X be the cotangent bundle. Let C ∞ (X, E) be the space of

10

Chapter 1. Demailly’s Holomorphic Morse Inequalities

smooth sections of E on X. Let Ωr (X, E) be the spaces of smooth r-forms on X with values in E, and set C ∞ (X) := C ∞ (X, C), Ω• (X) := Ω• (X, C). Let d : Ω• (X) → Ω•+1 (X) be the exterior differential. It is characterized by a) d2 = 0; b) for ϕ ∈ C ∞ (X), dϕ is the one form such that (dϕ)(U ) = U (ϕ) for a vector field U ; c) (Leibniz rule) for any α ∈ Ωq (X), β ∈ Ω(X), then d(α ∧ β) = dα ∧ β + (−1)q α ∧ dβ.

(1.1.1)

Then we verify that for any 1-form α, vector fields U, V on X, we have dα(U, V ) = U (α(V )) − V (α(U )) − α([U, V ]),

(1.1.2)

here [U, V ] is the Lie bracket of U and V . A linear map ∇E : C ∞ (X, E) → C ∞ (X, T ∗ X ⊗ E) is called a connection on E if for any ϕ ∈ C ∞ (X), s ∈ C ∞ (X, E) and U ∈ T X, we have E ∇E U (ϕs) = U (ϕ) s + ϕ∇U s .

(1.1.3)

Connections on E always exist. Indeed, let {Vk }k an open covering of X such that E|Vk is trivial. If {ηkl }l is a local frame of E|Vk , any section s ∈ C ∞ (Vk , E) has the form s = l sl ηkl with uniquely determined sl ∈ C ∞ (Vk ). We define a E connection on E|Vk by ∇k s := l dsl ⊗ ηkl . Consider now a partition of unity ∞ {ψk }k subordinated to {Vk }k . Then ∇E s := k ∇E k (ψk s), s ∈ C (X, E), defines a connection on E. E E 1 If ∇E 1 is another connection on E, then by (1.1.3), ∇1 −∇ ∈ Ω (X, End(E)). E If ∇ is a connection on E, then there exists a unique extension ∇E : • Ω (X, E) → Ω•+1 (X, E) verifying the Leibniz rule: for α ∈ Ωq (X), s ∈ Ωr (X, E), we have ∇E (α ∧ s) = dα ∧ s + (−1)q α ∧ ∇E s . (1.1.4) From (1.1.2), for s ∈ C ∞ (X, E) and vector fields U, V on X, we have E E E E (∇E )2 (U, V )s = ∇E U ∇V s − ∇V ∇U s − ∇[U,V ] s .

(1.1.5)

Then (∇E )2 (U, V )(ϕs) = (∇E )2 (U, ϕV )s = (∇E )2 (ϕU, V )s = ϕ(∇E )2 (U, V )s for any ϕ ∈ C ∞ (X). We deduce that: Definition and Theorem 1.1.1. The operator (∇E )2 defines a bundle morphism (∇E )2 : E → Λ2 (T ∗ X) ⊗ E, called the curvature operator. Therefore, there exists RE ∈ Ω2 (X, End(E)), called the curvature of ∇E , such that (∇E )2 is given by multiplication with RE , i.e., (∇E )2 s = RE s ∈ Ω2 (X, E) for s ∈ C ∞ (X, E). Let hE be a Hermitian metric on E, i.e., a smooth family {hE x }x∈X of E sesquilinear maps hE x : Ex × Ex → C such that hx (ξ, ξ) > 0 for any ξ ∈ Ex
{0}. We call (E, hE ) a Hermitian vector bundle on X. There always exist Hermitian metrics on E by using the partition of unity argument as above.

1.1. Connections on vector bundles

11

Definition 1.1.2. A connection ∇E is said to be a Hermitian connection on (E, hE ) if for any s1 , s2 ∈ C ∞ (X, E), ds1 , s2 hE = ∇E s1 , s2 hE + s1 , ∇E s2 hE .

(1.1.6)

There always exist Hermitian connections. In fact, let ∇E 0 be a connection E E on E, then ∇E 1 s1 , s2 hE = ds1 , s2 hE − s1 , ∇0 s2 hE defines a connection ∇1 on 1 E E E E E. Now ∇ = 2 (∇0 + ∇1 ) is a Hermitian connection on (E, h ). Let {ξl }m l=1 be a local frame of E. Denote by h = (hlk = ξk , ξl hE ) the matrix E l E of h with respect to {ξl }m l=1 . The connection form θ = (θk ) of ∇ with respect m l to {ξl }l=1 is defined by, with local 1-forms θk , ∇E ξk = θkl ξl .

(1.1.7)

Remark 1.1.3. If E is a real vector bundle on X, certainly, everything still holds, especially, a connection ∇E is said to be an Euclidean connection on (E, hE ) if it preserves the Euclidean metric hE .

1.1.2 Chern connection Let E be a holomorphic vector bundle over a complex manifold X. Let hE be a Hermitian metric on E. We call (E, hE ) a holomorphic Hermitian vector bundle. The almost complex structure J induces a splitting T X ⊗R C = T (1,0) X ⊕ (0,1) (1,0) X, where T X and√T (0,1) X are the eigenbundles of J corresponding to T √ the eigenvalues −1 and − −1, respectively. Let T ∗(1,0) X and T ∗(0,1) X be the corresponding dual bundles. Let Ωr,q (X, E) := C ∞ (X, Λr (T ∗(1,0) X) ⊗ Λq (T ∗(0,1) X) ⊗ E) be the spaces of smooth (r, q)-forms on X with values in E. E

0,1 The operator ∂ : C ∞ (X, E) → defined. Any section Ω (X, E) is well ∞ s ∈ C (X, E) has the local form s = l ϕl ξl where {ξl }m is a local holomorphic l=1 E frame of E and ϕl are smooth functions. We set ∂ s = l (∂ϕl ) ξl , here ∂ϕl = ∂ j dz j ∂z j ϕl in holomorphic coordinates (z1 , · · · , zn ).

Definition 1.1.4. A connection ∇E on E is said to be a holomorphic connection if E (0,1) ∇E X and s ∈ C ∞ (X, E). U s = iU (∂ s) for any U ∈ T Theorem 1.1.5. There exists a unique holomorphic Hermitian connection ∇E on (E, hE ), called the Chern connection . With respect to a local holomorphic frame, the connection matrix is given by θ = h−1 · ∂h. (1,0) Proof. By Definition 1.1.4, we have to define ∇E X. Relation U just for U ∈ T (1,0) ∞ (1.1.6) implies for U ∈ T X, s1 , s2 ∈ C (X, E), E U s1 , s2 hE = ∇E U s1 , s2 hE + s1 , ∇U s2 hE .

(1.1.8)

12

Chapter 1. Demailly’s Holomorphic Morse Inequalities E

Since ∇E s = iU (∂ s2 ), the above equation defines ∇E U uniquely. Moreover, if U 2 m {ξl }l=1 is a local holomorphic frame, from (1.1.6) we deduce that θ = h−1 ·∂h.
E

Since E is holomorphic, similar to (1.1.4), the operator ∂ extends naturally E to ∂ : Ω•,• (X, E) −→ Ω•,• +1 (X, E) and (∂ )2 = 0. Let ∇E be the holomorphic Hermitian connection on (E, hE ). Then we have a decomposition of ∇E after bidegree E

∇E = (∇E )1,0 + (∇E )0,1 ,

E

(∇E )0,1 = ∂ ,

(∇E )1,0 : Ω• , • (X, E) −→ Ω• +1 , • (X, E) .

(1.1.9)

E

By (1.1.8), (1.1.9) and (∂ )2 = 0 we have 2

E E E (∂ )2 = (∇E )1,0 = 0, (∇E )2 = ∂ (∇E )1,0 + (∇E )1,0 ∂ .

(1.1.10)

is trivial and Thus the curvature RE ∈ Ω1,1 (X, End(E)). If rk(E) = 1, End(E) √ RE is canonically identified to a (1, 1)-form on X, such that −1RE is real. In general, let us introduce an auxiliary Riemannian g T X metric on X, compatible with the complex structure J (i.e., g T X (·, ·) = g T X (J·, J·)). Then RE (1,0) induces a Hermitian matrix R˙ E ∈ End(T (1,0) X ⊗ E) such that for u, v ∈ Tx X, ξ, η ∈ Ex , and x ∈ X, RE (u, v)ξ, ηhE = R˙ E (u ⊗ ξ), v ⊗ η.

(1.1.11)

Definition 1.1.6. We say that (E, hE ) is Nakano positive (resp. semi-positive) if R˙ E ∈ End(T (1,0) X ⊗ E) is positive definite (resp. semi-definite), and Griffiths positive (resp. semi-positive) if RE (v, v)ξ, ξhE = R˙ E (v ⊗ ξ), v ⊗ ξ > 0 (resp. (1,0)  0) for all non-zero v ∈ Tx X and all non-zero ξ ∈ Ex . Certainly, these definitions do not depend on the choice of g T X .

1.2 Connections on the tangent bundle On the tangent bundle of a complex manifold, we can define several connections: the Levi–Civita connection, the holomorphic Hermitian (i.e., Chern) connection and Bismut connection. In this section, we explain the relation between them. We shall see that these three connections coincide, if X is a K¨ ahler manifold. We start by recalling in Section 1.2.1 some facts about the Levi–Civita connection. In Section 1.2.2, we study in detail the holomorphic Hermitian connection on the tangent bundle. In Section 1.2.3, we define the Bismut connection. Let (X, J) be a complex manifold with complex structure J and dimC X = n. Let Th X be the holomorphic tangent bundle on X, and let T X be the corresponding real tangent bundle. Let g T X be any Riemannian metric on T X compatible with J, i.e., g T X (Ju, Jv) = g T X (u, v) for any u, v ∈ Tx X, x ∈ X. We will shortly express this relation by g T X (J·, J·) = g T X (·, ·).

1.2. Connections on the tangent bundle

13

1.2.1 Levi–Civita connection The results of this section apply for any Riemannian manifold (X, g T X ). We denote TX by ·, · the C-bilinear form on T X ⊗R C induced by the metric g T X . Let ∇T X be

TX the Levi–Civita connection on (T X, g ). By the explicit equation for ∇ ·, · , for any U, V, W, Y vector fields on X,

2 ∇TU X V, W = U V, W  + V U, W  − W U, V  − U, [V, W ] − V, [U, W ] + W, [U, V ] . (1.2.1) ∇T X is the unique connection on T X which preserves the metric (satisfies (1.1.6)) and is torsion free, i.e., ∇TU X V − ∇TV X U = [U, V ].

(1.2.2)

The curvature RT X ∈ Λ2 (T ∗ X) ⊗ End(T X) of ∇T X is defined by X RT X (U, V ) = ∇TU X ∇TV X − ∇TV X ∇TU X − ∇T[U,V ].

(1.2.3)

Then we have the following well-known facts RT X (U, V )W + RT X (V, W )U + RT X (W, U )V = 0, RT X (U, V )W, Y  = RT X (W, Y )U, V .

(1.2.4)

i 2n ∗ Let {ei }2n i=1 be an orthonormal frame of T X and {e }i=1 its dual basis in T X. The Ricci curvature Ric and scalar curvature rX of (T X, g T X ) are defined by



RT X (·, ej )·, ej , rX = − RT X (ei , ej )ei , ej . Ric = − (1.2.5) j

ij

The Riemannian volume form dvX of (T X, g T X ) has the form dvX = e1 ∧ · · · ∧ e2n if the orthonormal frame {ei } is oriented. If α is a 1-form on X, the function Tr(∇α) is given by the formula Tr(∇α) = i ei (α(ei )) − α(∇TeiX ei ). (1.2.6) The following formula is quite useful. Proposition 1.2.1. For any C 1 1-form α with compact support, we have  Tr(∇α)dvX = 0.

(1.2.7)

X

Proof. Let W be the vector field on X corresponding to α under the Riemannian metric g T X , so that W, Y  = (α, Y ) for any Y ∈ T X. We denote by LW the Lie derivative of the vector field W . Recall that for any vector field Y on X, LW Y = [W, Y ] = ∇TWX Y − ∇TY X W.

(1.2.8)

14

Chapter 1. Demailly’s Holomorphic Morse Inequalities



Thus by (1.2.8) and ∇TWX ej , ej = 0, we get

LW dvX = LW ej , ej dvX = − ej , LW ej  dvX      = ∇TejX W, ej dvX = ej W, ej  − W, ∇TejX ej dvX = Tr(∇α)dvX .

(1.2.9)

We will denote by ∧ and i the exterior and interior product respectively. E. Cartan’s homotopy formula tells us that on the bundle of exterior differentials Λ(T ∗ X), LW = d · iW + iW · d. From (1.2.9) and (1.2.10), we get   LW dvX = Tr(∇α)dvX . 0= X

(1.2.10)

(1.2.11)

X




The proof of Proposition 1.2.1 is complete.

For x0 ∈ X, W ∈ Tx0 X, let R  u → xu = expX x0 (uW ) be the geodesic u | = W . For ε > 0, we denote by B X (x0 , ε) in X such that xu |u=0 = x0 , dx du u=0 Tx0 X and B (0, ε) the open balls in X and Tx0 X with center x0 and radius ε, respectively. Then the map Tx0 X  Z → expX from x0 (Z) ∈ X is a diffeomorphism B Tx0 X (0, ε) onto B X (x0 , ε) for ε small enough; by identifying Z = Zi ei ∈ Tx0 X with (Z1 , . . . , Z2n ) ∈ R2n , it yields a local chart for X around x0 , called normal coordinate system at x0 . We will identify B Tx0 X (0, ε) with B X (x0 , ε) by this map. Let {ei }i be an oriented orthonormal basis of Tx0 X. We also denote by {ei }i the dual basis of {ei }. Let ei (Z) be the parallel transport of ei with respect to ∂ . ∇T X along the curve [0, 1]  u → uZ. Then ej = ∂Z j The radial vector field R is the vector field defined by R = i Zi ei with (Z1 , . . . , Z2n ) the coordinate functions. Proposition 1.2.2. The following identities hold:

R= Zj ej = Zj ej (Z), j

j

(1.2.12)

R, ej  = Zj . u Proof. Note that xu : [0, 1]  u → uZ is a geodesic, and R(xu ) = u dx du , thus by T X dxu the geodesic equation ∇ dxu du = 0, we get du

∇TRX R = u∇TdxXu (u du

dxu dxu )=u = R. du du

(1.2.13)

1.2. Connections on the tangent bundle

15

Thus we have RR, ej  = ∇TRX R, ej  + R, ∇TRX  ej  = R, ej . This means that R, ej  is homogeneous of order 1. But
R, ej  = Zk ek ,  ej  = Zj + O(|Z|2 ).

(1.2.14)

(1.2.15)

k

Thus from (1.2.14) and (1.2.15), we infer the first equation of (1.2.12). Since the Levi–Civita connection ∇T X is torsion free and [R, ei ] = −ei , we have 1 ei R, R − R, ei . 2

R, ∇TRX ei  = R, ∇TeiX R + R, [R, ei ] =

(1.2.16)

From (1.2.13) and (1.2.16), we obtain RR, ei  = ∇TRX R, ei  + R, ∇TRX ei  = But R, ei  = (1.2.12).

j

1 ei R, R = Zi . 2

(1.2.17)

Zj ej , ei  = Zi + O(|Z|2 ). Thus we get the second equation of


α2n . For α = (α1 , . . . , α2n ) ∈ N2n , set Z α = Z1α1 . . . Z2n

Lemma 1.2.3. If  ei (Z) is written in the basis {ei }, its Taylor expansion up to order r is determined by the Taylor expansion up to order r − 2 of Rmqkl = RT X (eq , em )ek , el Z . Moreover we have ei (Z) = ei −


 ∂α  Zα

1
TX . ei (0) Rx0 (R, ei )R, ej x0 ej + 6 j ∂Z α α!

(1.2.18)

|α|3

Thus the Taylor expansion up to order r of gij (Z) = g T X (ei , ej )(Z) = ei , ej Z is a polynomial of the Taylor expansion up to order r − 2 of Rmqkl ; moreover gij (Z) = δij +

1 TX Rx0 (R, ei )R, ej x0 + O(|Z|3 ). 3

(1.2.19)

Proof. Let ΓT X be the connection form of ∇T X with respect to the frame { ei } of T X. Then ∇T X = d + ΓT X . Let ∂i = ∇ei be the partial derivatives along ei . By the definition of our fixed frame, we have iR ΓT X = 0. Thus LR ΓT X = [iR , d]ΓT X = iR (dΓT X + ΓT X ∧ ΓT X ) = iR RT X .  Let θ(Z) = (θji (Z))2n i,j=1 be the 2n × 2n-matrix such that
j  −1 )k ek . θi (Z) ej (Z), ej (Z) = (θ(Z) ei = j j

(1.2.20)

(1.2.21)

16

Chapter 1. Demailly’s Holomorphic Morse Inequalities

Set θj (Z) =



j i i θi (Z)e

θ=

and


ej ⊗ ej =




j

θj ej ∈ T ∗ X ⊗ T X.

(1.2.22)

j

As ∇T X is torsion free, ∇T X θ = 0, thus the R2n -valued 1-form θ = (θj (Z)) satisfies the structure equation, dθ + ΓT X ∧ θ = 0.

(1.2.23)

Observe first that under our trivialization by { ei }, by (1.2.12), for the R2n -valued function iR θ,
Zj ej = (Z1 , . . . , Z2n ) =: Z. (1.2.24) iR θ = j

Substituting (1.2.12), (1.2.24) and (LR − 1)Z = 0, into the identity iR (dθ + ΓT X ∧ θ) = 0, from (1.2.20), we obtain (LR − 1)LR θ = (LR − 1)(dZ + ΓT X Z) = (LR ΓT X )Z = (iR RT X )Z.

(1.2.25)

2n Where we consider the curvature RT X as a matrix of two-forms

iand θ is a R TX TX valued one-form. The ith component of R Z, θ is R R, ei , θ , from (1.2.25), we get

iej (LR − 1)LR θi (Z) = RT X (R, ej )R, ei (Z). (1.2.26)

By (1.2.12), LR ej = ej . Thus from the Taylor expansion of θji (Z), we get


(|α|2 + |α|)(∂ α θji )(0)

|α|1



Zα = RT X (R, ej )R, ei (Z). α!

(1.2.27)

Now by (1.2.21) and θji (x0 ) = δij , (1.2.27) determines the Taylor expansion up to order m in terms of the Taylor expansion of the coefficients of RT X of up to order m − 2. And θji (Z)

(θ−1 )ij = δij −

1 TX Rx0 (R, ei )R, ej x0 + O(|Z|3 ). 6

(1.2.28)

By (1.2.21), (1.2.27), we infer (1.2.18). From (1.2.21), gij (Z) = θik (Z)θjk (Z).

(1.2.29)

Thus the rest of Lemma 1.2.3 follows from (1.2.28) and (1.2.29). The proof of Lemma 1.2.3 is complete.


1.2. Connections on the tangent bundle

17

Let E be a complex vector bundle on X, and let ∇E be a connection on E (U, Z1 , . . . , Z2n ) be a local chart of X such that with curvature RE := (∇E )2 . Let ∂ 0 ∈ U represents x0 ∈ X. Set R = i Zi ∂Z . Now we identify EZ to Ex0 by parallel i transport with respect to the connection ∇E along the curve [0, 1]  u → uZ; this gives a trivialization of E near 0. We denote by ΓE the connection form with ∂ , ΓE becomes respect to this trivialization of E near 0. Then in the frame ej = ∂Z j a function with values in R2n ⊗ End(Crk(E) ) and ∇E = d + ΓE . Lemma 1.2.4. The Taylor coefficients of ΓE (ej )(Z) at x0 up to order r are determined by Taylor coefficients of RE up to order r − 1. More precisely,


(∂ α ΓE )x0 (ej )

|α|=r

1 Zα = α! r+1




(∂ α RE )x0 (R, ej )

|α|=r−1

Zα . α!

(1.2.30)

Especially, ΓE Z (ej ) =

1 E R (R, ej ) + O(|Z|2 ). 2 x0

(1.2.31)

Proof. By the definition of our fixed frame, we have RE = dΓE + ΓE ∧ ΓE and iR ΓE = 0,

LR ΓE = [iR , d]ΓE = iR (dΓE + ΓE ∧ ΓE ) = iR RE .

(1.2.32)

Using LR dZ j = dZ j and expanding both sides of the second equation of (1.2.32) in Taylor’s series of at Z = 0, we obtain


(|α| + 1)(∂ α ΓE )x0 (ej )

α


Zα Zα = . (∂ α RE )x0 (R, ej ) α! α! α

(1.2.33)


By equating coefficients of Z α of both sides, we get Lemma 1.2.4.

1.2.2 Chern connection (1,0)

T X Recall that T (1,0) X is a holomorphic vector bundle √with Hermitian metric h induced by g T X . The map Th X  Y → 12 (Y − −1JY ) ∈ T (1,0) X induces the natural identification of Th X and T (1,0) X. We will denote by ·, · the C-bilinear form on T X ⊗R C induced by g T X . Note that ·, · vanishes on T (1,0) X × T (1,0) X and on T (0,1) X × T (0,1) X. For U ∈ T X ⊗R C, we will denote by U (1,0) , U (0,1) its components in T (1,0) X and T (0,1) X. Let {wj }nj=1 be a local orthonormal frame of T (1,0) X with dual frame {wj }nj=1 . Then

e2j−1 =

√1 (wj 2

+ w j ) and e2j =

√ √−1 (wj 2

− wj ) ,

j = 1, . . . , n ,

(1.2.34)

form an orthonormal frame of T X. We fix this notation throughout the book and use it without further notice.

18

Chapter 1. Demailly’s Holomorphic Morse Inequalities (1,0)

X Let ∇T be the holomorphic Hermitian connection on (T (1,0) X, hT (1,0) X . For v ∈ C ∞ (X, T (0,1) X), we define with curvature RT

∇T Then ∇T

(0,1)

X

(0,1)

X

(1,0)

X

)

v := ∇T (1,0) X v.

defines a connection on T (0,1) X. Set  T X = ∇T (1,0) X ⊕ ∇T (0,1) X . ∇

(1.2.35)

 T X is a connection on T X ⊗R C and it preserves T X; we still denote by Then ∇ TX   T X preserves the metric g T X . ∇ the induced connection on T X. Then ∇  T X . Then T ∈ Λ2 (T ∗ X) ⊗ T X is Let T be the torsion of the connection ∇ defined by TXV − ∇  T X U − [U, V ], T (U, V ) = ∇ U V

(1.2.36)

for vector fields U and V on X. Hence T maps T (1,0) X ⊗ T (1,0) X (resp. T (0,1) X ⊗ T (0,1) X) into T (1,0) X (resp. T (0,1) X) and vanishes on T (1,0) X ⊗ T (0,1) X. Set

 T X − ∇T X , S=∇

S=




S(ei )ei .

(1.2.37)

(1.2.38)

i

Then S is a real 1-form on X taking values in the skew-adjoint endomorphisms of T X. Since ∇T X is torsion free, we have for U, V ∈ T X, T (U, V ) = S(U )V − S(V )U.

(1.2.39)

 T X preserves g T X we obtain Moreover, from (1.2.1), (1.2.36), (1.2.38) and since ∇ directly 2 S(U )V, W  − T (U, V ), W  − T (W, U ), V  + T (V, W ), U  = 0.

(1.2.40)

By (1.2.37), (1.2.39) and (1.2.40), we get S(wi )wk , wj  = 0, 2 S(wi )wk , wj  = 2 S(w k )wi , wj  = − T (wi , wj ), w k  .

(1.2.41)

Since T (wi , wj ) = 0, S(w j )wi = S(wi )w j , and so S = 2S(wj )w j = T (wi , wj ), w j  w i + T (wi , wj ), wj  wi = T (ei , ej ), ej  ei ,

(1.2.42)

2 S(·)wj , wj  = T (wi , wj ), w j  w − T (wi , w j ), wj  w . i

i

 T X on T X induces naturally a covariant derivative on the The connection ∇ ∗  T X . For any differential forms exterior bundle Λ(T X) and we still denote it by ∇ α, β and vector field Y , it satisfies  TY X (α ∧ β) = (∇  TY X α) ∧ β + α ∧ ∇  TY X β. ∇

(1.2.43)

1.2. Connections on the tangent bundle

19

 T X α, V ) = U (α, V )−(α, ∇  T X V ). For a 1-form α and vector fields U, V , we have (∇ U U TX TX ∗ induces naturally a connection ∇ on Λ(T X). We denote by ε Likewise, ∇ the exterior product T ∗ X ⊗ Λ• (T ∗ X) → Λ•+1 (T ∗ X). Lemma 1.2.5. For the exterior differentiation operator d acting on smooth sections of Λ(T ∗ X), we have  T X + iT , d=ε◦∇

d = ε ◦ ∇T X .

(1.2.44)

 T X + iT . Then by using (1.2.43), we know that for any Proof. We write d := ε ◦ ∇ homogeneous differential forms α, β, we have d(α ∧ β) = (dα) ∧ β + (−1)deg α α ∧ dβ.

(1.2.45)

From Leibniz’s rule (1.2.45), it suffices to show that d agrees with d on functions (which is clear) and 1-forms. Now, for any smooth function f on X, we have    Te X df, ej  = ei ∧ ej ei (ej (f )) − df, ∇  Te X ej   T X df = ei ∧ ej ∇ ε◦∇ i i    1 i  T X ej  − ej (ei (f )) − df, ∇  T X ei  = e ∧ ej ei (ej (f )) − df, ∇ (1.2.46) ei ej 2 1 = − ei ∧ ej df, T (ei , ej ) = −iT df. 2 Thus d coincides also d on 1-forms. Thus we get the first equation of (1.2.44). As ∇T X is torsion free, from the above argument, we obtain the second equation of (1.2.44).
If B ∈ Λ2 (T ∗ X) ⊗ T X we will denote by Bas the anti-symmetrization of the tensor V, W, Y → B(V, W ), Y . Then Bas (V, W, Y ) = B(V, W ), Y  + B(W, Y ), V  + B(Y, V ), W .

(1.2.47)

Especially from (1.2.37), we infer Tas =

1 T (ei , ej ), ek ei ∧ ej ∧ ek 2

1 1 T (wi , wj ), w k wi ∧ wj ∧ wk + T (wi , w j ), wk w i ∧ w j ∧ wk 2 2 (1,0) (0,1) =: Tas + Tas . =

(1,0)

(1.2.48)

(0,1)

Here Tas , Tas are the anti-symmetrizations of the components T (1,0) , T (0,1) of T in T (1,0) X and T (0,1) X. Let Θ be the real (1, 1)-form defined by Θ(X, Y ) = g T X (JX, Y ).

(1.2.49)

Note that the exterior differentiation operator d acting on smooth sections of Λ(T ∗ X) has the decomposition d = ∂ + ∂.

(1.2.50)

20

Chapter 1. Demailly’s Holomorphic Morse Inequalities

Proposition 1.2.6. We have the identity of 3-forms on X, √ Tas = − −1(∂ − ∂)Θ. √ Proof. By (1.2.34), we know that Θ = −1 i wi ∧ w i . Thus √  T X Θ = −1((∇  T X wi ) ∧ w i + wi ∧ ∇  T X wi ) ∇   √  T X wi , w j  − wi , ∇  T X w j  wi ∧ w j = 0. = −1 −∇

(1.2.51)

(1.2.52)

From (1.2.44), (1.2.48) and (1.2.52) we have √ (1,0) (0,1) dΘ = iT Θ = −1(Tas − Tas ).

(1.2.53)

The relations (1.2.48) and (1.2.53) yield √ √ (1,0) (0,1) ∂Θ = −1Tas , ∂Θ = − −1Tas .

(1.2.54)


(1.2.54) imply (1.2.51).

Definition 1.2.7. We call Θ as in (1.2.49) a Hermitian form on X and (X, J, Θ) a ahler complex Hermitian manifold. The metric g T X = Θ(·, J·) on T X is called a K¨ metric if Θ is a closed form, i.e., dΘ = 0. In this case, the form Θ is called a K¨ ahler form on X, and the complex manifold (X, J) is called a K¨ ahler manifold . Let ∇X J ∈ T ∗ X ⊗ End(T X) be the covariant derivative of J induced by the Levi–Civita connection ∇T X . Theorem 1.2.8. (X, J, Θ) is K¨ ahler if and only if the bundle T (1,0) X and T (0,1) X are preserved by the Levi–Civita connection ∇T X , or in other words, if and only if ∇X J = 0. In this case, TX, ∇T X = ∇

S = 0,

T = 0.

(1.2.55)

Proof. As Θ is a (1, 1)-form, by (1.2.41), (1.2.48) and (1.2.51), dΘ = 0 is equivalent to Tas = 0 and equivalent to S(wk )wi ∈ T (1,0) X for any i, k. But this means that the bundles T (1,0) X and T (0,1) X are preserved by ∇T X . Hence (1.2.55) is equivalent to (X, Θ) being K¨ ahler. Moreover, as J acts by multiplication with √ −1 on T (1,0) X, we get for U ∈ T X, √

1 S(U )wi , wj  = − ∇TU X wi , wj = − ∇TU X (1 − −1J)wi , wj 2 √ (1.2.56)

−1 X = (∇U J)wi , wj , 2 by (1.2.38). Now, from J 2 = −1 we deduce J(∇X J) + (∇X J)J = 0.

(1.2.57)

This means that (∇X J) exchanges T (1,0) X and T (0,1) X. By (1.2.44), and (1.2.56), ∇X J = 0 is equivalent to S(w k )wi ∈ T (1,0) X for any i, k. The proof of Theorem 1.2.8 is complete.


1.3. Spinc Dirac operator

21

1.2.3 Bismut connection Let S B denote the 1-form with values in the antisymmetric elements of End(T X) which satisfies for U, V, W ∈ T X, √   −1 1 (∂ − ∂)Θ (U, V, W ) = − Tas (U, V, W ). S (U )V, W  = 2 2 B

(1.2.58)

By (1.2.40), (1.2.47), (1.2.58), we have for U, V, W ∈ T X, B





(S − S)(U )V, W = − T (U, V ), W + T (U, W ), V .

(1.2.59)

Relations (1.2.41), (1.2.48), and (1.2.58) yield 1 1 S B (ej )ωl , ω m  = − T (ej , ωl ), ω m  + T (ej , ωm ), ωl  2 2 = −S(ej )ωl , ωm , 1 S B (ej )ωl , ωm  = − T (ωl , ωm ), ej  = S(ej )ωl , ωm . 2

(1.2.60)

Definition 1.2.9. The Bismut connection ∇B on T X is defined by  T X + S B − S. ∇B := ∇T X + S B = ∇

(1.2.61)

In view of (1.2.58), the torsion of ∇B is 2S B which is a skew-symmetric tensor. The connection ∇B will be used in the Lichnerowicz formula (1.4.29). Lemma 1.2.10. The connection ∇B preserves the complex structure of T X.

Proof. Using (1.2.60), we find that for V, W ∈ T (1,0) X, (S B − S)(U )V, W = 0, for any U ∈ T X. Equivalently, (S B − S)(U ) is a complex endomorphism of T X. Using (1.2.61), we find that ∇B preserves the complex structure of T X.


1.3 Spinc Dirac operator This section is organized as follows. In Section 1.3.1, we define the Clifford connection. In Section 1.3.2, we define the spinc Dirac operator on a complex manifold and prove the related Lichnerowicz formula. In Section 1.3.3, we obtain the Lichnerowicz formula for the modified Dirac operator. In Section 1.3.4, we explain also the Atiyah–Singer index theorem for the modified Dirac operator. In this section, we work on a smooth manifold with an almost complex structure J.

22

Chapter 1. Demailly’s Holomorphic Morse Inequalities

1.3.1 Clifford connection Let (X, J) be a smooth manifold with J an almost complex structure on T X. 0,• Let g T X be any Riemannian metric on T X compatible with J. Let hΛ be the ∗(0,1) TX Hermitian metric on Λ(T X) induced by g . The fundamental Z2 spinor bundle induced by J is given by Λ(T ∗(0,1) X), whose Z2 -grading is defined by Λ(T ∗(0,1) X) = Λeven(T ∗(0,1) X) ⊕ Λodd (T ∗(0,1) X). For any v ∈ T X with decomposition v = v (1,0) + v (0,1) ∈ T (1,0) X ⊕ T (0,1) X, let v (1,0),∗ ∈ T ∗(0,1) X be the metric dual of v (1,0) . Then √ (1.3.1) c(v) = 2(v (1,0),∗ ∧ − iv(0,1) ) defines the Clifford action of v on Λ(T ∗(0,1) X), where ∧ and i denote the exterior and interior product, respectively. We verify easily that for U, V ∈ T X, c(U )c(V ) + c(V )c(U ) = −2U, V .

(1.3.2)

For a skew-adjoint endomorphism A of T X, from (1.3.1), using the notation of (1.2.34),



1 1 Aei , ej c(ei )c(ej ) = − Awj , w j + Awl , w m wm ∧ iwl 4 2

1 1 + Awl , wm iwl iwm + Aw l , w m w l ∧ w m ∧ . 2 2

(1.3.3)

Let ∇det be a Hermitian connection on det(T (1,0) X) endowed with metric (1,0) X induced by g T X . Let Rdet be its curvature. Let P T be the natural projection (1,0) (1,0) X T X T (1,0) X X. Then the connection ∇1,0 = P T ∇ P from T X ⊗R C onto T on T (1,0) X induces naturally a connection ∇det1 on det(T (1,0) X). Let ΓT X ∈ T ∗ X ⊗ End(T X), Γdet be the connection forms of ∇T X , ∇det associated to the frames {ej }, w1 ∧ · · · ∧ wn , i.e., ∇TeiX ej = ΓT X (ei )ej ,

∇det (w1 ∧ · · · ∧ wn ) = Γdet w1 ∧ · · · ∧ wn ,   TX ∇det1 (w1 ∧ · · · ∧ wn ) = wj , wj  w1 ∧ · · · ∧ wn . j Γ

(1.3.4)

The Clifford connection ∇Cl on Λ(T ∗(0,1) X) is defined for the frame {wj1 ∧ · · · ∧ wjk , 1  j1 < · · · < jk  n} by the local formula ∇Cl = d +

1 TX 1 Γ ei , ej c(ei )c(ej ) + Γdet . 4 2

(1.3.5)

Proposition 1.3.1. ∇Cl defines a Hermitian connection on Λ(T ∗(0,1) X) and preserves its Z2 -grading. For any V, W vector fields of T X on X, we have   Cl (1.3.6) ∇V , c(W ) = c(∇TV X W ).

1.3. Spinc Dirac operator

23

Proof. At first, by (1.3.4) and (1.3.5), we have  Cl

 1  T X  ∇V , c(ek ) = Γ (V )ei , ej c(ei )c(ej ), c(ek )

4 = ΓT X (V )ek , ej c(ej ) = c(∇TV X ek ).

(1.3.7)

Thus if ∇Cl is well defined, we get (1.3.6) from (1.3.7). Now we observe that c(wj1 ) . . . c(wjk ) 1, (1  j1 < · · · < jk  n) generate a frame of Λ(T ∗(0,1) X). Taking into account (1.3.7), to verify that ∇Cl does not depend on the choice of our frame {wj }nj=1 , we only need to verify that ∇Cl 1 is well defined. Relations (1.2.38), (1.3.3), (1.3.4) and (1.3.5) entail

1 ∇Cl = d + (∇det − ∇det1 ) + ΓT X wl , wm w m ∧ iwl 2 (1.3.8)

1 1 − Swl , wm iwl iwm − Swl , wm wl ∧ w m ∧ . 2 2 From (1.3.8), we know

1 1
∇Cl 1 = (∇det − ∇det1 ) − Swl , wm wl ∧ wm . (1.3.9) 2 2 lm

Clearly, ∇det − ∇det1 is a 1-form on X, and the right-hand side of (1.3.9) does not depend on the choice of the frame wj . Thus ∇Cl 1 is well defined. Let c(ei )∗ be the adjoint of c(ei ) with respect to the Hermitian product on ∗(0,1) Λ(T X). By (1.3.1), we have c(ei )∗ = −c(ei ).



(1.3.10)

Using (1.3.5), (1.3.10) and the anti-symmetry of ΓT X ei , ej in i, j, we see that ∇Cl preserves the Hermitian metric on Λ(T ∗(0,1)X). Finally, from (1.3.5), ∇Cl preserves the Z2 -grading on Λ(T ∗(0,1) X). The proof of Proposition 1.3.1 is complete.
Let RCl be the curvature of ∇Cl . Proposition 1.3.2. We have the following identity:

1 1 RCl = RT X ei , ej c(ei )c(ej ) + Rdet . (1.3.11) 4 2 Proof. At first, observe that if i, j, k, l are different, then [c(ei )c(ej ), c(ek )c(el )] = 0. Thus from (1.3.2),  T X

 Γ (W )ei , ej c(ei )c(ej ), ΓT X (V )ek , el c(ek )c(el )




ΓT X (W )ei , ej ΓT X (V )ek , ej [c(ei )c(ej ), c(ek )c(ej )] =4 i=j=k (1.3.12) TX

TX = 4 Γ (W )ei , Γ (V )ek (c(ei )c(ek ) − c(ek )c(ei ))

= 4 (ΓT X ∧ ΓT X )(W, V )ei , ek c(ei )c(ek ).

24

Chapter 1. Demailly’s Holomorphic Morse Inequalities

Moreover, we have RT X = dΓT X + ΓT X ∧ ΓT X , Cl

R (el , em ) =

Cl ∇Cl el ∇em

−

Cl ∇Cl em ∇el

(1.3.13) −

∇Cl [el ,em ] .

Finally, (1.3.5), (1.3.12) and (1.3.13) yield (1.3.11).




1.3.2 Dirac operator and Lichnerowicz formula Let (E, hE ) be a Hermitian vector bundle on X. Let ∇E be a Hermitian connection on (E, hE ) with curvature RE . Set Eq = Λq (T ∗(0,1) X)⊗ E, E = ⊕nq=0 E q . We still denote by ∇Cl the connection on Λ(T ∗(0,1) X) ⊗ E induced by ∇Cl and ∇E . Let Ω0,q (X, E) := C ∞ (X, Eq ) be the set of smooth sections of Eq on X. Along the fibers of Λ(T ∗(0,1) X) ⊗ E, we consider the pointwise Hermitian product ·, ·Λ0,• ⊗E induced by g T X and hE . The L2 -scalar product on Ω0,• (X, E) is given by  s1 , s2  =

s1 (x), s2 (x)Λ0,• ⊗E dvX (x) .

(1.3.14)

X

We denote the corresponding norm with ·L2 , and by L2 (X, Λ(T ∗(0,1)X) ⊗ E) or 0,• (X, E) L20,• (X, E), the L2 completion of Ω0,• 0 (X, E), which is the subspace of Ω consisting of elements with compact support. Definition 1.3.3. The spinc Dirac operator Dc is defined by Dc =

2n


0,• c(ej )∇Cl (X, E) −→ Ω0,• (X, E) . ej : Ω

(1.3.15)

j=1

Ω

By Proposition 1.3.1 and equation (1.3.1), Dc interchanges Ω0,even (X, E) and (X, E). We write

0,odd

c = Dc |Ω0,even (X,E) , D+

c D− = Dc |Ω0,odd (X,E) .

(1.3.16)

Lemma 1.3.4. Dc is a formally self-adjoint, first order elliptic differential operator on Ω0,• (X, E). Proof. Let s1 , s2 ∈ Ω0,• (X, E) with compact support and let α be the 1-form on X given by α(Y ) = c(Y )s1 , s2 Λ0,• ⊗E , for any vector field Y on X. Proposition 1.3.1 and (1.3.10) imply that for x ∈ X, s1 , Dc s2 Λ0,• ⊗E,x = Dc s1 , s2 Λ0,• ⊗E,x − Tr(∇α)x .

(1.3.17)

The integral over X of the last term vanishes by Proposition 1.2.1. Thus Dc is formally self-adjoint.

1.3. Spinc Dirac operator

25

For ζ ∈ T ∗ X, let ζ ∗ ∈ T X be the metric dual of ζ. The principal symbol σ(D ) of Dc is √ (1.3.18) σ(Dc )(ζ) = −1c(ζ ∗ ). c

By (1.3.2), (σ(Dc )(ζ))2 = |ζ|2 , which means, that σ(Dc )(ζ) is invertible for any ζ = 0. Thus Dc is a first order elliptic differential operator.
Let (F, hF ) be a Hermitian vector bundle on X and let ∇F be a Hermitian connection on F . Then the usual Bochner Laplacians ∆F , ∆ are defined by ∆F := −

2n  
2 F (∇F ) − ∇ T X ei ∇e ei , i=1

∆ = ∆C .

(1.3.19)

i

Let s1 , s2 ∈ C ∞ (X, F ), with compact support and let α be the 1-form on X given by α(Y )(x) = ∇F Y s1 , s2 (x), for any Y ∈ Tx X. Then by (1.2.6), (1.2.7), we get the following useful equation:    F F F ∆ s1 , s2 dvX = ∇ s1 , ∇ s2 dvX − Tr(∇α)dvX X X X (1.3.20) = ∇F s1 , ∇F s2 dvX . X

∇

Cl

We denote by ∆Cl the Bochner Laplacian on Λ(T ∗(0,1) X) ⊗ E associated to as in (1.3.19). Now we prove the Lichnerowicz formula for Dc .

Theorem 1.3.5. (Dc )2 = ∆Cl +

1  E 1 det  rX (ei , ej )c(ei )c(ej ). + R + R 4 2 2

Proof. By (1.3.2), (1.3.6) and (1.3.15),  1
 Cl Cl Cl (Dc )2 = c(ei )∇Cl ei c(ej )∇ej + c(ej )∇ej c(ei )∇ei 2 ij  Cl  Cl 1
 Cl (c(ei )c(ej ) + c(ej )c(ei ))∇Cl = ei ∇ej + c(ei ) ∇ei , c(ej ) ∇ej 2 ij   Cl Cl Cl +c(ej )[∇Cl , c(e )]∇ + c(e )c(e ) ∇ , ∇ i j i ej ei ej ei



Cl 2 TX Cl =− ∇ei ej , ek c(ei )c(ek )∇ej (∇ei ) + i

(1.3.21)

(1.3.22)

ijk

  1
Cl + c(ej )c(ei ) ∇Cl , ∇ ej ei . 2 ij But we have

TX



∇ei ej , ek = − ej , ∇TeiX ek .

(1.3.23)

26

Chapter 1. Demailly’s Holomorphic Morse Inequalities

In view of (1.3.2), (1.3.23), we obtain

TX Cl ∇ei ej , ek c(ei )c(ek )∇Cl X ej = −c(ei )c(ek )∇∇T ei ek   1
Cl Cl = ∇Cl − c(e )c(e ) ∇ − ∇ T X T X T X i k ∇e ei ∇e ek ∇e ei i i k 2 i=k

(1.3.24)

1 − c(ei )c(ek )∇Cl = ∇Cl X [ei ,ek ] . ∇T ei ei 2 Comparing to (1.3.13), we have here Cl Cl Cl Cl (RCl + RE )(el , em ) = ∇Cl el ∇em − ∇em ∇el − ∇[el ,em ] .

(1.3.22)–(1.3.25) yield  1
 2 Cl (Dc )2 = − (∇Cl + c(ej )c(ei )(RCl + RE )(ej , ei ). Xe ei ) − ∇∇T i ei 2 i

(1.3.25)

(1.3.26)

To simplify the notation, set Rijkl := RT X (ej , ei )ek , el .

(1.3.27)

By Proposition 1.3.2, we get 1 c(ej )c(ei )RCl (ej , ei ) = − Rijkl c(ei )c(ej )c(ek )c(el ) 4 1 + c(ei )c(ej )Rdet (ei , ej ). 2

(1.3.28)

By the second equation of (1.2.4) and (1.3.2),

Rijkl c(ei )c(ej )c(ek ) = 2 (Rijkl + Rjkil + Rkijl )c(ei )c(ej )c(ek ) = 0. i=k=j

i<j
Thus Rijkl c(ei )c(ej )c(ek )c(el ) = −Rijjl c(ei )c(el ) + Rijil c(ej )c(el ) = 2c(ej )c(el )Rijil = −2Rijij .

(1.3.29)

In the last equation of (1.3.29), we use that Rijil is symmetric in j, l (which follows by the first equation of (1.2.4)). By (1.2.5) and (1.3.27), we get the right-hand side of (1.3.29) equals −2rX . Hence (1.3.26)–(1.3.29) imply (1.3.21).


1.3.3 Modified Dirac operator For any Z2 -graded vector space V = V + ⊕ V − , the natural Z2 -grading on End(V ) is defined by End(V )+ = End(V + ) ⊕ End(V − ),

End(V )− = Hom(V + , V − ) ⊕ Hom(V − , V + ),

1.3. Spinc Dirac operator

27

and we define deg B = 0 for B ∈ End(V )+ , and deg B = 1 for B ∈ End(V )− . For B, C ∈ End(V ), we define their supercommutator (or graded Lie bracket) by [B, C] = BC − (−1)deg B·deg C CB.

(1.3.30)



For B, B , C ∈ End(V ), the Jacobi identity holds:   

(−1)deg C·deg B B  , [B, C] + (−1)deg B ·deg B B, [C, B  ]   + (−1)deg B·deg C C, [B  , B] = 0. (1.3.31) We will apply the above notation for spaces Λ(T ∗(0,1) X) and Ω0,• (X, E) with natural Z2 -grading induced by the parity of the degree. For i1 < · · · < ij , we define c

(ei1 ∧ · · · ∧ eij ) = c(ei1 ) . . . c(eij ).

(1.3.32)

Then by extending C-linearly, c B is defined for any B ∈ Λ(T ∗ X ⊗R C). For A ∈ Λ3 (T ∗ X), set |A|2 = i<j
for U ∈ T X

(1.3.33)

be the Hermitian connection on Λ(T ∗(0,1) X) ⊗ E induced by ∇Cl and A. Let ∆A be the Bochner Laplacian defined by ∇A as in (1.3.19). c,A Definition 1.3.6. The modified Dirac operators Dc,A , D± are defined by

Dc,A := Dc + c A,

c,A c D± := D± + c A.

(1.3.34)

Theorem 1.3.7. The modified Dirac operator Dc,A is formally self-adjoint and (Dc,A )2 = ∆A +

rX c E 1 det + (R + R ) + c (dA) − 2|A|2 . 4 2

(1.3.35)

Proof. By Lemma 1.3.4 and (1.3.10), the operator Dc +c A is formally self-adjoint. c c TX By (1.3.6), ∇Cl ei A = (∇ei A). From (1.2.44) and (1.3.2) and since A is odd degree, we have [c(ei ), c A] = −2 c(iei A), c Cl c c i TX c c(ei )(∇Cl ei A) − (∇ei A)c(ei ) = 2 (e ∧ ∇ei A) = 2 (dA).

By (1.3.19), (1.3.33) and the first equation of (1.3.36),  1  Cl ∆A = ∆Cl + ∇ei [c(ei ), c A] + [c(ei ), c A]∇Cl ei 2
1 1 2 − [c(∇TeiX ei ), c A] − [c(ei ), c A] 2 4 i
1 2 Cl c c = ∆Cl − 2c (iei A)∇Cl (iei A) . ei + [c(ei ), ∇ei A] − 2 i

(1.3.36)

(1.3.37)

28

Chapter 1. Demailly’s Holomorphic Morse Inequalities

Then Theorem 1.3.5, (1.3.33), (1.3.36) and (1.3.37) imply Cl c c 2 (Dc + c A)2 =(Dc )2 + [c(ei ), c A]∇Cl ei + c(ei )(∇ei A) + ( A)
2 c c =∆A + (c A)2 + (iei A) + c(ei )(∇Cl ei A) i

(1.3.38)

X

r 1 1 c + c (RE + Rdet ). − [c(ei ), (∇Cl ei A)] + 2 4 2 Relations (1.3.36) and (1.3.38) yield (Dc + c A)2 =∆A + (c A)2 +




c

(iei A)2 + c (dA) +

i

rX c E 1 det + (R + R ). 4 2 (1.3.39)

Let I = {i1 , . . . , im } be an ordered subset of {1, . . . , 2n}, and assume that all ij ∈ I are distinct. Let |I| be the cardinal of I. Set c eI = c(ei1 ) . . . c(eim ). Take k  2n, and let I, J be two ordered subsets of {k + 1, . . . , 2n} such that I ∩ J = ∅. Then c

e1...k c eI c e1...k c eJ = (−1)k|I| (c e1...k )2 c eI c eJ = (−1)k|I|+

k(k+1) 2

c

eI c eJ .

(1.3.40)

Since A is odd degree, (1.3.40) imply c

2

(iei A) =

2





(−1)

k(k−1) 2

c

((iei1 . . . ieik iei A)2 ),

k=0 i1 <···
2

(A) =

3





(1.3.41)

(−1)

k(k+1) 2

c

2

((iei1 . . . ieik A) ).

k=0 i1 <···
Observe that since A ∈ Λ3 (T ∗ X), A2 = 0 and (iei1 iei2 A)2 = 0. Thus (c A)2 +


i

c

2

(iei A) = −2




(iei1 iei2 iei3 A)2 = −2|A|2 .

(1.3.42)

i1



From (1.3.39) and (1.3.42), we infer (1.3.35).

1.3.4 Atiyah–Singer index theorem Theorem 1.3.8. If X is compact, the modified Dirac operator Dc,A is an essentially self-adjoint Fredholm operator, thus its kernel Ker(Dc,A ) is a finite-dimensional complex vector space. Proof. At first, if sk ∈ L20,• (X, E), Dc,A sk = 0 and lim sk = s ∈ L20,• (X, E), then k→∞

Dc,A s = 0 in the sense of distributions. By Theorem A.3.4, s ∈ Ω0,• (X, E) and

1.4. Lichnerowicz formula for
E

29

s ∈ Ker(Dc,A ). Thus the space Ker(Dc,A ) is closed, so a Hilbert space. Since X is compact, Theorems A.3.1, A.3.2 and Lemma 1.3.4 imply that Dc,A is essentially self-adjoint and the unit ball B = {s ∈ L20,• (X, E) : sL2  1, Dc,A s = 0} ⊂ Ker(Dc,A ) is compact. Thus Ker(Dc,A ) is finite-dimensional and Dc,A is Fredhlom.

(1.3.43)


c,A c,A ) of D+ as When X is compact, we define the index Ind(D+ c,A c,A c,A ) := dim Ker(D+ ) − dim Coker(D+ ) Ind(D+ c,A c,A ) − dim Ker(D− ). = dim Ker(D+

(1.3.44)

For any Hermitian (complex) vector bundle (F, hF ) with Hermitian connection ∇F and curvature RF on X, set    −RF √ , ch(F, ∇F ) := Tr exp 2π −1   −RF √ c1 (F, ∇F ) := Tr , (1.3.45) 2π −1 √   RF /(2π −1) F √ Td(F, ∇ ) := det . exp(RF /(2π −1)) − 1 By Appendix B.5 these are closed real differential forms on X and their cohomology classes do not depend on the choice of the metric hF and connection ∇F . The corresponding cohomology classes are called the Chern class of F , the first Chern class of F , the Todd class of F , respectively, and we denote them by ch(F ), c1 (F ), Td(F ) ∈ H ∗ (X, R) (see Example B.5.5) . c,A Theorem 1.3.9 (Atiyah–Singer index theorem). If X is compact, Ind(D+ ) is a topological invariant given by  c,A )= Td(T (1,0) X) ch(E). (1.3.46) Ind(D+ X

1.4 Lichnerowicz formula for
E This section is organized as follows. In Section 1.4.1, we exhibit the relation beE E,∗ and the Dirac operator Dc . In Section 1.4.2, we prove tween the operator ∂ + ∂ Bismut’s Lichnerowicz formula for the Kodaira Laplacian
E . In Section 1.4.3, we establish the Bochner–Kodaira–Nakano formula for
E . In Section 1.4.4, we prove the Bochner–Kodaira–Nakano formula with boundary term. We will use the notation from Sections 1.2, 1.3.

30

Chapter 1. Demailly’s Holomorphic Morse Inequalities E

1.4.1 The operator ∂ + ∂

E,∗

Let (X, J) be a complex manifold with complex structure J and dimC X = n, and let g T X be any Riemannian metric on T X compatible with J. We consider a holomorphic Hermitian vector bundle (E, hE ) on X. Let ∇E be the holomorphic E Hermitian (i.e., Chern) connection on (E, hE ) whose curvature is RE . Let ∂ be 0,• 0,q the Dolbeault operator acting on Ω (X, E) := ⊕q Ω (X, E). Then E

(∂ )2 = 0.

(1.4.1)

E

The complex (Ω0,• (X, E), ∂ ) is called the Dolbeault complex and its cohomology, called Dolbeault cohomology of X with values in E, is denoted by H 0,• (X, E). By the Dolbeault isomorphism (Theorem B.4.4), H 0,• (X, E) is canonically isomorphic to the qth cohomology group H q (X, OX (E)) of the sheaf OX (E) of holomorphic sections of E over X. We shortly denote H q (X, E) := H q (X, OX (E)). Especially for q = 0, H 0,0 (X, E) = H 0 (X, OX (E)) = H 0 (X, E). E,∗

(1.4.2)

E

Let ∂ be the formal adjoint of ∂ on the Dolbeault complex Ω0,• (X, E) with respect to the scalar product ·, · in (1.3.14). Set √ E E,∗  , D= 2 ∂ +∂ (1.4.3) E E,∗ E,∗ E
E = ∂ ∂ +∂ ∂ . Then
E is called the Kodaira Laplacian and D 2 = 2
E .

(1.4.4)

Thus D2 preserves the Z-grading of Ω0,• (X, E). It is a fundamental result, that the elements of Ker(
E ), called harmonic forms, represent the Dolbeault cohomology. The following theorem follows from the more general Theorem 3.1.8 on noncompact manifolds (cf. Remark 3.1.10). Theorem 1.4.1 (Hodge theory). If X is a compact complex manifold, then for any q ∈ N, we have the following direct sum decomposition: Ω0,q (X, E) = Ker(D|Ω0,q ) ⊕ Im(
E |Ω0,q ) E

= Ker(D|Ω0,q ) ⊕ Im(∂ |Ω0,q−1 ) ⊕ Im(∂

E,∗

|Ω0,q+1 ).

(1.4.5)

Thus for any q ∈ N, we have the canonical isomorphism, Ker(D|Ω0,q ) = Ker(D2 |Ω0,q )  H 0,q (X, E). Especially, H q (X, E)  H 0,q (X, E) is finite-dimensional.

(1.4.6)

1.4. Lichnerowicz formula for
E

31

Definition 1.4.2. The Bergman kernel of E is P (x, x ), (x, x ∈ X), the Schwartz kernel of P , the orthogonal projection from (L2 (X, Λ(T ∗(0,1) X) ⊗ E),  ) onto Ker(D), the kernel of D acting on Ω0,• (X, E) ∩ L2 (X, Λ(T ∗(0,1) X) ⊗ E), with respect to the Riemannian volume form dvX (x ). Especially, P (x, x ) ∈ (Λ(T ∗(0,1) X) ⊗ E)x ⊗ (Λ(T ∗(0,1) X) ⊗ E)∗x
. Remark 1.4.3. From Theorem 1.4.1, the Bergman kernel P (x, x ) is smooth on x, x ∈ X when X is compact. In general, by the ellipticity of D and Schwartz kernel theorem, we know P (x, x ) is C ∞ (cf. Problem 1.5). Recall that the tensors S, T, S, Tas were defined in (1.2.38) and (1.2.48). E

Lemma 1.4.4. For the operators ∂ , (∇E )1,0 acting on Ω•,• (X, E) in (1.1.9), we have ∂

E

(∇E )1,0

 TwX + iT (0,1) = wj ∧ ∇ j j T X  w + 1 T (wj , wk ), wm w j ∧ w k ∧ iwm , =w ∧∇ j 2 j T X  w + iT (1,0) =w ∧∇ j  T X + 1 T (wj , wk ), w m wj ∧ wk ∧ iwm . = wj ∧ ∇ wj 2

For the formal adjoints ∂ (1.3.14), we have ∂

E,∗

(∇E )1,0∗

Proof. The operator ∂

E

E,∗

and (∇E )1,0∗ of ∂

E

(1.4.7)

(1.4.8)

and (∇E )1,0 with respect to

 TwX − T (wj , wk ), w k iwj = −iwj ∇ j 1 + T (wj , wk ), wm w m ∧ iwk ∧ iwj , 2  TwX − T (w j , wk ), wk iwj = −iwj ∇ j 1 + T (w j , wk ), wm wm ∧ iwk iwj . 2

(1.4.9)

(1.4.10)

on E is given by E

∂ =

n


w i ∧ ∇E wi .

(1.4.11)

i=1

 T X ⊗ 1 + 1 ⊗ ∇E and by iT the operator  T X the connection ∇ We still denote by ∇ •,• ∗ iT ⊗ 1 on Λ (T X) ⊗ E. From (1.2.44), we deduce  T X + iT . ∇E = ε ◦ ∇

(1.4.12)

32

Chapter 1. Demailly’s Holomorphic Morse Inequalities

Relations (1.2.37) and (1.4.12) imply (1.4.7) and (1.4.8), by decomposition after bidegree and the definition of T . Observe that from (1.2.38), the (0, 1) and (1, 0)components of S are  

  T X w i , wj − ∇T X ek , wj w j , S (0,1) = ∇ wi ek   (1.4.13)

  TwX wi , w j − ∇Te X ek , w j wj . S (1,0) = ∇ i k Let s1 , s2 ∈ Ω•,• 0 (X, E) and let α be the (0, 1)-form on X given for any vector field U = U (1,0) ⊕U (0,1) ∈ T (1,0) X⊕T (0,1)X on X, by α(U ) = −iU (0,1) s1 ,s2 Λ•,• ⊗E . Note that from (1.2.6), Tr(∇α) = wj α(w j ) + w j α(wj ) − α(∇TekX ek ).

(1.4.14)

Proceeding as in the proof of (1.3.17), (1.4.13) and (1.4.14) entail the following relation between pointwise scalar products:  TwX s2 Λ•,• ⊗E,x = −iwi ∇  TwX s1 , s2 Λ•,• ⊗E,x s1 , w i ∇ i i − Tr(∇α)x + iS (0,1) α. (1.4.15) The integral of the last term vanishes by Proposition 1.2.1, so integrating (1.4.15) and (1.2.42) over X, we infer (1.4.9).

Let β be the (1, 0) form on X given by β(U ) = − iU (1,0) s1 , s2 Λ•,• ⊗E . Then as in (1.4.15),  TwX s2 Λ•,• ⊗E,x = −iwj ∇  TwX s1 , s2 Λ•,• ⊗E,x s1 , wj ∇ j j − Tr(∇β)x + iS (1,0) β. (1.4.16) Integration of (1.4.16) and (1.2.42) gives (1.4.10).




In this section, in the definition (1.3.15) of the spinc Dirac operator Dc , we choose ∇det to be the holomorphic Hermitian connection on det(T (1,0) X). Consequently D is a modified Dirac operator. Theorem 1.4.5. We have the following identity: D = Dc −

1c (Tas ). 4

(1.4.17)

Proof. In view of (1.3.1), (1.4.7) and (1.4.9), we have √ E  T X − 1 c(wi )c(wj )c(T (w i , w j )), 2 ∂ =c(wi )∇ wi 4√ √ E,∗  TwX + 2 T (wi , wj ), w k iw iw ∧ wk 2∂ =c(w i )∇ j i i 2  TwX + 1 c(w j )c(w i )c(T (wi , wj )). =c(w i )∇ i 4

(1.4.18)

1.4. Lichnerowicz formula for
E

33

Taking into account (1.4.3) and (1.4.18), we get  T X + c(w i )∇ TX D =c(wi )∇ wi wi 1 1 − c(wi )c(wj )c(T (w i , w j )) − c(w i )c(w j )c(T (wi , wj )). 4 4

(1.4.19)

X X Let ΓT ∈ T ∗ X ⊗ End(T (1,0) X) be the connection form of ∇T associated j1 jk to the frames {wj }. Note that for the frame {w ∧· · ·∧w , 1  j1 < · · · < jk  n},

 T X = d + ΓT (1,0) X wl , wm w m ∧ iw , ∇ l (1.4.20) det T (1,0) X ]. Γ = Tr[Γ (1,0)

(1,0)

Comparing with (1.2.38), (1.3.3), (1.3.5), we obtain


 T X = ∇Cl + 1 ∇ S(·)ei , ej c(ei )c(ej ). 4 ij

(1.4.21)

Clearly, by (1.2.38), 

1 1  S(ei )ei , ej (c(ei ))2 c(ej ) + S(ei )ej , ei c(ei )c(ej )c(ei ) = − c(S). (1.4.22) 4 2 Thus (1.2.39), (1.4.21), (1.4.22) imply  TwX + c(w i )∇  TwX c(wi )∇ i i

1
1 c S(ei )ej , ek c(ei )c(ej )c(ek ) = D − c(S) + 2 4 j=i=k

(1.4.23)

1 1 = Dc − c(S) + c (Tas ). 2 4 Using (1.2.42), we get 1 1 c(w i )c(w j )c(T (wi , wj )) + c(wi )c(wj )c(T (w i , w j )) 4 4

1 1 1 = T (ei , ej ), ek c(ei )c(ej )c(ek ) = c (Tas ) − c(S). 4 2 2 Finally (1.4.19), (1.4.23) and (1.4.24) imply (1.4.17).

(1.4.24)


When X is compact, the Euler number χ(X, E) of the holomorphic vector bundle E is defined by χ(X, E) =

n


(−1)q dim H q (X, E).

(1.4.25)

q=0

From Theorems 1.3.9, 1.4.1, 1.4.5, we obtain: Theorem 1.4.6 (Riemann–Roch–Hirzebruch theorem). If X is compact, then  χ(X, E) = Td(Th X) ch(E). (1.4.26) X

34

Chapter 1. Demailly’s Holomorphic Morse Inequalities

1.4.2 Bismut’s Lichnerowicz formula for
E Recall that the Bismut connection ∇B preserves the complex structure on T X by Lemma 1.2.10, thus, as in (1.2.43), it induces a natural connection ∇B on 0,• 0,• Λ(T ∗(0,1)X) which preserves its Z-grading. Let ∇B,Λ , ∇B,Λ ⊗E be the connections on Λ(T ∗(0,1) X), Λ(T ∗(0,1) X) ⊗ E defined by

= ∇B + S(·)wj , wj ,

0,•

∇B,Λ

0,•

∇B,Λ

⊗E

0,•

= ∇B,Λ

(1.4.27)

⊗ 1 + 1 ⊗ ∇E .



0,• By (1.2.42), S(·)wj , wj is a purely imaginary form, thus ∇B,Λ ⊗E is a Hermitian connection on Λ(T ∗(0,1) X) ⊗ E which preserves its Z-grading. We denote by 0,• 0,• RB,Λ the curvature of ∇B,Λ . By (1.2.60), (1.3.3) and (1.3.8), as in (1.4.21), we get for U ∈ T X, 0,•

∇B,Λ U

⊗E

1c B 1c (S (U )) = ∇Cl (iU Tas ). U − 2 4

= ∇Cl U +

0,•

As in (1.3.19), we denote by ∆B,Λ ∇B,Λ ⊗E .

⊗E

(1.4.28)

the Bochner Laplacian defined by

0,•

Theorem 1.4.7. 0,•

D2 = ∆B,Λ

⊗E

+

(1,0) rX c E 1 X + (R + Tr[RT ]) 4 2 √ −1 c 1 + (∂∂Θ) − |(∂ − ∂)Θ|2 . (1.4.29) 2 8

Proof. Let Rdet be the curvature of the holomorphic Hermitian connection on det(T (1,0) X). Then Rdet = Tr[RT

(1,0)

X

].

(1.4.30)

Theorem 1.3.7 and relations (1.2.51), (1.4.17) and (1.4.30) entail (1.4.29).




∗(0,1)

X)⊗E Remark 1.4.8. If (X, Θ) is K¨ ahler, then ∇B,E coincides with ∇Λ(T , the ∗(0,1) connection on Λ(T X)⊗E induced by the holomorphic Hermitian connections (1,0) X and ∇E . Moreover, rX = 2Rdet (wi , w i ). (1.4.29) reads ∇T ∗(0,1)

X)⊗E D2 =∆Λ(T − RE (wj , w j )   (1,0) 1 X ] (wi , wj )w j ∧ iwi . + 2 RE + Tr[RT 2

(1.4.31)

1.4. Lichnerowicz formula for
E

35

1.4.3 Bochner–Kodaira–Nakano formula Let Θ be the real (1, 1)-form associated to g T X as in (1.2.49). We define the Lefschetz operator L = (Θ ∧ ) ⊗ 1 on Λ•,• (T ∗ X) ⊗ E and its adjoint Λ = i(Θ) with respect to the Hermitian product ·, ·Λ•,• ⊗E induced by g T X and hE . For {wj }nj=1 a local orthonormal frame of T (1,0) X, we have L=

√ −1wj ∧ wj ∧ ,

√ Λ = − −1iwj iwj .

(1.4.32) E,∗

Let us define the formal adjoints (∇E )1,0∗ of (∇E )1,0 and (∇E )0,1∗ = ∂ of E E 0,1 (∇ ) = ∂ with respect to (1.3.14) as in Lemma 1.4.4. We use next the supercommutator defined in (1.3.30), and we apply it on Ω•,• (X, E) endowed with natural Z2 -grading induced by the parity of degree. Definition 1.4.9. The holomorphic and anti-holomorphic Kodaira Laplacians are defined by:   E
= (∇E )1,0 , (∇E )1,0∗ ,  E E,∗  .
E = ∂ , ∂

(1.4.33)

The Hermitian torsion operator is defined by T := [Λ, ∂Θ] = [i(Θ), ∂Θ] .

(1.4.34)

 TX. Let us express now T in terms of the torsion T of the connection ∇ Lemma 1.4.10. We have T =

  1 T (wj , wk ), w m  2 wk ∧ w m ∧ iwj − 2 δjm wk − wj ∧ wk ∧ iwm . (1.4.35) 2

Proof. From (1.2.48), (1.2.54) and (1.4.34), we obtain T =

√  −1 T (wj , wk ), w m  [Λ, ω j ] ∧ ω k ∧ wm 2

 + ω j ∧ [Λ, ω k ] ∧ wm + ω j ∧ ω k ∧ [Λ, w m ] . (1.4.36)

By the formula (1.4.32) for Λ, we easily get √ √ [Λ, ω j ] = − −1iwj , [Λ, ω m ] = −1iwm .

(1.4.37)

Now, (1.4.36), (1.4.37) together with T (wj , wk ) = −T (wk , wj ) imply the desired relation (1.4.35).
We have the following generalization of the usual K¨ ahler identities in the presence of torsion.

36

Chapter 1. Demailly’s Holomorphic Morse Inequalities

Theorem 1.4.11 (generalized K¨ahler identities).  E,∗  √ E 1,0  ∂ , L = −1 (∇ ) + T , √ E  E 1,0∗   (∇ ) , L = − −1 ∂ + T , √

  E Λ, ∂ = − −1 (∇E )1,0∗ + T ∗ ,   √ E,∗ ∗ +T . Λ, (∇E )1,0 = −1 ∂

(1.4.38a) (1.4.38b) (1.4.38c) (1.4.38d)

Proof. Remark that the third and forth formulas are the adjoints of the first two. Thus it suffices to prove (1.4.38a), (1.4.38b). Using (1.4.9) we find  E,∗   TwX , L] − T (wj , wk ), w k [iw , L] ∂ , L = [−iwi ∇ j i (1.4.39) 1 + T (wj , wk ), wm [w m ∧ iwk iwj , L] . 2 By (1.4.32),



√  iwj , L = − −1 wj ∧ ,

[iwj , L] =

 T X so from (1.4.40)  T X L = L∇ By (1.2.52), ∇ wi wi

√ −1 wj ∧ .

(1.4.40)

√ TX . −1 wj ∧ ∇ wj

(1.4.41)

 [wm ∧ iwk iwj , L] = w m ∧ [iwk , L]iwj + iwk [iwj , L] √ = − −1w m ∧ (ω k ∧ iwj + iwk ω j ).

(1.4.42)

 T X , L] = −[iw , L]∇ TX = [−iwj ∇ j wj wj By (1.4.40), we infer

Relations (1.4.39)–(1.4.42) yield finally √  E,∗  √  TwX + −1 T (wj , wk ), w k wj ∂ , L = −1 wj ∧ ∇ j (1.4.43) √ + −1 T (wj , wk ), w m wk ∧ w m ∧ iwj .  √

Adding (1.4.8) and (1.4.35) shows that −1 (∇E )1,0 + T equals the right-hand side of (1.4.43), hence (1.4.38a) holds. Formula (1.4.38b) can be proved along similar lines as (1.4.38a). Alternatively, as the computation is local, we can choose a local holomorphic frame of E and using (1.4.40), we reduce the proof to the case of a trivial line bundle E. But then (1.4.38b) follows from (1.4.38a) by conjugation.
Theorem 1.4.12 (Bochner–Kodaira–Nakano formula). √ ∗ E (1.4.44)
E =
+ [ −1RE , Λ] + [(∇E )1,0 , T ∗ ] − [(∇E )0,1 , T ] . √ E,∗ ∗ = − −1 [Λ, (∇E )1,0 ] − T . Thus Proof. From (1.4.38d) we deduce that ∂ √  E  E E,∗ E ∗
E = [ ∂ , ∂ ] = − −1 ∂ , [Λ, (∇E )1,0 ] − [∂ , T ] . (1.4.45)

1.4. Lichnerowicz formula for
E

37

The Jacobi identity (1.3.31) implies  E     E  E ∂ , [Λ, (∇E )1,0 ] = Λ, [(∇E )1,0 , ∂ ] + (∇E )1,0 , [∂ , Λ] .

(1.4.46)

E

Since (∂ )2 = 0, ((∇E )1,0 )2 = 0, we have E

RE = (∇E )2 = [(∇E )1,0 , ∂ ].

(1.4.47)

E

Using the expression of [∂ , Λ] given in (1.4.38c) we find  E 1,0 E  √    √  (∇ ) , [∂ , Λ] = −1 (∇E )1,0 , (∇E )1,0∗ + −1 (∇E )1,0 , T ∗ . Taking into account the definition of
(1.4.45)–(1.4.48).

E

(1.4.48)

(cf. (1.4.33)), we conclude (1.4.44) from


Corollary 1.4.13. Assume that (X, g T X ) is K¨ ahler. Then √ E
E =
+ [ −1RE , Λ] , ∆ = 2
= 2
.

(1.4.49a) (1.4.49b)

C

Here
:=
= ∂∂ ∗ + ∂ ∗ ∂;
:=
C are usual ∂-Laplacian and ∂-Laplacian, ∆ = dd∗ + d∗ d is the Bochner Laplacian on Λ(T ∗ X) and d∗ is the adjoint of d. Therefore, the Hodge decomposition holds for the de Rham cohomology group H • (X, C): p (a) H j (X, C) ∼ = ⊕p+q=j H q (X, O ) ∼ = ⊕p+q=j H p,q (X), X

(b) H p,q (X) ∼ = H q,p (X). We denote here by H p,q (X) := H p,q (X, C) the Dolbeault cohomology groups. Proof. Indeed, by Theorem 1.2.8, g T X is K¨ahler if and only if T = 0, so (1.4.49a) follows trivially from (1.4.44). By taking E = C with a trivial metric, we obtain
=
. Moreover ∗

∗

∆ = [d, d∗ ] = [∂ + ∂, ∂ ∗ + ∂ ] =
+
+ [∂, ∂ ] + [∂, ∂ ∗ ],

(1.4.50)

and the two latter brackets vanish (Problem 1.6). By the real analogue of Theorem
1.4.1 (Hodge theory), H • (X, C)  Ker(∆). This completes the proof. Theorem 1.4.14 (Nakano’s inequality). For any s ∈ Ω•,• 0 (X, E), √

3 E 
s, s  [ −1RE , Λ]s, s 2 1 ∗ − (T s2L2 + T ∗ s2L2 + T s2L2 + T s2L2 ) . (1.4.51) 2 If (X, g T X ) is K¨ ahler, then

√ 
E s, s ≥ [ −1RE , Λ]s, s.

(1.4.52)

38

Chapter 1. Demailly’s Holomorphic Morse Inequalities •,•

Proof. Let s ∈ Ω0 (X, E). Since E


E s, s = ∂ s2L2 + ∂

E,∗

s2L2 ,

E


s, s = (∇E )1,0 s2L2 + (∇E )1,0∗ s2L2 ,

(1.4.53)

we deduce from (1.4.44) that E

E,∗

∂ s2L2 + ∂ s2L2 =(∇E )1,0 s2L2 + (∇E )1,0∗ s2L2 √



E ∗

+ [ −1RE , Λ]s, s + [(∇E )1,0 , T ∗ ]s, s − [∂ , T ]s, s .

(1.4.54)

By the Cauchy–Schwarz inequality, we find  

 1   (∇E )1,0 s2L2 + (∇E )1,0∗ s2L2 + T s2L2 + T ∗ s2L2 ,  [(∇E )1,0 , T ∗ ]s, s   2  E ∗ 

 1 E 2  [∂ , T ]s, s   ∂ s 2 + ∂ E,∗ s2 2 + T s2 2 + T ∗ s2 2 . L L L L 2 Therefore  1

 3 E 2 E,∗ ∂ sL2 + ∂ s2L2  (∇E )1,0 s2L2 + (∇E )1,0∗ s2L2 2 2 √  1

∗ + [ −1RE , Λ]s, s − T s2L2 + T ∗ s2L2 + T s2L2 + T s2L2 , 2

(1.4.55)




whereby the conclusion.

For the purpose of proving vanishing theorems and the spectral gap for forms of bidegree (0, q) with values in a positive bundle (especially on non-compact manifolds or with boundary), we derive sometimes another form of the Bochner–  = E ⊗ K ∗ where Kodaira–Nakano formula. Set E X ∗ KX = Λn (T (1,0) X) = det(T (1,0) X). ∗ ∼ Since KX ⊗ KX = C, there exists a natural isometry

 Ψ =∼ : Λ0,q (T ∗ X) ⊗ E −→ Λn,q (T ∗ X) ⊗ E, Ψ s = s = (w1 ∧ · · · ∧ wn ∧ s) ⊗ (w1 ∧ · · · ∧ wn ),

(1.4.56)

where {wj }nj=1 a local orthonormal frame of T (1,0) X. Theorem 1.4.15. For any s ∈ Ω0,• (X, E), we have  E

∗


E s = Ψ−1
Ψs + RE⊗KX (wj , w k )w k ∧ iwj s  E ∗   +Ψ−1 (∇E )1,0 T ∗ Ψs − ∂ , Ψ−1 T Ψ s.

(1.4.57)

1.4. Lichnerowicz formula for
E

39

Proof. We apply (1.4.44) for s:  E




E s =
s +

√

  E ∗      −1RE , Λ s + (∇E )1,0 , T ∗ s − ∂ , T s.

(1.4.58)

∗ is a holomorphic bundle, Since KX  E

E

∂ s = (∂ s)∼ ,

∂

 E,∗

s = (∂

E,∗

s)∼ ,




E s = (
E s)∼ .

(1.4.59)



Hence Ψ−1
E s =
E s. Likewise  E ∗   E ∗  Ψ−1 ∂ , T s = ∂ , Ψ−1 T Ψ s,     Ψ−1 (∇E )1,0 , T ∗ s = Ψ−1 (∇E )1,0 T ∗ s,  E



(1.4.60)



Ψ−1
Ψs = Ψ−1 (∇E )1,0 (∇E )1,0∗ Ψs . By (1.4.37) we have √

   [ −1RE , Λ] = RE (wj , w k ) wj ∧ iwk − iwj w k ∧ , thus Ψ−1

 √ ∗  −1RE , Λ s = RE⊗KX (wj , w k )w k ∧ iwj s.

(1.4.61)

(1.4.62)


From(1.4.59), (1.4.60) and (1.4.62), we obtain (1.4.57).

 T X on Remark 1.4.16. Assume now that g T X is K¨ahler. Then T = 0, and ∇ (1,0) X , Λ(T ∗(0,1)X) ⊗ E is induced by the holomorphic Hermitian connections ∇T E 0,• 0,• TX  TX TX   = − i (∇wi ∇wi − ∇∇T X wi ). From (1.4.8) and ∇ . On Ω (X, E), set ∆ (1.4.10), for s ∈ Ω

0,•

(X, E), we obtain Ψ

−1

∗

 E

wi


Ψs = ∆


E s = ∆0,• s + RE⊗KX (wj , w k )w k ∧ iwj s

0,•

s. We infer from (1.4.57):

for s ∈ Ω0,• (X, E) .

(1.4.63)



∗ 3 E 2 E,∗ ∂ sL2 + ∂ s2L2  RE⊗KX (wj , w k )w k ∧ iwj s, s 2  1

∗ − T ∗ s2L2 + T s2L2 + T s2L2 . 2

(1.4.64)

Corollary 1.4.17. For any s ∈ Ω0,q 0 (X, E),

 we obtain Proof. By applying (1.4.59) and (1.4.62) to (1.4.51) with s ∈ Ωn,q (X, E), (1.4.64). Alternatively, we can repeat the proof of Theorem 1.4.14 by replacing (1.4.44) with (1.4.57).


40

Chapter 1. Demailly’s Holomorphic Morse Inequalities

1.4.4 Bochner–Kodaira–Nakano formula with boundary term Keeping the same notations as before, let M be a smooth, relatively compact domain in X. We set M = {x ∈ X : (x) < 0} where  ∈ C ∞ (X) satisfies |d| = 1 on ∂M . (This is always possible by replacing  by /|d| near ∂M and using a partition of unity argument.) Let M be the closure of M . Let −en ∈ T M be the metric dual of d. Then en is the inward pointing unit (1,0) (0,1) ∈ T (1,0) X ⊕ T (0,1)X. Then normal at ∂M . We decompose en as en = en + en we have (1,0)

en

= −wj ()wj ,

(0,1)

en

= −wj ()w j .

(1.4.65)

•,• To simplify the in the rest

 of this section, for s1 , s2 ∈ Ω (M , E), notation we will denote by s1 , s2 the integral M s1 , s2 Λ•,• ⊗E,x dvX (x).

Lemma 1.4.18. For s1 , s2 ∈ Ω•,• (M , E), we have  E



E,∗ ∂ s 1 , s2 − s 1 , ∂ s 2 = ∂ ∧ s1 , s2 Λ•,• ⊗E dv∂M , ∂M 

E 1,0∗



E 1,0 s1 , ∂ ∧ s2 Λ•,• ⊗E dv∂M . s 1 , s2 = s1 , (∇ ) s2 − (∇ )

(1.4.66)

∂M

Proof. Let γ be a 1-form on X. From (1.2.9), (1.2.10) and Stokes theorem (remark that dvX = d ∧ dv∂M on ∂M ), we get   Tr(∇γ)dvX = − γ(en )dv∂M . (1.4.67) M

∂M

In view of (1.4.15), (1.4.65) and (1.4.67), we get  E,∗

E s 1 , ∂ s 2 − ∂ s 1 , s2 = α(en )dv∂M ∂M  



ie(0,1) s1 , s2 Λ•,• ⊗E dv∂M = s1 , ∂ ∧ s2 Λ•,• ⊗E dv∂M . (1.4.68) =− ∂M

n

∂M

Similarly, from (1.4.10), (1.4.16) and (1.4.67), we obtain



s1 , (∇E )1,0 s2 − (∇E )1,0∗ s1 , s2  



=− ie(1,0) s1 , s2 Λ•,• ⊗E dv∂M = s1 , ∂ ∧ s2 Λ•,• ⊗E dv∂M . (1.4.69) ∂M

n

∂M




The proof of Lemma 1.4.18 is complete. E,∗

E

Let ∂ H be the Hilbert space adjoint of ∂ on M . By definition, s ∈ E,∗ Dom(∂ H ) if and only if there exists s1 ∈ L2 (M, Λ•,• (T ∗ X) ⊗ E) such that for E E E,∗ any s2 ∈ Dom(∂ ), s, ∂ s2  = s1 , s2  and then ∂ H s = s1 . Let us set B 0,q (M, E) = {s ∈ Ω0,q (M , E) : ie(0,1) s = 0 on ∂M }. n

(1.4.70)

1.4. Lichnerowicz formula for
E

41 E,∗

E,∗

Proposition 1.4.19. We have B 0,q (M, E) = Dom(∂ H ) ∩ Ω0,q (M , E) and ∂ H = E,∗ ∂ on B 0,q (M, E). E,∗

Proof. For s1 ∈ Dom(∂ H ) ∩ Ω0,q (M , E), s2 ∈ Ω0,q−1 (M , E),  E,∗



E,∗ E ie(0,1) s1 , s2 Λ•,• ⊗E dv∂M . ∂ H s1 , s2 = s1 , ∂ s2  = ∂ s1 , s2 − ∂M

n

E,∗

E,∗

If s2 ∈ Ω0,q−1 (M, E), the boundary term vanishes, thus ∂ H s1 ,s2 = ∂ s1 ,s2 . 0 E,∗

E,∗

Since Ω0,q−1 (M, E) is dense in L20,q−1 (M, E), it follows that ∂ H s1 = ∂ s1 . This 0 implies that the boundary term vanishes for all s2 ∈ Ω0,q−1 (M , E), so ie(0,1) s1 = 0 n on ∂M .
Definition 1.4.20. The Levi form of ∂M is the restriction of ∂∂ to the holomorphic tangent bundle of ∂M . For s ∈ Ω0,q (M , E), at y ∈ ∂M , set L (s, s) = (∂∂)(wk , w j )w j ∧ iwk s, sΛ•,• ⊗E,y .

(1.4.71)

Theorem 1.4.21. For any s ∈ B 0,• (M, E), we have E

∂ s2L2 + ∂

E,∗



∗  s2L2 = (∇E )1,0∗ s2L2 + RE⊗KX (wj , w k )w k ∧ iwj s, s ∗

E

− ∂ s, Ψ−1 T s − Ψ−1 T s, ∂

E,∗



s + T ∗ s, (∇E )1,0∗ s  + L (s, s) dv∂M . (1.4.72) ∂M

E,∗

Proof. Since s ∈ B 0,• (M, E) = Dom(∂ H ) ∩ Ω0,q (M , E), by (1.4.66), we have ∂

E,∗

E E,∗

s2L2 =  ∂ ∂ s, s,  E,∗ E   E E ∂ s2L2 = ∂ ∂ s, s + ∂ s, ∂ ∧ sΛ•,• ⊗E dv∂M ,  E   E ∂M    ∗  ∗ E,∗ ∂ , Ψ−1 T Ψ s, s = ∂ s, Ψ−1 T s + Ψ−1 T s, ∂ s , (1.4.73)        (∇E )1,0 T ∗ s, s = T ∗ s, (∇E )1,0∗ s + ∂ ∧ T ∗ s, sΛ•,• ⊗E dv∂M , ∂M   E   1,0∗ 2  E
s, s = (∇ ) sL2 + ∂ ∧ (∇E )1,0∗ s, sΛ•,• ⊗E dv∂M . ∂M

Thus (1.4.57), (1.4.73) yield E

∂ s2L2 + ∂

E,∗



∗  s2L2 = (∇E )1,0∗ s2L2 + RE⊗KX (wj , w k )w k ∧ iwj s, s

E

 + ∂M

∗

−  ∂ s, Ψ−1 T s − Ψ−1 T s, ∂

 E ∂ s, ∂ ∧ sΛ•,• ⊗E

E,∗



s + T ∗ s, (∇E )1,0∗ s (1.4.74)   + ∂ ∧ ((∇E )1,0∗ + T ∗ ) s, sΛ•,• ⊗E dv∂M .

42

Chapter 1. Demailly’s Holomorphic Morse Inequalities

To conclude our theorem, we need to compute the last two terms in (1.4.74). By (1.4.59) and (1.4.65), we infer  E     ∂ s, ∂ ∧ s + ∂ ∧ ((∇E )1,0∗ + T ∗ ) s, s Λ•,• ⊗E Λ•,• ⊗E   E  = − ie(0,1) ∂ s + Ψ−1 ∂ ∧ ((∇E )1,0∗ + T ∗ )Ψs, s •,• . (1.4.75) n

⊗E

Λ

Recall that on T X ⊗R C, we denote also by  ,  the C-bilinear form induced  T X the connection on by g T X . As in the proof of Lemma 1.4.4, we denote by ∇ •,• ∗ E TX  Λ (T X) ⊗ E induced by ∇ and ∇ . From (1.4.35), we get   1 T ∗ = T (w j , wk ), wm  2 wj ∧ iwm ∧ iwk − 2δjm iwk − wm ∧ iwk iwj . (1.4.76) 2 By (1.4.10) and (1.4.76), we obtain   TwX + T (w j , wk ), wm w j ∧ iwm ∧ iw . (∇E )1,0∗ + T ∗ = −iwj ∇ k j

(1.4.77)

Thus from (1.4.7), (1.4.65), (1.4.77) and ie(0,1) s = 0 on ∂M , we have on ∂M , n



E

− ie(0,1) ∂ s + Ψ−1 ∂ ∧ ((∇E )1,0∗ + T ∗ )Ψs n   TwX + en(0,1) , wj ∇  TwX = − ie(0,1) wj ∇ j j n   1 (0,1) + T (w j , wk ), wm  − ie(0,1) w j ∧ w k ∧ iwm − en , wk w j ∧ iwm s 2 n   T X + en(0,1) , wj ∇  T X s. (1.4.78) = − i (0,1) w j ∇ wj

en

wj

To compute the term in (1.4.78), we use again our boundary condition. Relations (1.4.65) and (1.4.70) yield (iwj s)wj () = − ie(0,1) s = 0 n

on ∂M.

(1.4.79)

Especially, (iwj s)wj ∈ T ∂M ⊗ Λ0,• (T ∗ X) ⊗ E on ∂M . Thus at y ∈ ∂M , we have  TwX (i (0,1) s), iw sΛ•,• ⊗E,y = w j ∇  TwX (i (0,1) s), sΛ•,• ⊗E,y . 0 = ∇ j j j e e n

n

(1.4.80)

Now  T X i (0,1) = −i (0,1) w j ∇  T X + en wj ∇ wj e wj e

(0,1)

n

n

 T X + ∇  T X en , wj ∇ wj wj

(0,1)

, wk w j iwk . (1.4.81)

Moreover, from (1.4.7) and (1.4.65),  TwX en ∇ j

(0,1)

(0,1)

, wk  = w j en

(0,1)

, wk  − en

 TwX wk  ,∇ j

 TwX wk ) = −wj (∂, wk ) + (∂, ∇ j  TwX ∂)(wk ) = (∂∂)(wk , w j ). (1.4.82) = −(∇ j

1.5. Spectral gap



43

Using (1.4.78), (1.4.80), (1.4.81) and (1.4.82), we get at y ∈ ∂M ,  E  − ie(0,1) ∂ s + Ψ−1 ∂ ∧ ((∇E )1,0∗ + T ∗ )Ψs, s Λ•,• ⊗E,y

n

= (∂∂)(wk , w j )w iwk s, sΛ•,• ⊗E,y = L (s, s). (1.4.83) j

Finally, (1.4.74), (1.4.75) and (1.4.83) imply (1.4.72).




Similarly to Corollary 1.4.17, we obtain: Corollary 1.4.22. For any s ∈ B 0,q (M, E),  1

∗ 3 E 2 E,∗  ∂ sL2 + ∂ s2L2  (∇E )1,0∗ s2L2 + RE⊗KX (wj , w k )w k ∧ iwj s, s 2 2   1

∗ + L (s, s) dv∂M − T ∗ s2L2 + T s2L2 + T s2L2 . (1.4.84) 2 ∂M Our proof of the Bochner–Kodaira–Nakano formula with boundary term (1.4.72) and of the estimate (1.4.84) takes a different route as the usual proof, which consists in integrating by parts starting with the left-hand side and deriving at the end also the Bochner–Kodaira–Nakano formula without boundary. We integrate here directly (1.4.57) and we can easily identify the boundary term. It is remarkable that the curvature and the torsion do not contribute to the boundary integral.

1.5 Spectral gap As a direct application of the Lichnerowicz formula and Bochner–Kodaira–Nakano formula, we obtain various vanishing theorems and exhibit the spectral gap for the modified Dirac operators. The spectral gap property will play an essential role in our approach to the Bergman kernel. This section is organized as follows. In Section 1.5.1, we obtain the vanishing theorems and the spectral gap property for the Kodaira Laplacian. In Section 1.5.2 we establish the spectral gap property for a modified Dirac operator on symplectic manifolds.

1.5.1 Vanishing theorem and spectral gap Lemma 1.5.1 (∂∂-Lemma). Let ϕ be a smooth, real, d-exact, (q, q)-form on a compact K¨ ahler manifold M ; then there exists a smooth, real, (q − 1, q − 1)-form ρ on M such that √ (1.5.1) ϕ = −1∂∂ρ. ∗

Proof. Let ∂ , ∂ ∗ , d∗ be the adjoint of ∂, ∂, d associated to the K¨ahler metric g T M . From (1.4.49b) and (1.4.50), Ker(
) = Ker(d) ∩ Ker(d∗ ).

44

Chapter 1. Demailly’s Holomorphic Morse Inequalities

As ϕ is d-exact, ϕ is orthogonal to Ker(d∗ ), thus ϕ is orthogonal to Ker(
). By Hodge theory (Theorem 1.4.1) for E = Λq (T ∗(1,0) M ), there exists a (q, q)-form ϕ1 such that ϕ = 2
ϕ1 = (d d∗ + d∗ d)ϕ1 .

(1.5.2)

Again using ϕ is d-exact and Im(d) ∩ Im(d∗ ) = 0, we get ϕ = d d∗ ϕ1 . Let ψ q−1,q (resp. ψ q,q−1 ) be the (q −1, q) (resp. (q, q −1))-component of d∗ ϕ1 . As ϕ is a (q, q)-form, we get ϕ = ∂ψ q−1,q + ∂ψ q,q−1 ,

∂ψ q−1,q = 0,

∂ψ q,q−1 = 0.

(1.5.3)

(If q = 1, we get directly (1.5.3) from the d-exactness of ϕ). We claim that if θ is a (q − 1, q)-form and ∂θ = 0, then there exists a (q − 1, q − 1)-form η such that ∂θ = ∂∂η.

(1.5.4)

By Hodge theory (Theorem 1.4.1) for E = Λq−1 (T ∗(1,0) M ), there exists a smooth (q − 1, q − 1)-form η such that ∗

∗

∗

∂ θ = (∂ ∂ + ∂ ∂ )η.

(1.5.5)

∗

(1.4.5) shows that Im(∂) ∩ Im(∂ ) = 0. Thus we get ∗

∂ (θ − ∂η) = 0,

∗

∂ ∂ η = 0.

(1.5.6)

But from ∂(θ − ∂η) = 0, (1.4.49b) and (1.5.6) we know ∗

θ − ∂η ∈ Ker(∂) ∩ Ker(∂ ) = Ker(
) = Ker(∂) ∩ Ker(∂ ∗ ).

(1.5.7)

Thus we get (1.5.4) for θ and η. For ψ q,q−1 , we will apply (1.5.4) for ψ q,q−1 . Thus there exists ρ such that (1.5.1) holds. As ϕ is real, we can take ρ as real.
For a holomorphic Hermitian line bundle (F, hF ) on a complex manifold M , we will call the curvature RF associated to the holomorphic Hermitian connection ∇F on (F, hF ) simply the curvature RF associated to hF . √ The curvature RF is a (1, 1)-form on M and −1RF is real. For any holomorphic local frame s of F on an open set U , RF (x) = ∂∂ log |s(x)|2hF

on U.

(1.5.8)

Definition 1.5.2. A holomorphic line bundle F on a complex manifold M is positive (resp. semi-positive) if there is a metric hF on F with associated curvature RF √ F such that −1R is a positive (resp. semi-positive) (1, 1)-form on M . F is negative if F ∗ is positive.

1.5. Spectral gap

45

Certainly, the notions of positivity (Definition 1.5.2), Griffiths positivity and Nakano positivity (Definition 1.1.6) are equivalent for holomorphic line bundles. Proposition 1.5.3. Let F be a holomorphic line bundle on a compact K¨ ahler manifold M . If Ω is a real, closed (1, 1)-form on M with [Ω] = c1 (F ) ∈ H 2 (M, R),

(1.5.9)

then, up to multiplication by positive constants, there exists a unique metric hF √ −1 F on F such that Ω = 2π R , where RF is the curvature associated to hF . Thus F is positive if and only if its first Chern class may be represented by a positive form in H 2 (M, R). metric on F and let R0F be the curvature associated Proof. Let hF 0 be a Hermitian √ −1 F F to h0 . Then by (1.5.9), Ω− 2π R0 is a real, d-exact, (1, 1)-form on M . By Lemma 1.5.1, there exists a real function ρ on M such that √ √ −1 F −1 Ω= R0 + ∂∂ρ. (1.5.10) 2π 2π √ From (1.5.8) and (1.5.10), we know −2π −1Ω is the curvature associated to the metric eρ hF 0 on F . √ −1 F Let hF 1 be another metric on F such that Ω = 2π R1 . Then there is a real F ρ1 F function ρ1 such that h1 = e h . By (1.5.8), we have ∂∂ρ1 = 0.

(1.5.11)

Taking the trace of both sides in (1.5.11) and using (1.4.49b), we get ∆ρ1 = 0. Thus ρ1 is a constant function on X (cf. Problem 1.9).
For a variant of Proposition 1.5.3 for singular Hermitian metrics, see Lemma 2.3.5. Theorem 1.5.4. Let X be a compact complex manifold of dimension n and F be a positive holomorphic line bundle on X. Then: (a) (Kodaira vanishing theorem) H q (X, F ⊗ KX ) = 0, if q > 0. (b) (Nakano vanishing theorem) H r,q (X, F ) = 0, if r + q > n. Proof. Let hF be a metric on F with associated curvature RF such that ω = √ −1 F is a positive (1, 1)-form. Let g T X := ω(·, J·) be the associated K¨ahler 2π R √ metric on T X. Then the Hermitian torsion T = 0. Moreover, as ω = −1wj ∧ w j , by (1.4.37), we have [ω, Λ] = wk ∧ iwk − iwk w k ∧ . Thus for s ∈ Ω

r,q

(1.5.12)

(X, F ), we have [ω, Λ]s = (r + q − n)s.

(1.5.13)

Now the Nakano inequality (1.4.51) implies that if s is harmonic, i.e.,
F s = 0, it follows that s = 0 wherever r + q > n. By Hodge theory (Theorem 1.4.1) for the

46

Chapter 1. Demailly’s Holomorphic Morse Inequalities

holomorphic vector bundle Λr (T ∗(1,0) X) ⊗ F , we get (b). (a) is a particular case of (b) for r = n.
Now we will study the spectral gap property for Kodaira Laplacians. Let (X, J) be a compact complex manifold with complex structure J and dimC X = n. Consider a holomorphic Hermitian line bundle (L, hL ) on X, and a holomorphic Hermitian vector bundle (E, hE ) on X. Let ∇E , ∇L be the holomorphic Hermitian (i.e., Chern) connections on (E, hE ), (L, hL ) with curvatures RE , RL . Choose any Riemannian metric g T X on T X, compatible with the complex structure J. Set √ −1 L R , ω := Θ(·, ·) := g T X (J·, ·). (1.5.14) 2π Then ω, Θ are real (1, 1)-forms on X, and ω is the Chern–Weil representative of the first Chern class c1 (L) of L. Then the Riemannian volume form dvX of (T X, g T X ) is Θn /n!. We will identify the two-form RL with the Hermitian matrix R˙ L ∈ End(T (1,0) X) such that for W, Y ∈ T (1,0) X, RL (W, Y ) = R˙ L W, Y .

(1.5.15)

Let {wj }nj=1 be a local orthonormal frame of T (1,0) X with dual frame {wj }nj=1 . Set

RL (wl , w m ) w m ∧ iwl , τ (x) = RL (wj , w j ) . (1.5.16) ωd = − j

l,m

Then ωd ∈ End(Λ(T ∗(0,1) X)) and RL acts as the derivative ωd on Λ(T ∗(0,1) X). By (1.3.32), we have 1
L c (RL ) = R (ei , ej )c(ei )c(ej ) = −2ωd − τ. (1.5.17) 2 ij If we choose {wj }nj=1 to be an orthonormal basis of T (1,0) X such that R˙ L (x) = diag(a1 (x), . . . , an (x)) ∈ End(Tx(1,0) X), then ωd (x) = −




aj (x)w j ∧ iwj ,

τ (x) =

j




aj (x).

(1.5.18)

(1.5.19)

j

For p ∈ N, we denote by Lp := L⊗p . By replacing E by Lp ⊗ E in (1.4.3), we get √ Lp ⊗E Lp ⊗E,∗  , +∂ Dp = 2 ∂ (1.5.20) Lp ⊗E Lp ⊗E,∗ Lp ⊗E,∗ Lp ⊗E
p = ∂ ∂ +∂ ∂ . Dp2 = 2
p preserves the Z-grading on Ω0,• (X, Lp ⊗ E).

1.5. Spectral gap

47

We make the following basic assumption in the rest of this section. √ Assumption: −1RL is a positive (1, 1)-form on X, equivalently, for any 0 = Y ∈ T (1,0)X, we have RL (Y, Y ) > 0. (1.5.21) In the notation of (1.5.15)–(1.5.18), the condition (1.5.21) is equivalent to: R˙ L ∈ End(T (1,0) X) is positive-definite, i.e., aj (x) > 0 for any x ∈ X, 1  j  n. (1.5.22) Theorem 1.5.5. There  exist C0 , CL > 0 such that for any p ∈ N and any s ∈ Ω0,>0 (X, Lp ⊗ E) = q1 Ω0,q (X, Lp ⊗ E), Dp s2L2  (2C0 p − CL )s2L2 .

(1.5.23)

The spectrum Spec(
p ), of the Kodaira Laplacian
p , is contained in the set {0}∪ ]pC0 − 12 CL , +∞[. Proof. By (1.4.64) and (1.5.16), we get for any s ∈ Ω0,• (X, Lp ⊗ E),

Lp ⊗E 2 4 Lp ⊗E,∗ 2  Dp s2L2 = 2 ∂ sL2 + ∂ sL2  −ωd s, s p − Cs2L2 . (1.5.24) 3 Hence (1.5.18) and (1.5.22) yield (1.5.23). If s ∈ C ∞ (X, Lp ⊗ E) is an eigensection of Dp2 with Dp2 s = λs and λ = 0, then 0 = Dp s ∈ Ω0,1 (X, Lp ⊗ E), and Dp2 Dp s =
λDp s. Thus λ  2C0 p − CL . This finishes the last part of Theorem 1.5.5. By Theorems 1.4.1, 1.5.5, we conclude: Theorem 1.5.6 (Kodaira–Serre vanishing theorem). If L is a positive line bundle, then there exists p0 > 0 such that for any p  p0 , H q (X, Lp ⊗ E) = 0

for any q > 0.

(1.5.25)

1.5.2 Spectral gap of modified Dirac operators Let (X, J) be a compact manifold with almost complex structure J and dimR X = 2n. Let (L, hL ) be a Hermitian line bundle on X, and let (E, hE ) be a Hermitian vector bundle on X. Let ∇E , ∇L be Hermitian connections on (E, hE ), (L, hL ). Let RL = (∇L )2 , RE = (∇E )2 be the curvatures of ∇L , ∇E . Let g T X be any Riemannian metric on T X compatible with the almost complex structure J. We use the notation from (1.5.14)–(1.5.19) now. Assumption: (1.5.21) holds for RL . Set µ0 =

inf (1,0)

u∈Tx

X, x∈X

RxL (u, u)/|u|2gT X = inf aj (x) > 0. x∈X, j

(1.5.26)

48

Chapter 1. Demailly’s Holomorphic Morse Inequalities

Let ∇det be a Hermitian connection on det(T (1,0) X) with curvature Rdet . c the spinc Dirac operator defined in (1.3.15) associated to We denote by Dpc , D±,p p det L ⊗ E and ∇ . For A ∈ Λ3 (T ∗ X), by (1.3.34), set Dpc,A = Dpc + c A,

c,A c D±,p = D±,p + c A.

(1.5.27)

Theorem 1.5.7. There  exists CL > 0 such that for any p ∈ N and any s ∈ Ω0,>0 (X, Lp ⊗ E) = q1 Ω0,q (X, Lp ⊗ E), Dpc,A s2L2  (2µ0 p − CL )s2L2 .

(1.5.28)

Especially, for p large enough, c,A Ker(D−,p ) = 0.

(1.5.29)

Proof. At first, we claim that there exists a constant C > 0 such that for any p ∈ N, s ∈ C ∞ (X, Lp ⊗ E), we have ∇L

p

⊗E

s2L2 − pτ s, s  −Cs2L2 .

(1.5.30)

For s ∈ C ∞ (X, Lp ⊗ E), by Lemma 1.3.4, Theorem 1.3.5, (1.3.20) and (1.5.17), we get Dpc s2L2 =(Dpc )2 s, s = ∇Cl s2L2 − pτ s, s  X   r 1  F 1 det  (ei , ej )c(ei )c(ej ) s, s . + R + R + 4 2 2

(1.5.31)

From (1.3.8), for s ∈ C ∞ (X, Lp ⊗ E), the following identity holds: ∇Cl s = ∇L

p

⊗E

1 1 s + (∇det − ∇det1 )s − Sw l , w m w l ∧ w m ∧ s. 2 2

(1.5.32)

From (1.5.31) and (1.5.32), we know there exists C > 0, which does not depend on p, such that    2  Lp ⊗E 1 det det1 2  + (∇ − ∇ ) s (1.5.33) 0 ∇  2 − pτ s, s + CsL2 . 2 L 1 E det − ∇det1 ) But (∇det − ∇det1 ) is a purely imaginary 1-form, thus ∇E 1 = ∇ − 2 (∇ E is a Hermitian connection on E. Applying ∇1 on E for (1.5.33), we get (1.5.30). Relations (1.3.35), (1.5.17) imply that for s ∈ Ω0,• (X, Lp ⊗ E) ,

Dpc,A s2L2 =∇A s2L2 − pτ s, s − 2pωd s, s  X   r 1 det c E c 2 + (R + R ) + (dA) − 2|A| s, s . + 4 2

(1.5.34)

Now we apply (1.5.30) for E replaced by Λ(T ∗(0,1) X) ⊗ E with the Hermitian connection ∇A in (1.3.33). Then we know that the sum of the first two terms of

1.6. Asymptotic of the heat kernel

49

(1.5.34) is bounded below by −Cs2L2 . For s ∈ Ω0,>0 (X, Lp ⊗ E) the third term of (1.5.34), −2p(ωds, s) is bounded below by 2µ0 ps2L2 , by (1.5.19) and (1.5.26), while the norm of the remaining terms of (1.5.34) is bounded by Cs2L2 . Hence we obtain (1.5.28). The proof of Theorem 1.5.7 is completed.
Theorem 1.5.8. There exists CL > 0 such that for p ∈ N, the spectrum of (Dpc,A )2 verifies Spec((Dpc,A )2 ) ⊂ {0}∪ ]2pµ0 − CL , +∞[. Proof. The operator Dpc,A changes the parity of Ω0,• (X, Lp ⊗ E), so Theorem 1.5.7 shows that (Dpc,A )2 is invertible on Ω0,odd (X, Lp ⊗ E) for p large enough and its spectrum is in ]2µ0 p − CL , +∞[. Now, if s ∈ Ω0,even(X, Lp ⊗ E) is an eigensection of (Dpc,A )2 with (Dpc,A )2 s = λs and λ = 0, then Dpc,A s = 0 and (Dpc,A )2 Dpc,A s = λDpc,A s.

(1.5.35)

As Dpc,A s ∈ Ω0,odd (X, Lp ⊗ E), Theorem 1.5.7 yields λ > 2µ0 p − CL . The proof of Theorem 1.5.8 is complete.
Remark 1.5.9. From Theorems 1.4.5, 1.5.7, 1.5.8, we get another proof of Theorem 1.5.5.

1.6 Asymptotic of the heat kernel This section is organized as follows. In Section 1.6.1, we explain the main result, Theorem 1.6.1, the asymptotic of the heat kernel. In the rest of this section, we prove Theorem 1.6.1. In Section 1.6.2, we explain that our problem is local. In Section 1.6.3, we do the rescaling operation on coordinates and compute the limit operators. In Section 1.6.4, we obtain the uniform estimate of the heat kernel. Finally, in Section 1.6.5, we prove Theorem 1.6.1.

1.6.1 Statement of the result Let (X,J) be a compact complex manifold with complex structure J and dimC X = n. Let (L, hL ) be a holomorphic Hermitian line bundle on X, and (E, hE ) be a holomorphic Hermitian vector bundle on X. Let ∇E , ∇L be the holomorphic Hermitian (i.e., Chern) connections on (E, hE ), (L, hL ). Let RL , RE be the curvatures of ∇L , ∇E . Let g T X be any Riemannian metric on T X compatible with J. We use the notation in Section 1.5.1, especially Dp was defined in (1.5.20). For p ∈ N, we write Epj := Λj (T ∗(0,1) X) ⊗ Lp ⊗ E,

Ep = ⊕j Epj .

We will denote by ∇B,Ep the connection on Ep defined by (1.4.27).

(1.6.1)

50

Chapter 1. Demailly’s Holomorphic Morse Inequalities

By (1.4.29), Dp2 = 2
p is a second order elliptic differential operator with principal symbol σ(Dp2 )(ξ) = |ξ|2 for ξ ∈ Tx∗ X, x ∈ X. The heat operator e−uDp is well defined for u > 0. Let exp(−uDp2 )(x, x ), (x, x ∈ X) be the smooth kernel of the heat operator exp(−uDp2 ) with respect to the Riemannian volume form dvX (x ). Then 2

exp(−uDp2 )(x, x ) ∈ (Ep )x ⊗ (Ep )∗x
.

(1.6.2)

Especially exp(−uDp2 )(x, x) ∈ End(Ep )x = End(Λ(T ∗(0,1) X) ⊗ E)x ,

(1.6.3)

where we use the canonical identification End(L ) = C for any line bundle L on X. Since Dp2 preserves the Z-grading of the Dolbeault complex Ω0,• (X, Lp ⊗ E),  j j ∗ we get from (D.1.7), that exp(−uDp2 )(x, x ) ∈ j ((Ep )x ⊗ (Ep )x
), especially  exp(−uDp2 )(x, x) ∈ j End(Λj (T ∗(0,1) X) ⊗ E)x . p

We will denote by det the determinant on T (1,0) X. The following result is the main result of this section, and the rest of the section is devoted to its proof. Theorem 1.6.1. For each u > 0 fixed and any k ∈ N we have as p → ∞ det(R˙ L ) exp(2uωd ) u ⊗ IdE pn + o(pn ) exp(− Dp2 )(x, x) = (2π)−n p det(1 − exp(−2uR˙ L )) 

(1.6.4) n  aj (x) 1 + (e−2uaj (x) − 1)wj ∧ iwj n n = p + o(p ) , ⊗ Id E 2π(1 − e−2uaj (x) ) j=1 in the C k -norm on C ∞ (X, End(Λ(T ∗(0,1) X) ⊗ E)). Here we use the convention that if an eigenvalue aj (x) (cf. (1.5.18)) of R˙ xL is zero, then its contribution for det(R˙ xL )/ det(1 − exp(−2uR˙ xL )) is 1/(2u). Finally, the convergence in (1.6.4) is uniform as u varies in any compact subset of R∗+ .

1.6.2 Localization of the problem Let injX be the injectivity radius of (X, g T X ), and ε ∈]0, injX /4[. N0 X 0 As X is compact, there exist {xi }N i=1 such that {Uxi = B (xi , ε)}i=1 is a covering of X. Now we use the normal coordinates as in Section 1.2.1. On Uxi , we ∗(0,1) ∗(0,1) identify EZ , LZ , Λ(TZ X) to Exi , Lxi , Λ(Txi X) by parallel transport with 0,• respect to the connections ∇E , ∇L , ∇B,Λ along the curve [0, 1]  u → uZ. This induces a trivialization of Ep on Uxi . Let {ei }i be an orthonormal basis of Txi X. Denote by ∇U the ordinary differentiation operator on Txi X in the direction U . Let {ϕi } be a partition of unity subordinate to {Uxi }. For l ∈ N, we define a Sobolev norm on the lth Sobolev space H l (X, Ep ) by s2H l (p) =

l

i

2n


k=0 i1 ...ik =1

∇ei1 . . . ∇eik (ϕi s)2L2 .

(1.6.5)

1.6. Asymptotic of the heat kernel

51

 Lemma 1.6.2. For any m ∈ N, there exists Cm > 0 such that for any s ∈ 2m+2 ∗ (X, Ep ), p ∈ N , H  4m+4 p sH 2m+2 (p)  Cm

m+1


p−4j Dp2j sL2 .

(1.6.6)

j=0

Proof. Let ei (Z) be the parallel transport of ei with respect to ∇T X along the curve [0, 1]  u → uZ. Then { ei }i is an orthonormal frame on T X. Let ΓE , 0,• 0,• ΓL , ΓB,Λ be the corresponding connection forms of ∇E , ∇L and ∇B,Λ with respect to any fixed frame for E, L, Λ(T ∗(0,1) X) which is parallel along the curve [0, 1]  u → uZ under the trivialization on Uxi . On Uxi , we have   0,• ej ) ∇ej + pΓL ( ej ) + ΓB,Λ ( ej ) + ΓE ( ej ) . (1.6.7) Dp = c( By Theorem A.1.7, (1.6.7), there exists C > 0 (independent on p) such that for 2 any p  1, s ∈ H 2 (X, Ep ), we have sH 1 (p)  C(sH 2 (p) + sL2 )sL2 , and sH 2 (p)  C(Dp2 sL2 + p2 sL2 ).

(1.6.8)

Let Q be a differential operator of order m ∈ N with scalar principal symbol and with compact support in Uxi . Then     0,• ej )ΓL ( ej ), Q] + c( ej ) ∇ej + ΓB,Λ ( ej ) + ΓE ( ej ) , Q (1.6.9) [Dp , Q] = p[c( which are differential operators of order m − 1, m respectively. By (1.6.8), (1.6.9), QsH 2 (p)  C(Dp2 QsL2 + p2 QsL2 )  C(QDp2 sL2 + p2 QsL2 + p2 sH 2m+1 (p) ).

(1.6.10)

 Using (1.6.10), for m ∈ N, there exists Cm > 0 such that for p  1,  sH 2m+2 (p)  Cm (Dp2 sH 2m (p) + p2 sH 2m+1 (p) ).

From (1.6.11), we get (1.6.6). Let f : R → [0, 1] be a smooth even function such that  1 for |v|  ε/2, f (v) = 0 for |v|  ε. Definition 1.6.3. For u > 0, ς  1, a ∈ C, set  +∞ √ v2 dv eiva exp(− )f ( uv) √ , Fu (a) = 2 2π −∞  +∞ 2 √ v dv Gu (a) = eiva exp(− )(1 − f ( uv)) √ , 2 2π −∞  +∞ 2 √ v dv . Hu,ς (a) = eiva exp(− )(1 − f ( ςv)) √ 2u 2πu −∞

(1.6.11)


(1.6.12)

(1.6.13)

52

Chapter 1. Demailly’s Holomorphic Morse Inequalities

The functions Fu (a), Gu (a) are even holomorphic functions. The restrictions of Fu , Gu to R lie in the Schwartz space S(R). Clearly,  υ2  (1.6.14) Gu (υa) = Hυ2 , u2 (a), Fu (υDp ) + Gu (υDp ) = exp − Dp2 . υ 2 Let Fu (υDp )(x, x ), Gu (υDp )(x, x ) (x, x ∈ X) be the smooth kernels associated to Fu (υDp ), Gu (υDp ), calculated with respect to the volume form dvX (x ). Proposition 1.6.4. For any m ∈ N, u0 > 0, ε > 0, there exists C > 0 such that for any x, x ∈ X, p ∈ N∗ , u > u0 ,   ε2 p   u  ). (1.6.15) G p ( u/pDp )(x, x ) m  Cp3m+8n+8 exp(− 16u C 0,•

Here the C m norm is induced by ∇L , ∇E , ∇B,Λ

and hL , hE , g T X . m

∂ iva Proof. Due to the obvious relation im am eiva = ∂v ), we can integrate by m (e m parts in the expression of a Hu,ς (a) given by (1.6.13) and obtain that for any m ∈ N there exists Cm > 0 (which depends on ε) such that for u > 0, ς  1,

sup |a|m |Hu,ς (a)|  Cm ς

m 2

exp(−

a∈R

ε2 ). 16uς

(1.6.16)

Here we use that z k exp(−z 2 ) is bounded on R+ . Let Q be a differential operator of order m ∈ N with scalar principal symbol and with compact support in Uxi . From





Dpm H up ,1 (Dp )Qs, s  = s, Q∗ H up ,1 (Dp )Dpm s , (C.2.5) (or Theorem D.1.3, or using the Fourier transform as in (1.6.16) and the boundedness of the wave operator eiuDp in L2 -norm implied by (D.2.16)), (1.6.6) and (1.6.16), we know that for m, m ∈ N, there exists Cm,m
> 0 such that for p  1, u > u0 > 0, ε2 p )sL2 . (1.6.17) 16u We deduce from (1.6.17) that if P, Q are differential operators of order m, m with compact support in Uxi , Uxj respectively, then there exists C > 0 such that for p  1, u  u0 ,


Dpm H up ,1 (Dp )QsL2  Cm,m
p2m+2 exp(−

ε2 p )sL2 . (1.6.18) 16u By using the Sobolev inequality and (1.6.14) on Uxi ×Uxj , we conclude Proposition 1.6.4.



P H up ,1 (Dp )QsL2  Cp2m+2m +4 exp(−

Using (1.6.13) and the finite propagation speed, Theorem D.2.1 and (D.2.17), it is clear that for x, x ∈ X, F up ( up Dp )(x, x ) only depends on the restriction of Dp to B X (x, ε), and is zero if d(x, x )  ε.

1.6. Asymptotic of the heat kernel

53

1.6.3 Rescaling of the operator Dp2 Let ρ : R → [0, 1] be a smooth even function such that ρ(v) = 1 if |v| < 2;

ρ(v) = 0 if |v| > 4.

(1.6.19)

Let ΦE be the smooth self-adjoint section of End(Λ(T ∗(0,1) X) ⊗ E) on X defined by √ rX c E 1 det 1 −1 c + (R + R ) + ΦE = (∂∂Θ) − |(∂ − ∂)Θ|2 , (1.6.20) 4 2 2 8 (compare (1.4.29)). We fix x0 ∈ X. From now on, we identify B Tx0 X (0, 4ε) with B X (x0 , 4ε) ∗(0,1) as in Section 1.2.1. For Z ∈ B Tx0 X (0, 4ε), we identify EZ , LZ , Λ(TZ X) to ∗(0,1) Ex0 , Lx0 , Λ(Tx0 X) by parallel transport with respect to the connections ∇E , 0,• along the curve [0, 1]  u → uZ. Thus on B X (x0 , 4ε), (E, hE ), ∇L , ∇B,Λ 0,• L (L, h ), (Λ(T ∗(0,1) X), hΛ ), Ep are identified to the trivial Hermitian bundles 0,• 0,• ∗(0,1) (Ex0 , hEx0 ), (Lx0 , hLx0 ), (Λ(Tx0 X), hΛx0 ), (Ep,x0 , hEp,x0 ). Let ΓE , ΓL , ΓB,Λ 0,• be the corresponding connection forms of ∇E , ∇L and ∇B,Λ on B X (x0 , 4ε). 0,• 0,• Then ΓE , ΓL , ΓB,Λ are skew-adjoint with respect to hEx0 , hLx0 , hΛx0 . Denote by ∇U the ordinary differentiation operator on Tx0 X in the direction U . From the above discussion,   0,• (Z), (1.6.21) ∇Ep,x0 = ∇ + ρ(|Z|/ε) pΓL + ΓE + ΓB,Λ defines a Hermitian connection on (Ep,x0 , hEp,x0 ) on R2n  Tx0 X where the identification is given by
Zi ei ∈ Tx0 X. (1.6.22) R2n  (Z1 , . . . , Z2n ) −→ i

Here {ei }i is an orthonormal basis of Tx0 X. Let g T X0 be a metric on X0 := R2n which coincides with g T X on B Tx0 X (0,2ε), and g Tx0 X outside B Tx0 X (0, 4ε). Let dvX0 be the Riemannian volume form of (X0 , g T X0 ). Let ∆Ep,x0 be the Bochner Laplacian associated to ∇Ep,x0 and dvX0 on X0 . Set Lp,x0 = ∆Ep,x0 − p ρ(|Z|/ε)(2ωd,Z + τZ ) − ρ(|Z|/ε)ΦE,Z .

(1.6.23)

Then Lp is a self-adjoint operator with respect to the scalar product (1.3.14) induced by hEp,x0 , g T X0 . Moreover, Lp,x0 coincides with Dp2 on B T X (0, 2ε). Let dvT X be the Riemannian volume form on (Tx0 X, g Tx0 X ). Let κ(Z) be the smooth positive function defined by the equation dvX0 (Z) = κ(Z)dvT X (Z), with k(0) = 1.

(1.6.24)

54

Chapter 1. Demailly’s Holomorphic Morse Inequalities

Let exp(−uLp,x0 )(Z, Z  ), (Z, Z  ∈ R2n ) be the smooth kernel of the heat operator exp(−uLp,x0 ) on X with respect to dvX0 (Z  ). Lemma 1.6.5. Under the notation in Proposition 1.6.4, the following estimate holds uniformly on x0 ∈ X:        ε2 p  u u   .  exp − Dp2 (x0 , x0 ) − exp − Lp,x0 (0, 0)  Cp8n+8 exp − 2p 2p 16u (1.6.25)  u, G  u, H  u,ς be the holomorphic functions on C such that Proof. Let F  u (a2 ) = Fu (a), F

 u (a2 ) = Gu (a), G

 u,ς (a2 ) = Hu,ς (a). H

(1.6.26)

 u,1 (a) still verifies (1.6.16). And on R2n , Lemma 1.6.2 still holds  u (ua) = H Then G uniformly on x0 ∈ X, if we replace Dp2 therein by Lp,x0 . Thus from the proof of  u (uLp,x0 ). Proposition 1.6.4, we still have (1.6.15) for G Now by the finite propagation speed (Theorem D.2.1), we know that !    u u  u u Dp Lp,x0 (0, ·). Fp (x0 , ·) = F p p p


Thus, we get (1.6.25) by (1.6.14).

Let SL be a unit vector of Lx0 . Using SL , we get an isometry Ep,x0  (Λ(T ∗(0,1) X) ⊗ E)x0 =: Ex0 . As the operator Lp,x0 takes values in End(Ep,x0 ) = End(E)x0 (using the natural identification End(Lp )  C, which does not depend on SL ), thus our formulas do not depend on the choice of SL . Now, under this identification, we will consider Lp,x0 acting on C ∞ (X0 , Ex0 ). For s ∈ C ∞ (R2n , Ex0 ), Z ∈ R2n and t = √1p , set (St s)(Z) = s(Z/t), ∇t = St−1 tκ1/2 ∇Ep,x0 κ−1/2 St ,

(1.6.27)

Lt2 = St−1 κ1/2 t2 Lp,x0 κ−1/2 St . Put 1 ∇0,· = ∇· + RxL0 (Z, ·),
2 0 (∇0,ei )2 − 2ωd,x0 − τx0 . L2 = −

(1.6.28)

i

Lemma 1.6.6. When t → 0, we have ∇t,· = ∇0,· + O(t), Proof. Let (gij (Z))ij .

Lt2 = L02 + O(t).

(1.6.29)

T X0 (ei , ej ), and let (g ij (Z))ij be the inverse of the matrix gij (Z) = gZ T X0 Let ∇ei ej = Γkij (Z)ek . By (1.3.19), we know that on B(0, 4ε),

  B,Ep B,Ep p . ∆B,Ep = −g ij (tZ) ∇B,E ∇ − ∇ T X ei ej ∇ ej ei

(1.6.30)

1.6. Asymptotic of the heat kernel

55

From (1.5.17), (1.6.21), (1.6.23), (1.6.27) and (1.6.30), we get   0,• 1 L E ∇t,· =κ1/2 (tZ) ∇· + ρ(|tZ|/ε)(tΓB,Λ Γ + + tΓ ) κ−1/2 (tZ), tZ tZ tZ t   Lt2 = − g ij (tZ) ∇t,ei ∇t,ej − tΓkij (tZ)∇t,ek

(1.6.31)

+ ρ(|tZ|/ε)(−2ωd,tZ − τtZ + t2 ΦE,tZ ).


Since g ij (0) = δij , (1.2.31) and (1.6.31) imply (1.6.29).

1.6.4 Uniform estimate on the heat kernel E Let hEx0 be the metric on Ex0 induced by hΛ x0 , hx0 . We also denote by ·, ·0,L2 2 and  · 0,L2 the scalar product and the L norm on C ∞ (X0 , Ex0 ) induced by g T X0 , hEx0 as in (1.3.14). For s ∈ C ∞ (Tx0 X, Ex0 ), set  s2t,0 := s20 = |s(Z)|2hEx0 dvT X (Z), 0,•

R2n

s2t,m =

m


2n


(1.6.32) ∇t,ei1 · · · ∇t,eil s2t,0 .

l=0 i1 ,...,il =1 

We denote by s , st,0 the inner product on C ∞ (X0 , Ex0 ) corresponding to  · 2t,0 . −1 Let H m t be the Sobolev space of order m with norm ·t,m . Let H t be the Sobolev −1 space of order −1 and let  · t,−1 be the norm on H t defined by st,−1 = m
 sup0=s
∈H 1t | s, s t,0 |/s t,1 . If A ∈ L (H m t , H t ) (m, m ∈ Z), we denote by


Am,m the norm of A with respect to the norms  · t,m and  · t,m
. t Since Lp,x0 is formally self-adjoint with respect to  · 0,L2 , Lt2 is also a formally self-adjoint elliptic operator with respect to  · 2t,0 , and is a smooth family of operators with parameter x0 ∈ X. Theorem 1.6.7. There exist constants C1 , C2 , C3 > 0 such that for t ∈]0, 1] and any s, s ∈ C0∞ (R2n , Ex0 ), t

L2 s, s t,0  C1 s2t,1 − C2 s2t,0 ,

(1.6.33) | Lt2 s, s t,0 |  C3 st,1 s t,1 . Proof. Now from (1.4.29) and (1.5.17),   Lp,x0 s, s0,L2 = ∇Ep,x0 s20,L2 + ρ( |Z| )(−2pω − pτ + Φ )s, s d E ε

0,L2

From (1.6.24), (1.6.27), (1.6.32) and (1.6.34),



t L2 s, s t,0 =∇t s2t,0 − ρ(|tZ|/ε)(−2ωd,tZ − τtZ + t2 ΦE,tZ )s, s t,0 . From (1.6.35), we get (1.6.33).

. (1.6.34)

(1.6.35)


56

Chapter 1. Demailly’s Holomorphic Morse Inequalities

Γ

i

0

−2C2

−i

Figure 1.1. Let Γ be the oriented path in C defined by Figure 1.1. Theorem 1.6.8. There exists C > 0 such that for t ∈]0, 1], λ ∈ Γ, and x0 ∈ X, (λ − Lt2 )−1 0,0 t  C, (λ − Lt2 )−1 −1,1  C(1 + |λ|2 ). t

(1.6.36)

Proof. As Lt2 is a self-adjoint differential operator, by (1.6.33), (λ−Lt2 )−1 exists for λ ∈ Γ. The first inequality of (1.6.36) comes from our choice of Γ. Now, by (1.6.33),  C11 . for λ0 ∈ R, λ0  −2C2 , (λ0 − Lt2 )−1 exists, and we have (λ0 − Lt2 )−1 −1,1 t Then, (λ − Lt2 )−1 = (λ0 − Lt2 )−1 − (λ − λ0 )(λ − Lt2 )−1 (λ0 − Lt2 )−1 . Thus (1.6.37) imply for λ ∈ Γ (λ − Lt2 )−1 −1,0  t

 1  1 1 + |λ − λ0 | . C1 C

(1.6.37)

(1.6.38)

Now we interchange the last two factors in (1.6.37), apply (1.6.38) and obtain  1 1 |λ − λ0 |  1 + |λ − λ (λ − Lt2 )−1 −1,1  + | (1.6.39) 0 t C1 C C1 2  C(1 + |λ|2 ). The proof of our theorem is complete.




Proposition 1.6.9. Take m ∈ N∗ . There exists Cm > 0 such that for t ∈]0, 1],  ∞ 2n Q1 , . . . , Qm ∈ {∇t,ei , Zi }2n i=1 and s, s ∈ C0 (R , Ex0 ),  

  (1.6.40)  [Q1 , [Q2 , . . . , [Qm , Lt2 ] . . . ]]s, s t,0   Cm st,1 s t,1 .

1.6. Asymptotic of the heat kernel

57

Proof. Note that [∇t,ei , Zj ] = δij . Thus by (1.6.31), we know that [Zj , Lt2 ] verifies (1.6.40). 0,•

0,•

Let RρΛ , RρL and RρE be the curvatures of the connections ∇+ρ(|Z|/ε)ΓB,Λ , ∇ + ρ(|Z|/ε)ΓL and ∇ + ρ(|Z|/ε)ΓE . Then by (1.6.21), (1.6.27),   0,• [∇t,ei , ∇t,ej ] = RρL + t2 RρΛ + t2 RρE (tZ)(ei , ej ).

(1.6.41)

Thus from (1.6.31) and (1.6.41), we know that [∇t,ek , Lt2 ] has the same structure as Lt2 for t ∈]0, 1], i.e., [∇t,ek , Lt2 ] is of the type


aij (t, tZ)∇t,ei ∇t,ej +

ij




di (t, tZ)∇t,ei + c(t, tZ),

(1.6.42)

i

and aij (t, Z), di (t, Z), c(t, Z) and their derivatives on Z are uniformly bounded for Z ∈ R2n , t ∈ [0, 1]; moreover, they are polynomials in t. Let (∇t,ei )∗ be the adjoint of ∇t,ei with respect to ·, ·t,0 (see (1.6.32)). Then (∇t,ei )∗ = −∇t,ei − t(κ−1 ∇ei κ)(tZ),

(1.6.43)

and the last term of (1.6.43) and its derivatives in Z are uniformly bounded in Z ∈ R2n , t ∈ [0, 1]. By (1.6.42) and (1.6.43), (1.6.40) is verified for m = 1. By iteration, we know that [Q1 , [Q2 , . . . , [Qm , Lt2 ] . . . ]] has the same structure
(1.6.42) as Lt2 . By (1.6.43), we get Proposition 1.6.9. Theorem 1.6.10. For any t ∈]0, 1], λ ∈ Γ, m ∈ N, the resolvent (λ − Lt2 )−1 maps m+1 . Moreover for any α ∈ N2n , there exist N ∈ N, Cα,m > 0 such Hm t into H t that for t ∈]0, 1], λ ∈ Γ, s ∈ C0∞ (X0 , Ex0 ), Z α (λ − Lt2 )−1 st,m+1  Cα,m (1 + |λ|2 )N







Z α st,m .

(1.6.44)

α

α 2n Proof. For Q1 , . . . , Qm ∈ {∇t,ei }2n i=1 , Qm+1 , . . . , Qm+|α| ∈ {Zi }i=1 , we can express t −1 Q1 . . . Qm+|α| (λ − L2 ) as a linear combination of operators of the type

[Q1 , [Q2 , . . . [Qm
, (λ − Lt2 )−1 ] . . . ]]Qm
+1 . . . Qm+|α|

m  m + |α|.

(1.6.45)

Let Rt be the family of operators Rt = {[Qj1 , [Qj2 , . . . [Qjl , Lt2 ] . . . ]]}. Clearly, any commutator [Q1 , [Q2 , . . . [Qm
, (λ−Lt2 )−1 ] . . . ]] is a linear combination of operators of the form (1.6.46) (λ − Lt2 )−1 R1 (λ − Lt2 )−1 R2 . . . Rm
(λ − Lt2 )−1 with R1 , . . . , Rm
∈ Rt .

58

Chapter 1. Demailly’s Holomorphic Morse Inequalities

By Proposition 1.6.9, the norm  1,−1 of the operators Rj ∈ Rt is unit formly bounded by C. By Theorem 1.6.8, we find that there exist C > 0 and N ∈ N such that the norm  · t0,1 of operators (1.6.46) is dominated by C(1 + |λ|2 )N .
Let e−uL2 (Z, Z  ) be the smooth kernels of the operators e−uL2 with respect to dvT X (Z  ). Note that Lt2 are families of differential operators with coefficients in End(Ex0 ) = End(Λ(T ∗(0,1) X) ⊗ E)x0 . Let π : T X ×X T X → X be the natural t projection from the fiberwise product of T X on X. Then we can view e−uL2 (Z, Z  ) ∗ ∗(0,1) End(E) X)⊗E)) on T X ×X T X. Let ∇ be the as smooth sections of π (End(Λ(T ∗(0,1) B,Λ0,• E End(E) connection on End(Λ(T X) ⊗ E) induced by ∇ and ∇ . Then ∇ induces naturally a C m -norm for the parameter x0 ∈ X. t

t

Theorem 1.6.11. Set u > 0 fixed; then for any m, m ∈ N, there exists C > 0, such that for t ∈]0, 1], Z, Z  ∈ Tx0 X, |Z|, |Z  |  1,   ∂ |α|+|α
| t   C.  α  α
e−uL2 (Z, Z  )  m
C (X) |α|,|α
|
m ∂Z ∂Z sup




(1.6.47)




Here C m (X) is the C m norm for the parameter x0 ∈ X. Proof. By (1.6.33) and (1.6.36), (cf. also (C.2.5)), for any k ∈ N∗ ,  t (−1)k−1 (k − 1)! e−uL2 = e−uλ (λ − Lt2 )−k dλ. 2πiuk−1 Γ

(1.6.48)

For m ∈ N, let Qm be the set of operators {∇t,ei1 . . . ∇t,eij }j
m . From Theorem 1.6.10, we deduce that if Q ∈ Qm , there are M ∈ N, Cm > 0 such that for any λ ∈ Γ,  Cm (1 + |λ|2 )M . (1.6.49) Q(λ − Lt2 )−m 0,0 t Observe that if an operator Qt has the structure and properties after (1.6.42) and {aij (t, Z)} is uniformly positive, then all the above arguments apply for Qt . t t∗ Next we study Lt∗ 2 , the formal adjoint of L2 with respect to (1.6.32). Then L2 t t∗ t has the same structure (1.6.31) as the operator L2 (in fact, L2 = L2 ), especially, −m 0,0 t  Cm (1 + |λ|2 )M . Q(λ − Lt∗ 2 )

(1.6.50)

After taking the adjoint of (1.6.50), we get (λ − Lt2 )−m Q0,0  Cm (1 + |λ|2 )M . t

(1.6.51)

From (1.6.48), (1.6.49) and (1.6.51), we have, for Q, Q ∈ Qm , Qe−uL2 Q 0,0  Cm . t t

h

Let  m Λ(Tx∗(0,1) X)⊗Ex0 0

(1.6.52)

be the usual Sobolev norm on C ∞ (R2n , Ex0 ) induced by hEx0 = and the volume form dvT X (Z) as in (1.6.32).

1.6. Asymptotic of the heat kernel

59

Observe that by (1.6.31), (1.6.32), for any m  0, there exists Cm > 0 such that for s ∈ C ∞ (X0 , Ex0 ), supp(s) ⊂ B Tx0 X (0, 1), 1 st,m  sm  Cm st,m . Cm

(1.6.53)

Now (1.6.52), (1.6.53) together with Sobolev’s inequalities implies that if Q, Q ∈ Qm , sup |Z|,|Z
|
1

|QZ QZ
e−uL2 (Z, Z  )|  C. t

Thus by (1.6.31), (1.6.54), we derive (1.6.47) for the case when m = 0. Finally, for U a vector on X,  (−1)k−1 (k − 1)! π ∗ End(E) −uLt2 π ∗ End(E) e = e−uλ ∇U (λ − Lt2 )−k dλ. ∇U 2πiuk−1 Γ

(1.6.54)

(1.6.55)

π ∗ End(E)

(λ − Lt2 )−k , where we replace We use a similar formula as (1.6.46) for ∇U π ∗ End(E) t π ∗ End(E) t Rt by {∇U L2 }. Moreover, we remark that ∇U L2 is a differential operator on Tx0 X with the same structure as Lt2 . Then the above argument yields (1.6.47) for m  1.
Theorem 1.6.12. There exists C > 0 such that for t ∈ [0, 1],


   (λ − Lt )−1 − (λ − L0 )−1 s  Ct(1 + |λ|4 ) Z α s0,0 . 2 2 0,0

(1.6.56)

|α|
3

Proof. Remark that by (1.6.31), (1.6.32), for t ∈ [0, 1], k  1,
Z α s0,k . st,k  C

(1.6.57)

|α|
k

An application of Taylor expansion for (1.6.31) leads to the following equation, if s, s have compact support: 


  Z α s0,1 . (1.6.58)  (Lt2 − L02 )s, s 0,0   Cts t,1 |α|
3

Thus we get
 t  (L − L0 )s  Ct Z α s0,1 . 2 2 t,−1

(1.6.59)

|α|
3

Note that (λ − Lt2 )−1 − (λ − L02 )−1 = (λ − Lt2 )−1 (Lt2 − L02 )(λ − L02 )−1 .

(1.6.60)

After taking the limit, we know that Theorems 1.6.8–1.6.10 still hold for t = 0. From Theorem 1.6.10, (1.6.59) and (1.6.60), we infer (1.6.56).


60

Chapter 1. Demailly’s Holomorphic Morse Inequalities

Theorem 1.6.13. For u > 0 fixed, there exists C > 0, such that for t ∈]0, 1], Z, Z  ∈ Tx0 X, |Z|, |Z  |  1,   0   −uLt2 − e−uL2 )(Z, Z  ) Ct1/(2n+1) . (1.6.61) (e Proof. Let Jx00 be the vector space of square integrable sections of Ex0 over {Z ∈ Tx0 X, |Z|  2}. If s ∈ Jx00 , put s2(1) = |Z|
2 |s|2hEx0 dvT X (Z). Let A(1) be the operator norm of A ∈ L (Jx00 ) with respect to  (1) . Let u > 0 fixed. By (1.6.48) and (1.6.56), we get: There exists C > 0, such that for t ∈]0, 1],  t 0 1 (e−uL2 − e−uL2 )(1)  |e−uλ | (λ − Lt2 )−1 − (λ − L02 )−1 (1) dλ 2π Γ  (1.6.62)   C t e−uRe(λ) (1 + |λ|4 )dλ  Ct. Γ

Let φ : R2n  → [0, 1] be a smooth function with compact support, equal 1 near 0, such that Tx X φ(Z)dvT X (Z) = 1. Take ν ∈]0, 1]. By the proof of Theorem 1.6.11, 0

e−uL2 verifies the similar inequality as in (1.6.47). Thus by Theorem 1.6.11, there exists C > 0 such that if |Z|, |Z  |  1, U, U  ∈ Ex0 , 0

  t 0   (e−uL2 − e−uL2 )(Z, Z  )U, U     t 0 (e−uL2 − e−uL2 )(Z − W, Z  − W  )U, U  − Tx0 X×Tx0 X

×

    φ(W/ν)φ(W /ν)dv (W )dv (W )   Cν|U ||U  |. (1.6.63) T X T X 4n

1 ν

On the other hand, by (1.6.62),   



(e−uL2 − e−uL2 )(Z − W, Z  − W  )U, U  t

0



Tx0 X×Tx0 X

×

 1 1    φ(W/ν)φ(W /ν)dv (W )dv (W )   Ct 2n |U ||U  |. (1.6.64) TX TX 4n ν ν

By taking ν = t1/(2n+1) , we get (1.6.61).




1.6.5 Proof of Theorem 1.6.1 √ Note that in (1.6.24), κ(0) = 1. Recall also that t = 1/ p. By (1.6.27), for s ∈ C0∞ (X0 , Ex0 ), (e−uL2 s)(Z) = (St−1 κ 2 e− p Lp κ− 2 St s)(Z)  1 1 u 2 = κ (tZ) exp(− Lp )(tZ, Z  )(St s)(Z  )κ 2 (Z  ) dvT X (Z  ). p 2n R t

1

u

1

(1.6.65)

1.7. Demailly’s holomorphic Morse inequalities

61

Thus, for Z, Z  ∈ Tx0 X, t u exp(− Lp,x0 )(Z, Z  ) = pn e−uL2 (Z/t, Z  /t)κ−1/2 (Z  )κ−1/2 (Z). p

(1.6.66)

By Theorem 1.6.13, (1.6.25), (1.6.66), we get that uniformly on x0 ∈ X, we have u exp(− Dp2 )(x0 , x0 ) − pn exp(−uL02,x0 )(0, 0) = o(pn ). p

(1.6.67)

By (1.5.19), (1.6.28), (E.2.4) and (E.2.5), we get with the convention in Theorem 1.6.1, exp(−uL02 )(0, 0) =

1 det(R˙ xL0 ) exp(2uωd,x0 ) . (2π)n det(1 − exp(−2uR˙ xL0 ))

(1.6.68)

Moreover, for any j fixed, exp(−2uaj (x0 )w j ∧ iwj ) = 1 + (exp(−2uaj (x0 )) − 1)w j ∧ iwj .

(1.6.69)

From (1.6.67)–(1.6.69), we get (1.6.4). If u varies in a compact set of R∗+ , the constant C in (1.6.47) and (1.6.61) is uniformly bounded, so the convergence of (1.6.4) is uniform. The proof of Theorem 1.6.1 is complete.

1.7 Demailly’s holomorphic Morse inequalities We will use the notation of Section 1.6.1 and (1.5.14)–(1.5.19). √ Let X(q) be the set of points x of X such that −1RxL is non-degenerate and (1,0) R˙ xL ∈ End(Tx X) has exactly q negative eigenvalues. Set X( q) = ∪qi=0 X(i), X( q) = ∪ni=q X(i). Theorem 1.7.1. Let X be a compact complex manifold with dim X = n, and let L, E be holomorphic vector bundles on X, rk(L) = 1. As p → ∞, the following strong Morse inequalities hold for every q = 0, 1, . . . , n: q
j=0

(−1)q−j dim H j (X, Lp ⊗ E)  rk(E)

pn n!

 (−1)q X(
q)

√

−1 L 2π R

n

+ o(pn ) ,

(1.7.1) with equality for q = n (asymptotic Riemann–Roch–Hirzebruch formula). In particular, we get the weak Morse inequalities  √ n pn −1 L q p (−1)q 2π R + o(pn ). (1.7.2) dim H (X, L ⊗ E)  rk(E) n! X(q)

62

Chapter 1. Demailly’s Holomorphic Morse Inequalities

Proof. For 0  q  n, set Bqp = dim H q (X, Lp ⊗ E).

(1.7.3)

Remark that the operator Dp2 preserves the Z-grading of the Dolbeault com2

2

plex Ω0,• (X, Lp ⊗ E). We will denote by Trq [e− p Dp ] the trace of e− p Dp acting on Ω0,q (X, Lp ⊗ E), then we have   u 2  −u Dp2 p Trq [e ]= Trq e− p Dp (x, x) dvX (x). (1.7.4) u

u

X

Lemma 1.7.2. For any u > 0, p ∈ N∗ , 0  q  n, we have q


(−1)q−j Bjp 

j=0

  u (−1)q−j Trj exp(− Dp2 ) , p j=0

q


(1.7.5)

with equality for q = n. Proof. If λ is an eigenvalue of Dp2 , set Fjλ be the corresponding finite-dimensional eigenspace in Ω0,j (X, Lp ⊗ E). We claim that ∂

Lp ⊗E

λ (Fjλ ) ⊂ Fj+1 ,

and ∂

Lp ⊗E,∗

In fact, if s ∈ Fjλ , then Dp2 s = λs. By (1.5.20), ∂ L ⊗E p

L ⊗E

L ⊗E

p

λ (Fj+1 ) ⊂ Fjλ .

Lp ⊗E

p

Dp2 ∂ s=∂ Dp2 s = λ∂ of (1.7.6). Thus we have the complex ∂

Lp ⊗E

(1.7.6)

commutes with Dp2 , thus

s. In the same way, we get the second equation

∂

Lp ⊗E

∂

Lp ⊗E

0 −→ F0λ −→ F1λ −→ · · · −→ Fnλ −→ 0.

(1.7.7)

If λ = 0, then Fj0  H j (X, Lp ⊗ E). If λ > 0, we claim that the complex (1.7.7) is exact. In fact, if ∂

Lp ⊗E

s = 0 and s ∈ Fjλ , then by (1.5.20),

s = λ−1 Dp2 s = λ−1 ∂ From (1.7.8), we know s ∈ ∂ exact and q


L ⊗E

Lp ⊗E

(∂

Lp ⊗E,∗

s).

(1.7.8)

p

λ (Fj−1 ). Thus for λ > 0, the complex (1.7.7) is

(−1)q−j dim Fjλ = dim(∂

Lp ⊗E

(Fqλ ))  0

(1.7.9)

j=0

with equality when q = n. Now
u u Trj [exp(− Dp2 )] = Bjp + e− p λ dim Fjλ . p

(1.7.10)

λ>0

(1.7.9) and (1.7.10) yield (1.7.5).




1.7. Demailly’s holomorphic Morse inequalities

63

We denote by TrΛ0,q the trace on Λq (T ∗(0,1) X). By (1.6.69), in the notation of (1.5.19), q  
exp − 2u aji (x) .




TrΛ0,q [exp(2uωd)] =

j1 <j2 <···<jq

(1.7.11)

i=1 ˙L

(det(R /(2π))) TrΛ0,q [exp(2uωd)] is uniThus by the second equality in (1.6.4), det(1−exp(−2u R˙ L )) formly bounded for x ∈ X, u > 1, 0  q  n, and for any x0 ∈ X, 0  q  n,

 R˙ L  det(R˙ L /(2π)) TrΛ0,q [exp(2uωd )] (x0 ) = 1X(q) (−1)q det (x0 ). (1.7.12) u→∞ 2π det(1 − exp(−2uR˙ L )) lim

The function 1X(q) is defined by 1 on X(q), 0 otherwise. From Theorem 1.6.1, (1.7.4) and (1.7.5), we have lim p−n

p→∞

q
j=0

(−1)q−j Bjp 

 rk(E)

det(R˙ L /(2π))

(1.7.13)

q

q−j j=0 (−1)

TrΛ0,j [exp(2uωd )] dvX (x) , det(1 − exp(−2uR˙ L))

X

for any q with 0  q  n and any u > 0. Using (1.7.12), (1.7.13) and dominate convergence, we get lim p−n

p→∞

q


 (−1)q−j Bjp  (−1)q rk(E)

j=0

det

 R˙ L 

∪qi=0 X(i)

2π

(x)dvX (x).

(1.7.14)

But (1.5.18) entails det

 R˙ L  2π

(x)dvX (x) =

 aj (x) j

2π

dvX (x) =

 √−1 2π

RL

n /n!.

(1.7.15)

Relations (1.7.14), (1.7.15) imply (1.7.1). For q = n, we apply (1.7.5) with equality, so we get (1.7.1) with equality. (1.7.2) follows by subtracting inequalities (1.7.1) for q and q + 1 (or directly from Theorem 1.6.1, (1.7.10) and (1.7.12)). The proof of Theorem 1.7.1 is complete.


Problems Problem 1.1. Verify (1.3.2), (1.3.31) and (1.3.41). With the notation from (1.3.44) verify that c,A c,A Ker(Dc,A ) = Ker((Dc,A )2 ) = Ker(D+ ) ⊕ Ker(D− ).

64

Chapter 1. Demailly’s Holomorphic Morse Inequalities

Problem 1.2. In Section 1.3.2, we can always assume that ∇Cl on Λ(T ∗(0,1) X) ⊗ E E is induced by ∇T X , ∇det1 and a Hermitian connection ∇E 1 on (E, h ). (Hint: 1 det det1 −∇ ) is a purely imaginary 1-form.) 2 (∇ Problem 1.3. In local coordinates (x1 , . . . , xn ) of a Riemannian manifold (X, g T X ), ∂ we set fj = ∂x , gij (x) = fi , fj gT X (x). Let (g ij (x)) be the inverse of the matrix j (gij (x)). Verify that in (1.3.19),  
F F ∆F = − g ij (x) ∇F fi ∇fj − ∇∇T X fj . fi

ij

Problem 1.4. In the context of (1.4.5) show that ∗

Ker(D) = Ker(∂) ∩ Ker(∂ ),

∗

Im(∂) ∩ Im(∂ ) = 0.

∗

Thus Ker(D), Im(∂) and Im(∂ ) are pairwise orthogonal. Problem 1.5. Verify Remark 1.4.3 (cf. also [9, §2]). By Theorem A.3.2 for k ∈ Z, and D2 is elliptic, for s a distribution with values in E, D2 s = 0 in the sense of distributions implies that s ∈ Ω0,• (X, E) (cf. also [148, Chap. 3], [238, §7.4]). Using this fact, show that Ker(D) ⊂ Ω0,• (X, E) ∩ L20,• (X, E) is closed in L20,• (X, E). By the Schwartz kernel theorem, P (x, y) is a distribution on X × X with values in (Λ(T ∗(0,1) X) ⊗ E)x ⊗ (Λ(T ∗(0,1) X) ⊗ E)∗y . Prove first Dx2 P (x, y) = 0,

Dy2 P (x, y) = 0,

in the sense of distributions. Here we identify (Λ(T ∗(0,1) X)⊗E)∗y to (Λ(T ∗(0,1) X)⊗ E)y by the Hermitian product ·, ·Λ0,• ⊗E . Now as Dx2 + Dy2 is an elliptic operator on X × X, (Dx2 + Dy2 )P (x, y) = 0 in the sense of distributions implies P (x, y) is C ∞ for x, y ∈ X. Problem 1.6. Let X be a K¨ ahler manifold. ∗

a) Show that [∂, ∂ ] = 0 , [∂, ∂ ∗ ] = 0. ∗

b) Show that ∆ commutes with all operators ∂, ∂, ∂ ∗ , ∂ , L, Λ. (Hint: Use Theorem 1.4.11 and (1.3.31).) Problem 1.7. Verify first (1.5.8). Now let (X, ω, J) be a K¨ahler manifold. Let KX := det(T ∗(1,0) X) be the canonical line bundle on X. Set Ricω = Ric(J·, ·). Using (1.2.55), verify that RT X is a (1, 1)-form with values in End(T X). Using √ √ ∗ (1,0) X ]. (1.2.5), verify that Ricω = −1RKX = −1 Tr[RT Problem 1.8. We will use the homogeneous coordinate (z0 , . . . , zn ) ∈ Cn+1 for CPn  (Cn+1 \ {0})/C∗ . Denote by Ui = {[z0 , . . . , zn ] ∈ CPn ; zi = 0}, (i = 0, . . . , n), the open subsets of CPn , and the coordinate charts are defined by φi : Ui  Cn , φi ([z0 , . . . , zn ]) = ( zz0i , . . . , zz"ii , . . . , zzni ). (A hat over a variable means that this variable is skipped.)

1.7. Demailly’s holomorphic Morse inequalities

65

Let O(−1) be the tautological line bundle of CPn , i.e., O(−1) = {([z], λz) ∈ n+1 , λ ∈ C}. For any α = (α0 , . . . , αn ) ∈ Nn+1 , the map Cn+1  z → CP #n × αCj ∗ j=0 zj is naturally identified with a holomorphic section of O(−|α|) = O(|α|) n on CP ; we denote it by sα . Let hO(−1) be the Hermitian metric on O(−1), as a subbundle of the trivial bundle Cn+1 on CPn , induced by the standard metric on Cn+1 . Let hO(1) be √ −1 the Hermitian metric on O(1) induced by hO(−1) . Let ωF S = 2π RO(1) be the (1, 1)-form associated to (O(1), hO(1) ) defined by (1.5.14). On Ui , the trivialization of the line bundle O(1) is defined by O(1)  s → s/zi , here zi is considered as a holomorphic section of O(1). We work now on Cn by using φ0 : U0 → Cn . Verify that for z ∈ Cn , n

n −1 
|s(1,0,...,0) |2hO(1) (z) = 1 + |zj |2 . j=1

From (1.5.8), verify that for z ∈ Cn , % $ n n n √ −1 j=1 dzj dz j j=1 z j dzj ∧ k=1 zk dz k n n ωF S (z) = − . 2π 1 + j=1 |zj |2 (1 + j=1 |zj |2 )2 ahler form on CPn . ωF S is called the Fubini–Study form, and the Thus ωF S is a K¨ n associated Riemannian metric gFT CP is the Fubini–Study metric on CPn . S Problem 1.9. Let f be a harmonic function on a connected compact   manifold X, i.e., ∆f = 0. Show that f is constant on X. (Hint: X |df |2 dvX = X f (∆f )dvX ). Problem 1.10. Consider a real (1, 1)-form√α ∈ Ω1,1 (X). Let us choose the local n n orthonormal frame {wj }j=1 such that α = −1 j=1 cj (x)wj ∧w j at a given point x ∈ X . For any u = I,J uIJ wI ∧ w J ∈ Ω•,• (X), from (1.4.37), (cf. (1.4.61)), we have n 



[α, Λ]u = cj (x) + cj (x) − cj (x) uIJ wI ∧ wJ . I,J

j∈I

j∈J

j=1

Problem 1.11. (a) (Nakano vanishing theorem) Let X be a compact K¨ ahler manifold and (E, hE ) be a Nakano-positive vector bundle over X (cf. Definition 1.1.6). Show that H q (X, E ⊗ KX ) = 0 for any q  1. (b) On T (1,0) C Pn we consider the Fubini–Study metric. Show that T (1,0) C Pn ⊗O(p) is Nakano-positive for p  1. Deduce that T (1,0) C Pn ⊗KC∗ Pn is Nakanopositive and that H q (C Pn , T (1,0) C Pn ) = 0 for q  1. Note: The case q = 1 in (b) implies that the complex structure of C Pn cannot be deformed (cf. [179, Ch.1, Th. γ]). Problem 1.12. (a) Let (E, hE ) be a holomorphic Hermitian vector bundle. Show that if (E, hE ) is Nakano-positive, then (E, hE ) is Griffiths-positive. (b) Define the vector bundle E = C Pn ×Cn+1 /O(−1) over C Pn . Show that E is Griffiths-positive but not Nakano-positive.

66

Chapter 1. Demailly’s Holomorphic Morse Inequalities

Note: The notion of Griffiths-positivity is more suitable for the study of ampleness than that of Nakano positivity. For more details see [79, Ch. VI], [217]. Problem 1.13. Verify that if M is a weakly pseudoconvex domain (i.e., the Levi form is positive semi-definite), and L is a positive line bundle on M , then the spectral gap property for Kodaira Laplacian similar to Theorem 1.5.5 still holds. Problem 1.14. For q = n, prove directly (1.7.1) with equality (use Theorem 1.4.6).

1.8 Bibliographic notes In Section 1.2.1 we basically follow [15, §1.2]. For basic material concerning manifolds, vector bundles and Riemannian geometry we refer to [85], [252], [140] and [179]. The proof of Lemmas 1.2.3 and 1.2.4 appeared in [10, Appendix II]. A good references for Section 1.3 is [148, Appendix D]. Instead of referring to [148, Appendix D], [160, §2] for a construction of the Clifford connection on Λ(T ∗(0,1)X), we define it here directly and verify its properties. The Atiyah–Singer index theorem was established in [11]. The Riemann–Roch–Hirzebruch theorem appears in Hirzebruch’s Habilitation thesis [130] for an algebraic variety X. In [15, Chap. 4], the readers can find a heat kernel approach to the Atiyah–Singer index theorem. Section 1.3.3 and Theorems 1.4.5, 1.4.7 are taken from [26], where Bismut used them to prove a local index theorem for modified Dirac operators. The K¨ ahler identities for K¨ ahler manifolds were proved by A. Weil [251] using the primitive decomposition theorem. Ohsawa [187] used the approach of Weil for non-K¨ahler metrics and showed the existence of the Hermitian torsion operator satisfying the generalized K¨ ahler identities. Theorem 1.4.11 and the Bochner– Kodaira–Nakano formula (1.4.44) were proved in this precise form by Demailly [73]. For (1.4.63) see also Kodaira–Morrow [179, Ch. 3, Th. 6.2]. Bochner–Kodaira–Nakano formulas with boundary term similar to (1.4.72) were proved by Andreotti–Vesentini [7, p. 113] and Griffiths [119, (7.14)]. Estimate (1.4.84) is a more geometric version of the famous Morrey–Kohn–H¨ormander estimate [143, 131, 108], which is essential in the solution of the ∂-Neumann problem (cf. also Section 3.5). Section 1.5. Theorems 1.5.7 and 1.5.8 are [160, Th. 1.1 and 2.5] if A = 0. If A = 0, Borthwick–Uribe [43] and Braverman [54] observed also (1.5.29). (1.5.23) was first proved by Bismut and Vasserot [35, Th. 1.1] by using the Bochner– Kodaira–Nakano formula [73, Th. 0.3]. Theorem 1.6.1 was first proved by Bismut in [25] by using probability theory. Demailly [74] and Bouche [48] gave a different approach. Our proof is new and is inspired by the analytic localization techniques of Bismut–Lebeau [33, §11]. Certainly, the argument here works well for the modified Dirac operator. Theorem 1.7.1 represents Demailly’s holomorphic Morse inequalities [72]. The proof in Section 1.7 is Bismut’s heat kernel proof of Theorem 1.7.1.

1.8. Bibliographic notes

67

Demailly’s work [72] was influenced by Witten’s seminal analytic proof of Morse inequalities [253] for a Morse function f with isolated critical points on a compact manifold. In [24], Bismut gave a heat kernel proof of Morse inequalities and of the degenerate Morse inequalities. Subsequently, in [25], he adapted his heat kernel proof of Morse inequalities for Demailly’s holomorphic Morse inequalities. Milnor’s book [176] is the standard reference for the classical Morse theory. For the analytic proof of classical Morse inequalities, we refer our readers to the interesting recent book [263]. In the literature, there exists another type of holomorphic Morse inequalities [254, 175, 256], which relate the Dolbeault cohomology groups of the fixed point set X G of a compact connected Lie group G acting on a compact K¨ ahler manifold X to the Dolbeault cohomology groups of X itself.

Chapter 2

Characterization of Moishezon Manifolds In this chapter we start some basic facts on analytic and complex geometry (divisors, blowing-up, big line bundles), we prove the theorem of Siegel–Remmert– Thimm, that the field of meromorphic functions on a connected compact complex manifold is an algebraic field of transcendence degree less than the dimension of the manifold. Then we study in more detail Moishezon manifolds and their relation to projective manifolds. In particular we prove that a Moishezon manifold is projective if and only if it carries a K¨ ahler metric. We end the section 2.2 by giving the solution of the Grauert–Riemenschneider conjecture as application of the holomorphic Morse inequalities from Theorem 1.7.1. Section 2.3 is devoted to the Shiffman–Ji–Bonavero–Takayama criterion, which states that Moishezon manifolds can be characterized in terms of integral K¨ ahler currents. We will present a proof using the singular holomorphic Morse inequalities of Bonavero in Section 2.3.2. As application of the singular holomorphic Morse inequalities, we present the computation of the volume of a big line bundle following Boucksom in Section 2.3.3. Moreover, we give some examples of non-projective Moishezon manifolds in Section 2.3.4, showing that the Shiffman– Ji–Bonavero–Takayama criterion is sharp. Section 2.4 provides an algebraic reformulation of the holomorphic Morse inequalities.

2.1 Line bundles, divisors and blowing-up In this section we review the basic facts on divisors and blow-ups. Let X be a complex manifold and dim X = n. Let OX be the sheaf of holomorphic functions ∗ on X. We denote by OX the sheaf whose sections over an open set U are the units ∗ of the ring OX (U ), and endow OX (U ) with multiplication as a group operation.

70

Chapter 2. Characterization of Moishezon Manifolds

Let SX ⊂ OX be the subsheaf defined by SX,x = OX,x
{0x }; for an open set U , SX (U ) consists of f ∈ OX (U ) which do not vanish identically on any connected component of U . Let MX be the sheaf associated to the pre-sheaf of rings of quotients U → SX−1 (U )OX (U ), U ⊂ X open. The sections of MX over an open set U are called meromorphic functions on U . By definition, f ∈ MX (U ) can be written in the neighborhood V of any point as f = g/h, where g ∈ OX (V ), h ∈ SX (V ). We denote by MX∗ the multiplicative sheaf of germs of non-zero meromorphic functions. If P, Q ∈ C[z0 , . . . , zn ] are two homogeneous polynomials of same degree, then P/Q defines a meromorphic function on C Pn , called a rational function. We denote by L (X) the set of holomorphic line bundles on X, which becomes a group, multiplication being given by tensor product and inverses by dual bundles. Let L ∈ L (X) and U = (Uα )α be an open cover of X such that L|Uα is trivial and choose holomorphic frames eα ∈ Γ(Uα , L); then the trivialization ϕα : L|Uα → C is defined by ϕα (s) = s/eα , and the transition functions ∗ gαβ := ϕα ◦ ϕ−1 β = eβ /eα ∈ OX (Uα ∩ Uβ )

on Uα ∩ Uβ = ∅.

(2.1.1)

−1 and gαβ gβγ gγα = 1 on Uα ∩ Uβ ∩ Uγ = ∅, so that It is easy to see that gαβ = gβα ∗ ˇ (gαβ ) represents a Cech 1-cocycle with values in OX , called the associated cocycle to L and the covering {Uα } and the set of frames {eα }. This defines a cohomology ∗ ). Observe that by choosing another set of class LU := {(gαβ )} ∈ H 1 (U , OX   holomorphic frames eα ∈ Γ(Uα , L), we get a cocycle (gαβ ) which differs from ˇ (gαβ ) by a Cech 1-coboundary. Thus the class LU := {(gαβ )} does not depend on the choice of frames. The same argument shows that two isomorphic line bundles ∗ L and L produce the same cohomology class in H 1 (U , OX ). 1 ∗ ) is the image Moreover, if we consider a refinement V of U , LV ∈ H (V , OX 1 ∗ 1 ∗ 1 ∗ of LU ∈ H (U , OX ) by the canonical map H (U , OX ) → H (V , OX ). Thus (the ∗ ∗ ) = lim H 1 (W , OX ) by isomorphism class of) L defines an element in H 1 (X, OX −→ W taking the direct limit.

Definition 2.1.1. The Picard group Pic(X) of X is the group of isomorphism classes of holomorphic line bundles over X. By the discussion above, we have a well-defined map ∗ Pic(X) −→ H 1 (X, OX ).

(2.1.2)

Actually this map is a group isomorphism. Indeed, it is well known that we can reconstruct a line bundle isomorphic to L using the cocycle (gαβ ). On the disjoint union α (Uα × C), we introduce the equivalence relation Uβ × C  (x, ξ) ∼ (y, η) ∈ Uα × C if x = y and η = gαβ (x)ξ. Then α (Uα × C)/ ∼ can be naturally given the structure of a holomorphic line bundle isomorphic to L.

2.1. Line bundles, divisors and blowing-up

71

The isomorphism (2.1.2) allows us to describe L and related objects in terms of the cocycle (gαβ ). For example, to any section s ∈ H 0 (X, L), we associate a collection (2.1.3) (sα ) , sα ∈ OX (Uα ) , sα = sβ gαβ on Uα ∩ Uβ , where sα is defined by s = sα eα on Uα . Conversely, any collection (sα ) as in (2.1.3) defines a holomorphic section s ∈ H 0 (X, L) by setting s = sα eα on Uα . If L and L are two line bundles, we choose a trivializing covering U for both.  If L (resp. L ) is given by the cocycle (gαβ ) (resp. (gαβ )), then L ⊗ L is given by −1  the cocycle (gαβ gαβ ) and L∗ , the dual bundle of L, is given by (gαβ ). Let L be a holomorphic line bundle and s ∈ H 0 (X, L ⊗ MX∗ ) be a meromorphic section of L on X. We define the divisor of s by Div(s) =

V

ordV (s) · V,

(2.1.4)

where the sum runs formally over all irreducible analytic hypersurfaces of X and ordV (s) ∈ Z is the order of s along V , as in (B.1.2). Locally there exist only a finite number of V ’s such that ordV (s) = 0. Conversely, a Weil divisor defines canonically a holomorphic line bundle. Definition 2.1.2. A (Weil) divisor on X is a locally finite linear combination D = i ci Vi , where Vi are irreducible analytic hypersurfaces of X and ci ∈ Z. The set of all divisors on X is denoted by Div(X). D is called effective if ci  0 for all i. The support of D is defined by supp(D) = ∪{Vi : ci = 0}. A divisor with normal crossings is a divisor of the form D = i Vi where Vi are distinct irreducible smooth hypersurfaces which intersect transversely; i.e., for any x ∈ supp(D), there is a local coordinate system (U, z1 , . . . , zn ) centered at x such that supp(D) ∩ U = {z1 z2 . . . zr = 0} for some 1  r  n. Let D = i ci Vi be a divisor on X. Choose an open covering U = (Uα ) such that every Vi has a local defining function fiα ∈ OX (Uα ) (i.e., which vanishes at order 1 along Vi ). Set fα :=



ci fiα ∈ MX∗ (Uα ) ,

∗ gαβ := fα /fβ ∈ OX (Uα ∩ Uβ ).

(2.1.5)

i

Definition 2.1.3. The holomorphic line bundle defined by the cocycle (gαβ ) in (2.1.5) is called the line bundle associated to D and is denoted by OX (D). Note that fα in (2.1.5) defines a meromorphic section sD of OX (D) and Div(sD ) = D, sD is called the canonical section of OX (D). Let M (D) be the space of meromorphic functions f on X which are holomorphic on X
∪Vi and ordVi (f )  −ci . By definition, we have Proposition 2.1.4. The map M (D) −→ H 0 (X, OX (D)), f −→ f · sD is bijective.

72

Chapter 2. Characterization of Moishezon Manifolds

For V a complex vector space, the projective space P(V ) is the set of complex lines through the original in V . Let OP(V ) (−1) be the universal line bundle on P(V ), then OP(V ) (−1) = {([z], λz) ∈ P(V ) × V ; λ ∈ C}. Especially CPn−1 = P(Cn ). A fundamental construction in complex and algebraic geometry is the blowup of X with center a point x ∈ X. The idea is to reduce the study of the ideal of holomorphic functions vanishing at x to the study of sections of a divisor, and hence of a line bundle. Let U be a coordinate neighborhood of the point x, with coordinates z = (z1 , . . . , zn ), where z = 0 corresponds to the point x. Consider the product U × C Pn−1 , where we assume that [t1 , . . . , tn ] are homogeneous coordinates for C Pn−1 . Then let W = {(z, [t]) ∈ U × C Pn−1 : tj zk − tk zj = 0, j, k = 1, . . . , n} ,

(2.1.6)

which is a submanifold of U × C Pn−1 . Then there is a holomorphic projection π : W −→ U given by π(z, [t]) = z. Moreover, π has the following properties: π −1 (0) = E = {0} × C Pn−1  C Pn−1 , π|W E : W
E −→ U
{0} is a biholomorphism .

(2.1.7)

 = X x := (X
{x}) π W by pasting X
{x} and W via the We define X biholomorphism π : W
E → U
{x}. Let NE/X be the holomorphic normal  bundle of E in X.  → X is called the blow-up of X with center the Definition 2.1.5. The map π : X point x ∈ X. E is called the exceptional divisor of π. x . We set Wi = {(z, [t]) ∈ Let us describe the local coordinates near E on X U × C Pn−1 : ti = 0}. Then {Wi }ni=1 is an open cover of W and in Wi , we have t z local coordinates wij = tji = zji , for j = i, and wii = zi . In these local coordinates, the map π is given by (wi1 , . . . , win ) → (wii · wi1 , . . . , wii , . . . , wii · win )

(2.1.8)

and E is given by E = {wii = 0} = {zi = 0}. It follows that the transition functions of OX (E) over W are gji = wij on Wi ∩ Wj . We consider the holomorphic map  : E −→ C Pn−1 , (0, [t]) = [t] and the line bundle ∗ OC Pn−1 (−1). Let us denote by N{x}/X  {x}×Cn the normal bundle to {x}. It is useful to regard C Pn−1 as the projectivized space P(N{x}/X ) = {(x, [t]) : [t] ∈ C Pn−1 } . We can describe the bundle ∗ OC Pn−1 (−1) in the following way; if we denote by OC Pn−1 (−1) ⊂ π ∗ (N{x}/X ) the tautological subbundle over E = P(N{x}/X ) such that the fiber above the point (x, [t]) ∈ E is Ct ⊂ N{x}/X , then ∗ OC Pn−1 (−1)  OP(N{x}/X ) (−1). Since the transition functions of OC Pn (−1) are gji = tj /ti on (Wi ∩ Wj ∩ E), we obtain:

2.1. Line bundles, divisors and blowing-up

73

Proposition 2.1.6. We have an isomorphism of line bundles NE/X  OX (E)|E  ∗ OC Pn−1 (−1)  OP(N{x}/X ) (−1) .

(2.1.9)

 KX := det(T ∗(1,0) X) be the canonical line bundles on Let KX := det(T ∗(1,0) X),  X. It is also important to calculate the canonical bundle of the blow-up. X, Proposition 2.1.7. KX = π ∗ (KX ) ⊗ OX ((n − 1)E). Proof. The bundle KX is generated on U by dz1 ∧· · · ∧dzn and π ∗ KX is generated on Wj by (2.1.10) π ∗ (dz1 ∧ · · · ∧ dzn ) = (wjj )n−1 dwj1 ∧ · · · ∧ dwjn . If s is the canonical section of OX (E), defined by wjj = 0 in Wj , we have a line bundle isomorphism π ∗ (KX ) −→ OX ((1 − n)E) ⊗ KX ,

ξ → s1−n ⊗ π ∗ (ξ) .

The proof of Proposition 2.1.7 is completed.

(2.1.11)


In replacing the point with a divisor, we have to deal with the following situation: if a line bundle L is positive near x, its pull-back π ∗ (L) is only semipositive near E. We show now how to regain positivity. Proposition 2.1.8. Let L be a positive line bundle over a compact complex manifold X. Then for any x ∈ X, there exists p0 (x) such that π ∗ Lp ⊗OX (−E) is positive for p  p0 (x). Moreover, there exists a neighborhood Ux of x such that p0 (y) = p0 (x), for any y ∈ Ux . Proof. We have OX (−E)|E  ∗ OC Pn−1 (1). We endow OX (−E)|E with the pullback of a Hermitian metric on OC Pn−1 (1) with positive curvature (cf. Problem 1.8) and then extend this metric (by a partition of unity) to a Hermitian metric on  OX (−E) over X. √ L Let h be a metric on L with associated curvature RL such that −1RL is a positive (1, 1)-form on X. Denote Fp = π ∗ Lp ⊗ OX (−E) with metric induced by the metrics on L and OX (−E). Then RFp = pRπ

∗

L

+ ROX& (−E) .

(2.1.12)

  compatible with the complex structure Let g T X be any Riemannian metric on T X  we will identify it as the J. Let NE/X be the holomorphic normal bundle of E in X,  as C ∞ -Hermitian vector bundles. orthogonal complement of T (1,0) E in T (1,0) X Fp  the sphere bundle We have to show that R (w, w) > 0 for any w ∈ ST (1,0) X, (1,0)   X is compact so ROX& (−E) has a lower of X. The unit normal bundle N  ∩ST E/X

 E −→ T X. Since bound on it. Moreover, let us consider the differential π∗ : T X|

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Chapter 2. Characterization of Moishezon Manifolds

π(E) = {x}, we have T E ⊂ Ker(π∗ ). Thus π∗ induces a map π∗ : NE/X −→ Tx X, which is actually injective. Indeed, in local coordinates {wij }nj=1 , the vector fields ∂ ∂ provide a basis for NE/X and using (2.1.8), we see that π∗ ( ∂w i ) = 0. Thus ∂w i i

i

Rπ

∗

L

(w, w) = RL (π∗ (w), π∗ (w)) > 0

 for w ∈ NE/X ∩ ST (1,0)X.

(2.1.13)

 On the other Thus, for p large enough, RFp (w, w) > 0, for w ∈ NE/X ∩ ST (1,0)X. hand, RFp (w, w) = ROX& (−E) (w, w) > 0

for w ∈ T (1,0) E ⊂ Ker(π∗ ).

(2.1.14)

 E , and thus in a neighborhood VE Summing up, RFp (w, w) > 0 for w ∈ T (1,0) X|  of E, for p  1. Using the compacity of X
VE , we finally find p0 such that  and p  p0 . To see that we can choose p0 RFp (w, w) > 0 for w ∈ ST (1,0) X uniformly in a neighborhood of x, we express the equations of the blow-up at y ∈ U in local coordinates centered at x, namely Wy = {(z, [t]) ∈ U × C Pn−1 : (zi − yi )tj = (zj − yj )ti }.

(2.1.15)

By denoting ψyx : Wy −→ Wx the isomorphism induced by the translation with ∗ y, we have ψyx (Ey ) = Ex , Fp (y) = ψyx Fp (x), where Fp (y) is the associated line bundle when we blow up at y. We can perform the construction of metrics on Wx , ∗ pull back to Wy and get RFp (y) = ψyx RFp (x) .
We consider now the blow-up with center a compact submanifold Y ⊂ X of codimension l  2. The holomorphic normal bundle of Y in X is defined by NY /X := (T (1,0) X)|Y /T (1,0)Y . The projectivized normal bundle P(NY /X ) −→ Y is the bundle with fibers the projective spaces P(NY /X,y ) associated to the fibers of NY /X at y ∈ Y .  together with a holomorphic map π : Theorem 2.1.9. There exists a manifold X −1  and π : E −→ Y is a  X −→ X such that E = π (Y ) is a smooth divisor in X 
E −→ holomorphic fiber bundle which is isomorphic to P(NY /X ). Moreover π : X X
Y is biholomorphic. Proof. We generalize the construction given for Y = {x}, x ∈ X. Let us denote by E = P(NY /X ), π  : P(NY /X ) −→ Y the canonical projection. Set  = (X
Y )  E, X

π = IdXY  π  .

(2.1.16)

 is obtained by replacing each point y ∈ Y by the projective This means that X space of the directions normal to Y .  as follows. First every chart on X
Y is taken to We define an atlas on X  be a chart of X, too. If y ∈ Y , let (U, z1 , . . . , zn ) be local coordinates on X such that Y ∩ U = {z ∈ U : z1 = · · · = zl = 0}. Hence zl+1 , . . . , zn are coordinates on

2.1. Line bundles, divisors and blowing-up

75

 ∂ Y ∩ U and ∂z∂ 1 , . . . , ∂z is a holomorphic frame of NY /X over U ∩ Y . We denote l by [ξ1 , . . . , ξl ] the corresponding fiber homogeneous coordinates on P(NY /X ); We set (2.1.17) Wi = {z ∈ U
Y : zi = 0} ∪ {(z, [ξ]) ∈ P(NY /X ) : ξi = 0}. Then {Wi }li=1 is a covering of π −1 (U ) and we define coordinates on Wi : wij =

zj zi

 for j = i, wii = zi and wik = zk for k > l. We can check that, with this atlas, X becomes a complex manifold. With respect to these coordinates, π can be written π : Wi −→ U , (wi1 , . . . , win ) −→ (wii wi1 , . . . , wii , . . . , wii wil , wil+1 , . . . , win ) . The proof of Theorem 2.1.9 is completed.




 −→ X is called the blow-up of X with center Y Definition 2.1.10. The map π : X and E = π −1 (Y ) is called the exceptional divisor. As before, let us introduce the subbundle OP(NY /X ) (−1) ⊂ π ∗ (NY /X ) over E = P(NY /X ) such that the fiber over (y, [ξ]) is C ξ ⊂ NY /X,y .  −→ X of X with center Y and l = Proposition 2.1.11. The blow-up π : X codim Y  2, has the following properties: (a) NE/X  OX (E)|E  OP(NY /X ) (−1), KX  π ∗ KX ⊗ OX ((l − 1)E). (b) If X is compact and L is a positive line bundle over X, then π ∗ (Lp )⊗OX (−E)  for p  0. is positive on X  Z) = H 2 (X, Z) ⊕ Zc1 (O  (E)) and Pic(X)  = π ∗ Pic(X) ⊕ ZO  (E). (c) H 2 (X, X X Proof. The same proof of Propositions 2.1.6, 2.1.7, 2.1.8 gives (a) and (b). Assertion (c) follows from the computation of Aeppli of the cohomology of modifications.
 −→ X is the blow-up with center Y and Z is a submanifold of If π : X  is called the strict transform X, Z  Y , then the closure of π −1 (Z ∩ (X
Y )) in X of Z. We remind the following important notion. Definition 2.1.12. Let A ⊂ C PN . A is called projective manifold if it is a complex submanifold of C PN . A is called an projective algebraic variety of C PN if there exist homogeneous polynomials f1 , · · · , fr ∈ C[z0 , . . . , zN ] such that A = {[z] ∈ C PN : f1 (z) = · · · = fr (z) = 0}. If A is also smooth, it is called projective algebraic manifold. We shall use the Hironaka embedded resolution of singularities theorem. Let Z be a closed complex space of a complex manifold X. Consider a sequence of

76

Chapter 2. Characterization of Moishezon Manifolds

transformations σj+1

−→Xj+1 −→ Zj+1 Ej+1

σ

1 Xj −→ · · · −→ X1 −→ X0 = X Zj Z1 Z0 = Z

Ej

E1

(2.1.18)

E0 = ∅

where, for each j, σj+1 : Xj+1 −→ Xj denotes a blowing-up with smooth center Cj ⊂ Xj , Zj+1 is the strict transform of Zj by σj+1 and Ej+1 is the set of −1 (Cj ), where Ej denotes exceptional hypersurfaces in Xj+1 ; i.e., Ej+1 = Ej ∪ σj+1 the set of strict transforms by σj+1 of all hypersurfaces in Ej . Theorem 2.1.13 (Hironaka). Let Z be a compact complex subspace of a complex manifold X. Then there exists a finite sequence of blowing-ups (2.1.18) with smooth centers Cj such that: 1. For each j, either Cj ⊂ (Zj )sing , the singular set of Zj , or Zj is smooth and Cj ⊂ Zj ∩ Ej . 2. Let Z  and E  denote the final strict transform of Z and exceptional set, respectively. Then Z  is smooth and E  has only normal crossings. We will need an even more general notion of blow up. Proposition 2.1.14. Let X be a compact complex space and I ⊂ OX be a coherent  and a sheaf of ideals. Then there exists a canonical compact complex space X  surjective holomorphic map π : X −→ X with the following properties:  is invertible, (a) the inverse image sheaf π −1 I · O  on X X


π −1 (Y ) −→ X
Y is biholomorphic. (b) if Y = supp(OX /I ), then π : X  is projective too. (c) If X is projective, then X

 is accomplished in the following way. We set I 0 = The construction of X OX and consider the graded OX -algebra A := ⊕q0 I q . Then A is of finite presentation, that is, for every point of X there exist a neighborhood U , an integer k and a finitely generated homogeneous ideal E ⊂ OU [w0 , w1 , . . . , wk ] such that A |U ∼ OU [w0 , w1 , . . . , wk ]/E . We consider the subspace Proj(A )U ⊂ U × C Pk defined by E . (For example, if I is generated on U by independent functions f0 , . . . , fk , then Proj(A )U is given in U × C Pk by the equations fi wj − fj wi = 0, 1  i, j  k.) By gluing together the subspaces Proj(A )U we obtain a complex  = Proj(A ). The space Proj(A ) comes with a natural projection π : space X Proj(A ) → X and a line bundle OProj(A ) (1) whose tensor powers OProj(A ) (q) satisfy π∗ OProj(A ) (q) = I q for q  0. If X is projective and L is an ample line bundle over X then π ∗ L m ⊗ OProj(A ) (1) is very ample for m sufficiently large,  is also projective. hence X  → X is called the blow up of X along the ideal I or with The map π : X center (Y, (OX /I )|Y ). If Y is smooth, we will talk about a blow-up with smooth center. The blow-up of Definition 2.1.10 are blow-ups with center (Y, (OX /I )|Y ), where Y is a submanifold and I the ideal of holomorphic functions vanishing on Y .

2.1. Line bundles, divisors and blowing-up

77

Blow-ups are used to resolve the singularities of complex spaces: Theorem 2.1.15 (Hironaka). Let X be a compact complex space. Then there exists a finite sequence of blow-ups with smooth centers σ

σj

σ

N 1 XN −→ · · · −→ Xj −→ Xj−1 −→ · · · −→ X1 −→ X0 = X ,

(2.1.19)

such that XN is non-singular (i.e. a complex manifold) and the center Cj−1 of the blow-up σj is contained in (Xj−1 )sing . If X is projective, XN is also projective. Combining Proposition 2.1.14 and Theorem 2.1.15 we obtain: Proposition 2.1.16. Let X be a compact complex space and I ⊂ OX be a coherent  and a surjective sheaf of ideals. Then there exists a compact complex manifold X  −→ X with the properties (a)–(c) from Proposition 2.1.14. holomorphic map π : X Since the composition of two blow-ups with smooth center is not in general a blowing-up, we introduce the notion of proper modification. Note, however, that the resulting complex space can be obtained from the original one by blowing up one ideal, by a classical theorem of Hironaka–Rossi. In the rest of the section we consider only reduced complex spaces, if not otherwise stated. Definition 2.1.17. Let ϕ : X −→ Y be a holomorphic map between complex irreducible spaces; ϕ is called a proper modification if it is proper, surjective and if there is an analytic set A ⊂ Y with the following properties: 1. A ⊂ Y and ϕ−1 (A) ⊂ X are nowhere dense analytic sets, 2. the restriction ϕ : X
ϕ−1 (A) −→ Y
A is biholomorphic. Theorem 2.1.18. If ϕ : X  −→ X is a proper modification, then the canonical " : MX → morphism ϕ  : OX → ϕ∗ OX
is injective and extends to an isomorphism ϕ ϕ∗ MX
. In particular, the map ϕ(X) " : MX (X) → MX
(X  ) is an isomorphism. More precisely, if h ∈ MX (X) has the local form h = f /g with f, g ∈ OX (U ), U open set, then ϕ(X)h " has the local form (f ◦ ϕ)/(g ◦ ϕ) on ϕ−1 (U ). Definition 2.1.19. Let X and Y be irreducible complex spaces. A map ϕ of X to the set of subsets of Y is called a meromorphic map of X into Y , written ϕ : X  Y , if the following conditions are fulfilled: (a) The graph Graph(ϕ) = {(x, y) ∈ X ×Y : y ∈ ϕ(x)} is an irreducible analytic subset of X × Y . (b) The projection on the first factor pr1 : Graph(ϕ) → X is a proper modification. To give a meromorphic map is equivalent to giving an analytic subset G ⊂ X × Y such that pr1 : G → X is a proper modification. Indeed, then we define ϕ through ϕ(x) = pr2 (pr−1 1 (x)), where pr2 : G → Y is the projection on the second factor.

78

Chapter 2. Characterization of Moishezon Manifolds

Example 2.1.20. (a) Assume that X and Y are projective algebraic varieties. Any rational map of X into Y , i.e., a map whose components are rational functions, is a meromorphic map. Conversely, by the GAGA principle, any meromorphic map of X into Y is actually a rational map. (b) Let f1 , . . . , fk be meromorphic functions on a complex space X, not all identically zero. Let A be a nowhere dense analytic set such that f1 , . . . , fk are holomorphic on U := X
A. We consider the holomorphic map ψ : U → Ck , ψ(x) = (f1 (x), . . . , fk (x)). One can show that the closure G of Graph(ψ) ⊂ U × Ck ⊂ X × C Pk in X × C Pk satisfies the condition that pr1 : G → X is a proper modification. Thus f1 , . . . , fk define a meromorphic map X  C Pk denoted x → [1, f1 (x), . . . , fk (x)]. (c) In particular, the Kodaira map (cf. Definition 2.2.3 and (2.2.12)) is a meromorphic map: if S1 , . . . , Sd is a basis of H 0 (X, L), the Kodaira map is defined by the meromorphic functions S2 /S1 , . . . , Sd /S1 . Definition 2.1.21. A meromorphic map ϕ : X  Y is called bimeromorphic map if the projection on the second factor pr2 : Graph(ϕ) → Y is also a proper modification. Two complex spaces X and Y are called bimeromorphically equivalent if there exists a bimeromorphic map ϕ : X  Y . In this case the analytic set G = {(y, x) ∈ Y × X : (x, y) ∈ Graph(ϕ)} ⊂ Y × X defines a meromorphic map ϕ−1 : Y  X. A holomorphic map between arbitrary complex spaces f : X −→ Y is called flat at a point x ∈ X if OX,x is flat when viewed as an OY,f (x) –module via the canonical morphism OY,f (x) → OX,x . f is called flat if it is flat at every x ∈ X. If f : X → Y is flat and X, Y are irreducible, then the fibres f −1 (y) with y ∈ Y have the same dimension. Theorem 2.1.22 (Hironaka’s flattening theorem). Let f : X → Y be a holomorphic map of compact complex spaces, where X is not necessarily reduced. Then there exists a commutative diagram  X

f

/ Y

πX

 X

(2.1.20) πY

f

 /Y

where πY is the blow-up along a coherent sheaf of ideals I ⊂ OY , πX is the blow-up of along the inverse image of I by f and f is a flat holomorphic map. Moreover, we can assume that πY is a composition of a finite succession of blow-ups with smooth centers. The flattening theorem has two important consequences: elimination of indeterminacies and the Chow lemma.

2.1. Line bundles, divisors and blowing-up

79

Definition 2.1.23. Let ϕ : X  Y be a meromorphic map. The smallest nowhere dense analytic set I(ϕ) ⊂ X such that ϕ : X
I(ϕ) → Y is holomorphic is called the set of points of indeterminacy of ϕ. Theorem 2.1.24 (elimination of points of indeterminacy). Let X, Y be irreducible compact complex spaces and ϕ : X  Y be a bimeromorphic map. There exists  → X obtained as a composition of a finite succession a proper modification ψ : X  → Y such that we of blow-ups with smooth centers and a holomorphic map ϕ :X have the commutative diagram:  X ?? ??ϕ ψ ?? ?  _ _ _ X ϕ /Y

(2.1.21)

Theorem 2.1.25 (Chow Lemma). Let X, Y be irreducible compact complex spaces and ϕ : X  Y be a bimeromorphic map. Then (i) there exists a proper modification πY : Y → Y obtained as a composition of a finite succession of blow-ups with smooth centers and a holomorphic map α : Y → X such that ϕ ◦ α = πY , and  → X obtained as a composition of (ii) there exists a proper modification πX : X a finite succession of blow-ups with smooth centers and a holomorphic map  → Y such that the following diagram is commutative: ϕ :X  X

ϕ 

/ Y

πX

(2.1.22) πY

  X _ ϕ_ _/ Y Proof. By Theorem 2.1.24 (for ϕ−1 ) there exists a composition of finitely many blow-ups with smooth centers πY : Y → Y such that the bimeromorphic map α := ϕ−1 ◦ πY is holomorphic: Y   α  πY      X _ _ _/ Y

(2.1.23)

ϕ

Applying again the same theorem, there exists a composition of finitely many blow → X such that ϕ  := α−1 ◦πX is holomorphic.
ups with smooth centers πX : X

80

Chapter 2. Characterization of Moishezon Manifolds

2.2 The Siu–Demailly criterion Our goal is to prove the Siu–Demailly criterion, which gives a positive answer to the Grauert–Riemenschneider conjecture. The conjecture is the generalization of the Kodaira embedding theorem in connection to the characterization of Moishezon varieties, that is, compact complex spaces with transcendence degree of the meromorphic function field equal to the complex dimension. They are important in algebraic geometry because most of the natural modifications of algebraic varieties can be performed in the category of Moishezon varieties but sometimes not in the category of algebraic varieties. This section is organized as follows. In Section 2.2.1, we introduce big line bundles and give a characterization in term of the growth of dim H 0 (X, Lp ) as p → ∞. In Section 2.2.2, we prove two fundamental results of Moishezon. First, that a Moishezon manifold can be transformed into a projective one by a finite succession of blowing-up along smooth centers. Using this theorem, we show that the obstruction for a Moishezon manifold to be projective is the existence of K¨ ahler metric. Finally, as an application of the holomorphic Morse inequalities, we establish Siu–Demailly criterion which gives a characterization of Moishezon manifolds.

2.2.1 Big line bundles Let X be a compact connected complex manifold of dimension n and let L be a holomorphic line bundle over X. It is well known that the space of holomorphic sections of L on X, L

H 0 (X, L) = {s ∈ C ∞ (X, L) : ∂ s = 0} is finite-dimensional. This follows from the Hodge theory, Theorem 1.4.1. But we start with an elementary proof of this fact which has two advantages: it can be refined to prove Siegel’s lemma and generalizes to Andreotti pseudoconcave manifolds. For x ∈ X, we denote by I (x) the sheaf of holomorphic functions vanishing at x and by mx ⊂ OX,x the maximal ideal of the ring of germs of holomorphic functions at x. For a positive integer k we have a canonical residue map L → L ⊗ (OX /I (x)k+1 ) which induces in cohomology a map which associates to each global holomorphic section of L its k-jet at x: Jxk : H 0 (X, L) −→ H 0 (X, L ⊗ (OX /I (x)k+1 )) = Lx ⊗ (OX,x /mk+1 x ).

(2.2.1)

The right-hand side of (2.2.1) is called the space of k-jets of holomorphic sections of L at x. For a point x ∈ X, we denote by P (x, r) the polydisc {z ∈ U : |zi | < r, 1 ≤ i ≤ n} where (U, z1 , . . . , zn ) is a holomorphic coordinate system centered at x. The Shilov boundary of P (x, r) is defined by S(P (x, r)) = {z ∈ U : |zi | = r, 1 ≤

2.2. The Siu–Demailly criterion

81

i ≤ n}. It is well known that any holomorphic function in the neighborhood of P (x, r), attains its maximum on P (x, r) on S(P (x, r)). Lemma 2.2.1. Let X be a compact complex manifold of dimension n and L be a holomorphic line bundle on X. Then for any points x1 , . . . , xm in X and r1 , . . . , rm −1 ∈ R∗+ such that L is trivial over each P (xi , 2ri ) and X ⊂ ∪m ), there i=1 P (xi , ri e 0 exists an integer k = k(L) such that if s ∈ H (X, L) vanishes at each point xi up to order k, then s vanishes identically. Hence dim H 0 (X, L)  m n+k k . Proof. We fix a trivialization of L over each P (xi , 2ri ). Assume that the line bundle L is given by the holomorphic transition functions cij : P (xi , ri ) ∩ P (xj , rj ) −→ C∗ .

(2.2.2)

Set ϕ(L) = sup{|cij (x)| : x ∈ P (xi , ri ) ∩ P (xj , rj ) , i, j = 1, . . . , m}.

(2.2.3)

0 Since cij = c−1 ji , we have ϕ(L)  1. Consider a section s ∈ H (X, L) which vanishes up to order k = log ϕ(L) + 1 at each xj (u is the integer part of u). Assume that s is given in the given trivialization of L over P (xj , 2rj ) by sj . Set

s = sup{|sj (x)| : x ∈ P (xj , rj ) , j = 1, . . . , m}.

(2.2.4)

There exists q ∈ {1, 2, . . . , m} such that for some y ∈ S(P (xq , rq )), |sq (y)| = s. We can find j = q such that y ∈ P (xj , rj e−1 ). Hence sq (y) = cqj (y)sj (y) so that s = |sq (y)| = |cqj (y)sj (y)|  ϕ(L) |sj (y)|.

(2.2.5)

By applying the Schwarz inequality (cf. Problem 2.3) to sj in P (xj , rj ), we get |sj (y)|  s |y|k0 rj−k

where |y|0 = sup{|zj (y)|; 1  j  n}  rj e−1 .

(2.2.6)

Consequently, s  s ϕ(L)e−k .

(2.2.7)

If s is not identically zero, this leads to a contradiction, by our choice of k. Consider the map  1
j
m

Jxkj : H 0 (X, L) −→



Lxj ⊗ (OX,xj /mk+1 xj ),

(2.2.8)

1
j
m

which sends every section in its k-jets at x1 , . . . , xm . By the preceding argument, (2.2.8) is injective. Since the dimension of the target space satisfies the desired estimate, we are done.


82

Chapter 2. Characterization of Moishezon Manifolds

There exists an important link between existence of meromorphic functions and the growth of the dimension of the space of holomorphic sections of a line bundle. We shall exhibit this link by means of Siegel’s lemma. For this purpose let us define the Kodaira map. We give first a geometric description. The linear system associated to V = H 0 (X, L) is |V | = {Div(s) : s ∈ V }. Since X is compact, Div(s) = Div(s ) if and only if s = λs for some λ ∈ C∗ , so that |V | is parameterized by the projective space P(V ). The base point locus of the linear system |V | is given by Bl|V | = ∩s∈V Div(s) = {x ∈ X : s(x) = 0

for all s ∈ V }.

(2.2.9)

Theorem 2.2.2 (Bertini). The generic element of the linear system |H 0 (X, L)| is smooth away from Bl|H 0 (X,L)| , i.e., Div(s)
Bl|H 0 (X,L)| is a smooth submanifold of X for a generic element s of H 0 (X, L). For x ∈ / Bl|V | , the space of divisors D ∈ |V | containing x forms a hyperplane in |V | or equivalently the space of sections vanishing at x is a hyperplane in V . Set d = dim H 0 (X, L) and let G(d − 1, V ) = G(d − 1, H 0 (X, L)) be the Grassmannian of hyperplanes of H 0 (X, L). Definition 2.2.3. The Kodaira map ΦV associated to L is defined by ΦV : X
Bl|V | −→ G(d − 1, H 0 (X, L)), ΦV (x) = {s ∈ H 0 (X, L) : s(x) = 0}.

(2.2.10)

We give now an algebraic definition. Let us identify P(V ∗ ) with G(d − 1, V ) by sending an equivalence class of non-zero forms on H 0 (X, L) to their (common) kernel. By composing ΦV with this identification, we obtain a map ΦV : X
Bl|V | −→ P(H 0 (X, L)∗ )

(2.2.11)

described as follows. For x ∈ / Bl|V | , we choose a vector ex ∈ Lx
{0}, then we obtain an element of V ∗ given by V  s → s(x)/ex ∈ C. The class of this element in P(V ∗ ) does not depend on the choice of ex and is exactly ΦV (x). To get an analytic description of ΦV , let us choose a basis S1 , . . . , Sd of V  V through which gives identifications V  Cd and P(V ∗ )  C Pd−1 . We define Φ the commutative diagram Φ

X
Bl|V | −−−V−→ P(V ∗ ) ⏐ ⏐ ⏐ ⏐ (Id (  Φ

X
Bl|V | −−−V−→ C Pd−1 . Let eL be a local holomorphic frame of L on U and set Si = fi eL with fi ∈ OX (U ). Then  V (x) = [f1 (x), . . . , fd (x)] , x ∈ U. Φ (2.2.12)

2.2. The Siu–Demailly criterion

83

 V and ΦV are holomorphic. If Si (x) = 0 This form immediately shows that Φ  V (x) = [S1 (x)/Si (x), . . . , Sd (x)/Si (x)] ∈ C Pd−1 which justifies the slightly then Φ  V (x) = [S1 (x), . . . , Sd (x)]. abusive notation Φ Definition 2.2.4. We say that 1. V = H 0 (X, L) separates two points x = y, if x, y ∈ Bl|V | and ΦV (x) = ΦV (y) (or equivalently, the restriction map H 0 (X, L) → Lx ⊕ Ly , s → s(x) ⊕ s(y) is surjective). 2. V = H 0 (X, L) gives local coordinates at a point x if x ∈ Bl|V | and ΦV is an immersion at x (i.e., Jx1 : H 0 (X, L) → Lx ⊗ (OX,x /m2x) is surjective). We will be interested in the Kodaira map associated to high tensor powers of L. We write Vp := H 0 (X, Lp ) and Φp := ΦVp . Definition 2.2.5. 1. The graded ring ⊕p0 H 0 (X, Lp ) separates two points x = y, if H 0 (X, Lp ) separates x, y for some p. 2. The graded ring ⊕p0 H 0 (X, Lp ) gives local coordinates at a point x, if H 0 (X, Lp ) gives local coordinates at x for some p. 3. Set p = max{rkx Φp : x ∈ X
Bl|Vp | } if Vp = {0}, and p = −∞ otherwise. The Kodaira–Iitaka dimension of L is κ(L) = max{p : p ∈ N∗ }. A line bundle L is called big if κ(L) = dim X. Big line bundles can be characterized in terms of the growth of dim H 0 (X, Lp ) as p → ∞. We introduce for this purpose a very useful general estimate. Lemma 2.2.6 (Siegel’s lemma). Let X be a compact connected complex manifold and L be a holomorphic line bundle on X. Then there exists C > 0 such that dim H 0 (X, Lp )  Cp p ,

for any p  1.

(2.2.13)

Proof. We modify the proof of Lemma 2.2.1 and use the same notation as there. The set of points where Φp has rank less than p is a proper analytic set of X, so {x ∈ X : rkx Φp = p for any p  1} is dense in X. Let x1 , . . . , xm ∈ X as in Lemma 2.2.1 such that Φp has rank p at each xj for any p  1. Since Φp is of constant rank in a neighborhood of xj , there exists a submanifold Mp,j in xj to the fiber

 in the neighborhood of xj which is transversal 0 Φ (x ) and dim M =  . Consider a section s ∈ H (X, Lp ) which vanΦ−1 p j p,j p p ishes up to order k = p(log ϕ(L) + 1) at each xj along Mp,j . But s vanishes on the fiber Φ−1 p (Φp (xj )) which passes through xj by definition of the Kodaira map, hence s vanishes up to order k at xj on X. Repeating the argument from Lemma 2.2.1, we obtain s  s ϕ(Lp )e−k . (2.2.14) By noting that ϕ(Lp ) = ϕ(L)p we infer that s = 0, so s vanish identically.

84

Chapter 2. Characterization of Moishezon Manifolds

Let mMp,j ,xj be the maximal ideal of the ring OMp,j ,xj . Consider the map 

H 0 (X, Lp ) −→

Lpxj ⊗ (OMp,j ,xj /mk+1 Mp,j ,xj ),

(2.2.15)

1
j
m

which sends every section to its jet of order k at xj along Mp,j , j = 1, . . . , m. The preceding argument shows that this map

is injective. Since the dimension of the
target space is less than or equal to m pk+k the desired estimate follows. Theorem 2.2.7. Let X be a compact connected complex manifold of dimension n, and let L be a holomorphic line bundle on X. Then L is big if and only if lim sup p−n dim H 0 (X, Lp ) > 0.

(2.2.16)

p→∞

Proof. We get immediately the “if” part of Theorem 2.2.7 from Lemma 2.2.6. Conversely, assume that L is big. Let us choose x0 and m with rkx0 Φm = n. There exist s0 , . . . , sn ∈ H 0 (X, Lm ) with s0 (x0 ) = 0 and d( ss10 ) ∧ · · · ∧ d( ssn0 )|x0 = 0. Let P be a polynomial of degree r in n variables. Then P ( ss10 , . . . , ssn0 ) = 0 implies that P = 0, as ( ss10 (x), . . . , ssn0 (x)) are local coordinates near x0 . Therefore each non-zero homogeneous polynomial Q of degree r in n + 1 variables produces a non-zero section Q(s0 , . . . , sn ) ∈ H 0 (X, Lmr ). In fact if Q(s0 , . . . , sr ) = 0, then P

s

1

s0

,...,

1 sn  = r Q(s0 , . . . , sn ) = 0, s0 s0

so P = 0 and Q = 0. Since the space r in  of nhomogeneous polynomials of0 degree mr  r /n!, we deduce that dim H (X, L ) n + 1 variables has dimension n+r r rn /n!, thus we get (2.2.16).


2.2.2 Moishezon manifolds Let X be a compact connected complex manifold of dimension n. The ring MX (X) of global sections of MX over X is called the ring of meromorphic functions on X. Certainly, the ring MX (X) forms a field. Let us describe concretely MX (X) for a connected projective manifold X ⊂ C PN . If P, Q ∈ C[z0 , . . . , zN ] are two homogeneous polynomials of same degree, such that Q does not vanish identically on X, then P/Q ∈ MX (X) is called a rational function on X. The set of rational functions forms a subfield KX ⊂ MX (X). Actually, by a theorem of Hurwitz, any meromorphic function on a projective manifold is rational, i.e KX = MX (X). Definition 2.2.8. We say that f1 , · · · , fk ∈ MX (X) are algebraically dependent, if there exists a non–trivial polynomial P ∈ C[z1 , · · · , zk ] such that P (f1 , · · · , fk ) = 0 wherever it is defined. The transcendence degree of MX (X) over C is the maximal number of algebraically independent meromorphic functions on X. We denote

2.2. The Siu–Demailly criterion

85

by a(X) the transcendence degree of MX (X) over C and call it the algebraic dimension of X. By an algebraic field of transcendence degree d, we mean a finite algebraic extension of the field C(t1 , . . . , td ) of all rational functions in d variables. We say that f1 , . . . , fk ∈ MX (X) are analytically dependent if df1 (x)∧· · ·∧dfk (x) = 0 for any point x ∈ X where all f1 , . . . , fk are holomorphic. Theorem 2.2.9. The functions f1 , . . . , fk+1 ∈ MX (X) are analytically dependent if and only if they are algebraically dependent. Proof. We prove first that algebraic dependence implies analytic dependence. If k + 1 > n = dim X there is nothing to prove, so we assume k + 1  n. Let U be the open set of X where all f1 , . . . , fk+1 are holomorphic. We may also assume without loss of generality that f1 , . . . , fk are algebraically independent. Let Q ∈ C[z1 , . . . , zk+1 ]
{0} be a polynomial of minimal degree > 0 in zk+1 such that Q(f1 , . . . , fk+1 ) = 0. By differentiation we obtain k+1
j=1

∂Q (f1 , . . . , fk+1 )dfj = 0 , ∂zj

on U .

(2.2.17)

Since Q is of minimal degree in zk+1 , ∂z∂Q (f1 , . . . , fk+1 ) is not identically zero, k+1 so we get from (2.2.17) a non-trivial linear relation between the differentials dfj . This implies df1 (x) ∧ · · · ∧ dfk+1 (x) = 0 for x ∈ U . Let us prove now the converse. Assume that f1 , . . . , fk+1 are analytically dependent. Without loss of generality we can assume that f1 , . . . , fk are analytically independent. By Problem 2.2 we find a holomorphic line bundle L on X and holomorphic sections s0 , s1 , . . . , sk with s0 not identically zero, such that fj = sj /s0 , j = 1, . . . , k. Let us choose a covering of X with polydiscs P (xi , 2ri ), i = 1, . . . , m as in Lemma 2.2.1 such that: (a) L|P (xi ,2ri ) is trivial, for i = 1, . . . , m; −1 (b) X = ∪m ); i=1 P (xi , ri e (i)

(i)

(c) f1 , . . . , fk are holomorphic at xi and f1 − f1 (xi ) = z1 , . . . , fk − fk (xi ) = zk can be taken among a set of local holomorphic coordinates at xi , for any i = 1, . . . , m. These conditions can be realized by translating the points xi in the coordinates of P (xi , 2ri ) and taking into account that the set of points where condition (c) is satisfied is an open dense set of X. Choose also a holomorphic line bundle F on X and holomorphic sections σ0 , σ1 with σ0 not identically zero, such that fk+1 = σ1 /σ0 . We can also assume that F |P (xi ,2ri ) is trivial and fk+1 is holomorphic at each xi , for i = 1, . . . , m. For p, q ∈ N we consider the set of all polynomials Q ∈ C[z1 , . . . , zk+1 ]
{0} of degree  p in z1 , . . . , zk and degree  q in zk+1 . To each such Q we associate the homogeneous polynomial  0 , z1 , . . . , zk , w0 , w1 ) = z p wq Q(z1 /z0 , . . . , zk /z0 , w1 /w0 ). Q(z 0 0

(2.2.18)

86

Chapter 2. Characterization of Moishezon Manifolds

 forms a vector space V (p, q) of dimension The set of polynomials Q We define the linear map α : V (p, q) → H 0 (X, Lp ⊗ F q ) ,

 → Q(s  0 , s1 , . . . , sk , σ0 , σ1 ) . Q

k+p k (q + 1). (2.2.19)

Set h = p log ϕ(L) + q log ϕ(F ) + 1, with ϕ(L), ϕ(F ) introduced in (2.2.3).  0 , s1 , . . . , sk , We consider the map ψ which associates to each section Q(s σ0 , σ1 ) the h-jets of Q(f1 , . . . , fk+1 ) at xi , i = 1, . . . , m. As f1 , . . . , fk+1 are ana(i) (i) lytically dependent, we know dz1 ∧ · · · ∧ dzk ∧ dfk+1 = 0 near xi , thus these jets (i) (i) (i) (i) involve only the coordinates z1 , . . . , zk , i = 1, . . . , m. Denote by C{z1 , . . . , zk } (i) (i) (i) the local ring of convergent power series in the variables z1 , . . . , zk and by mxi the maximal ideal of this ring. Consider the map  (i) (i) h+1 ψ : Im(α) → Lpxi ⊗ Fxqi ⊗ (C{z1 , . . . , zk }/(m(i) ). (2.2.20) xi ) 1
i
m

As in Lemma 2.2.1, we show that ψ is injective due to our choice of h. The dimension of the target space of ψ satisfies the estimate   k+h m m (2.2.21) d=m  (h + k)k = pk (log ϕ(L))k + o(pk ) . k k! k! If we choose q such that q + 1 > m(log ϕ(L))k and p sufficiently large, we obtain dim V (p, q) > dim Im(α) and consequently Ker(α) = 0, which implies that
f1 , . . . , fk+1 are algebraically dependent. Proposition 2.2.10. Let f1 , . . . , fk ∈ MX (X) be algebraically independent. Then there exists an integer q = q(f1 , . . . , fk ) such that any f ∈ MX (X) which is algebraically dependent on f1 , . . . , fk satisfies a non-trivial algebraic equation over C(f1 , . . . , fk ) of degree q. Proof. We modify the proof of Theorem 2.2.9. Let us choose a holomorphic line bundle L on X, holomorphic sections s0 , s1 , . . . , sk and polydiscs P (xi , 2ri ), i = 1, . . . , m satisfying conditions (a)–(c) as in Theorem 2.2.9. Note that the conditions (a)–(c) remain valid for small translations of P (xi , 2ri ) within their coordinate patch. Thus we can determine for each xi a closed neighborhood Vi such that (a)–(c) still hold for all P (yi , 2ri ) with yi ∈ Vi . Let us set Pi = P (yi , 2ri ), with yi ∈ Vi . We will calculate ϕ(L) with respect to the all possible covering {Pi }m i=1 . We find moreover a holomorphic line bundle F on X and holomorphic sections σ0 , σ1 with σ0 not identically zero, such that f = σ1 /σ0 . We can assume that F |Pi is trivial. Let us choose q ∈ N such that q + 1 > m(log ϕ(L))k . We can also choose the centers xi of P (xi , 2ri ) in Vi such that f is holomorphic at each xi , for i = 1, . . . , m. Proceeding as in the proof of Theorem 2.2.9 (with fk+1 = f ), we can pick p sufficiently large such that f satisfies a non-trivial algebraic equation
over C(f1 , . . . , fk ) of degree at most q.

2.2. The Siu–Demailly criterion

87

Theorem 2.2.11 (Siegel-Remmert-Thimm). Let X be a compact connected complex manifold. The field of meromorphic functions MX (X) is an algebraic field of transcendence degree a(X)  n = dim X. Therefore, there exists a projective algebraic variety V such that dim V  n and MX (X) is isomorphic to the rational function field MV (V ). Proof. Let f1 , . . . , fk be a maximal set of algebraically independent meromorphic functions. We choose fk+1 such that its degree d over C(f1 , . . . , fk ) is maximal, which is possible by Proposition 2.2.10. We show that MX (X) = C(f1 , . . . , fk , fk+1 ). It is obvious that C(f1 , . . . , fk , fk+1 ) ⊂ MX (X). Let g ∈ MX (X), then the field C(f1 , . . . , fk , fk+1 , g) is a finite algebraic extension of C(f1 , . . . , fk ), so there exists a primitive element h ∈ MX (X) such that C(f1 , . . . , fk , fk+1 , g) = C(f1 , . . . , fk , h). For a field extension K1 ⊂ K, we set [K : K1 ] = dimK1 K. We have d  [C(f1 , . . . , fk , h) : C(f1 , . . . , fk )] = [C(f1 , . . . , fk , h) : C(f1 , . . . , fk , fk+1 )] · [C(f1 , . . . , fk , fk+1 ) : C(f1 , . . . , fk )]. Since the last factor equals d, the first factor equals one, so C(f1 , . . . , fk , h) = C(f1 , . . . , fk , fk+1 ) and g ∈ C(f1 , . . . , fk , fk+1 ). Hence MX (X) is a finite algebraic extension of C(f1 , . . . , fk ). Since algebraic and analytic dependence coincide, we see immediately a(X)  n. Let Q ∈ C(f1 , . . . , fk )[t] be the minimal polynomial of fk+1 over C(f1 , . . . , fk ). We chase denominators of f1 , . . . , fk and divide off factors involving only in the variables f1 , . . . , fk in Q, so we may assume that Q ∈ C[f1 , . . . , fk , t] and is irreducible. Let A = {z ∈ Ck+1 : Q(z) = 0} be the affine algebraic variety defined by Q and let V ⊂ C Pk+1 be its projective closure. Then V is an irreducible projective algebraic variety with dim V = k and its field of rational functions is
isomorphic to C(f1 , . . . , fk , fk+1 ). Definition 2.2.12. A compact connected complex manifold X is called a Moishezon manifold if it possesses dim X algebraically independent meromorphic functions. Observe that z1 /z0 , · · · , zn /z0 are algebraically independent rational functions on C Pn , where [z0 , · · · , zn ] are homogeneous coordinates. Hence C Pn is Moishezon. Moreover, if X  is a connected compact complex manifold, and ϕ : X  −→ X is a proper modification, then by Theorem 2.1.18, X is Moishezon if and only if X  is Moishezon. Recall that a holomorphic map π : V → Y between complex spaces is called a ramified covering if there exists a nowhere dense analytic set A ⊂ Y such that π : V
π −1 (A) → Y
A is locally biholomorphic. Theorem 2.2.13. If V ⊂ C PN is a connected projective manifold of dimension n, then there exists a projection π of C PN on a projective n-plane C Pn ⊂ C PN such that π|V is finite ramified cover. Therefore, any connected projective manifold is Moishezon.

88

Chapter 2. Characterization of Moishezon Manifolds

Theorem 2.2.14. A compact connected complex manifold X is Moishezon if and only if there exists a projective algebraic variety Y with dim Y = dim X and a bimeromorphic map ϕ : X  Y . Proof. Assume that X is Moishezon. Consider the projective algebraic variety Y ⊂ C PN with dim Y = dim X = n constructed at the end of Theorem 2.2.11. By resolution of singularities (Theorem 2.1.15) and since the blow-up of a projective manifold is projective (Proposition 2.1.14), we can assume that Y is smooth. Let C[ξ0 , . . . , ξN ] be a homogeneous coordinate ring of Y (i.e., the quotient of the polynomial ring by the ideal of Y ). Then ξ1 /ξ0 , . . . , ξN /ξ0 ∈ MY (Y ) and we denote by ϕ1 , . . . , ϕN ∈ MX (X) the corresponding elements through the isomorphism MX (X) ∼ = MY (Y ). Consider the meromorphic mapping ψ : X  C PN , ψ = [1, ϕ1 , . . . , ϕN ] (cf. Example 2.1.20). Let G = Graph(ψ) and denote pr1 : G → X. Consider a resolution of singularities σ : X  → G and set ρ := pr2 ◦ σ : X  → Y . The holomorphic map ρ is called the algebraic reduction of X. It is clear that X  is smooth and bimeromorphically equivalent to X. Moreover, ρ∗ : MY (Y ) → MX
(X  ) is an isomorphism. Consider the Stein factorization (Theorem B.1.10) of ρ : X  → Y : ρ /Y (2.2.22) X A O AA AA β α AAA Z Then β is a finite holomorphic map which is bimeromorphic, since MZ (Z) and MY (Y ) are isomorphic via β. By Zariski’s Main Theorem B.1.11 we infer that β is biholomorphic. By the properties of the Stein factorization, the fibers of α are connected, hence the fibers of ρ are connected. Now, the map ψ : X  Y is generically locally biholomorphic so this holds for ρ also. We deduce that ρ is a proper modification, hence X  , Y are bimeromorphically equivalent and thus also X and Y .
We can also give a simple characterization of Moishezon manifolds in terms of order of growth of spaces of sections of line bundles. Theorem 2.2.15. A compact connected complex manifold X is Moishezon if and only if it carries a big line bundle. Proof. If X is Moishezon, then there exist n = dim X algebraically independent meromorphic functions. Using Problem 2.2, we can find a line bundle L such that these functions have the form s1 /s0 , . . . , sn /s0 where s0 , . . . , sn ∈ H 0 (X, L). By Theorem 2.2.9 these functions are also analytically independent, i.e., d(s1 /s0 ) ∧ · · · ∧ d(sn /s0 ) = 0 on the set where the left–hand side is defined. By completing {s0 , · · · , sn } to a basis of H 0 (X, L), we see that the Kodaira map Φ1 : X
Bl|H 0 (H,L)| −→ P(H 0 (H, L)∗ ) has maximal rank i.e. ρ1 = n and hence κ(L) = n.

2.2. The Siu–Demailly criterion

89

Conversely, if L is big, there exists p > 0 such that p = n. Thus there exist s0 , . . . , sn ∈ H 0 (X, Lp ) such that d(s1 /s0 ) ∧ · · · ∧ d(sn /s0 ) = 0 outside a nowhere dense analytic set. This means that s1 /s0 , . . . , sn /s0 are n analytically independent and so algebraically independent meromorphic functions.
Moishezon showed that even more than Theorem 2.2.14 holds: Theorem 2.2.16 (Moishezon). If X is a Moishezon manifold, then there exists a " −→ X, obtained by a finite number of blow-ups with proper modification π : X " is a projective algebraic manifold. smooth centers, such that X Proof. By Theorem 2.2.14 there exists a projective algebraic variety Y , dim Y = dim X, and a bimeromorphic map ϕ : Y  X. Applying Theorem 2.1.25 (ii)  → X, πY : Y → Y and (Chow Lemma) we find holomorphic maps πX : X   ϕ  : Y → X such that πX , πY are compositions of finitely many blow-ups with smooth centers, and the diagram Y

ϕ 

πY

/X 

(2.2.23)

πX

  Y _ _ϕ _/ X  commutes. Using Hironaka’s flattening theorem 2.1.22 for the map ϕ  : Y → X  we may assume that ϕ  is flat and πX : X → X is the composition of finitely many blow-ups with smooth centers. Now ϕ  is flat and bimeromorphic, therefore a finite holomorphic map which is a proper modification. By Zariski’s main theorem B.1.11 we deduce that ϕ  is a biholomorphic map. On the other hand, Y is the successive blow-up of Y along coherent ideal sheaves hence also projective (Proposition 2.1.14). This completes the proof.
Remark 2.2.17. Artin introduced the notion of algebraic space and showed that Moishezon manifolds carry a structure of proper algebraic space over C. Thus, it is possible to apply the algebraic versions of the elimination of indeterminacy points and Chow lemma in order to prove Theorem 2.2.16. Since Moishezon manifolds are not so far from projective ones, they share a lot of features of projective manifolds. We prove now that the Hodge decomposition holds on Moishezon manifolds. Let’s recall first the definition of the spectral sequence associated to a filtered complex. Let (E = ⊕kj=0 E j , d : E • → E •+1 ) be a complex of vector spaces, i.e., d2 = 0. The cohomology of the complex is H j (E) =

Ker(d) ∩ E j , Im(d) ∩ E j

H • (E) =

) j

H j (E).

(2.2.24)

90

Chapter 2. Characterization of Moishezon Manifolds

A subcomplex (V, d) of (E, d) is given by subspaces V j ⊂ E j with dV j ⊂ V j+1 . A filtered complex (F p E • , d) is a decreasing sequence of subcomplexes E • = F 0 E • ⊃ F 1 E • ⊃ · · · ⊃ F m E • ⊃ F m+1 E • = 0.

(2.2.25)

Certainly, the filtration F p E • on E • also induces a filtration F p H • (E) on the p q E cohomology by F p H q (E) = Ker(d)∩F Im(d)∩F p E q . The associated graded cohomology is GrH • (E) =

)

Grp H q (E), with Grp H q (E) =

p,q

F p H q (E) . F p+1 H q (E)

(2.2.26)

Let (F p E • , d) be a filtered complex, for r ∈ N, we define Erp,q =

{u ∈ F p E p+q : du ∈ F p+r E p+q+1 } , d(F p−r+1 E p+q−1 ) + F p+1 E p+q

Er =

)

Erp,q ,

(2.2.27)

p,q

and for [u] ∈ Erp,q , dr [u] = [du] ∈

{v ∈ F p+r E p+q+1 : dv ∈ F p+2r E p+q+2 } = Erp+r,q−r+1 . (2.2.28) d(F p+1 E p+q ) + F p+r+1 E p+q+1

Then we verify that dr : Erp,q → Erp+r,q−r+1 ,

d2r = 0,

H • (Er ) = Er+1 .

(2.2.29)

Moreover, from (2.2.27), we have E0p,q =

F p E p+q , F p+1 E p+q

Erp,q = Grp H p+q (E) for r  m + 1.

(2.2.30)

The above sequence (Er , dr ) (r  0) is called the spectral sequence associated to the filtered complex (F p E • , d). The last statement in (2.2.30) is usually written Er ⇒ H • (E), and we say that the spectral sequence abuts to H • (E). For a compact connected complex manifold X, we denote H p,q (X) = H p,q (X, C). We define a filtration on the de Rham complex (Ω(X), d) by ) )

Ωj (X) = Ωp,q (X), F p Ωj (X) = Ωp ,j−p (X).

(2.2.31)

p
p

p+q=j

ohlicher The corresponding spectral sequence (Er (X), dr ) (r  0) is called the Fr¨ spectral sequence which abuts to H • (X, C), the de Rham cohomology of X. As d = ∂ + ∂, by (2.2.27)-(2.2.30), we get E0p,q (X) = Ωp,q (X),

d0 = ∂,

E1p,q (X) = H p,q (X).

(2.2.32)

2.2. The Siu–Demailly criterion

91

Theorem 2.2.18. If X is a Moishezon manifold, then the Hodge decomposition holds: p (a) H j (X, C) ∼ = ⊕p+q=j H q (X, OX ) ∼ = ⊕p+q=j H p,q (X), (b) H p,q (X) ∼ = H q,p (X). " −→ X be a modification as in Theorem 2.2.16. Since X " is K¨ Proof. Let π : X ahler, " the Hodge decomposition holds for X. " induced by the pull-back of Let us remark first that π ∗ : H p,q (X) → H p,q (X) forms is an injective morphism . To see this, consider a ∂-closed form α ∈ Ωp,q (X) " Take an arbitrary ∂-closed form γ ∈ Ωn−p,n−q (X), such that π ∗ (α) = ∂β on X.  where n = dim X. By Stokes formula we have X" π ∗ (α) ∧ π ∗ (γ) = 0. Since π is a  modification, we have X α ∧ γ = X" π ∗ (α) ∧ π ∗ (γ) = 0. By Serre duality we know that the linear map  α ∧ [·] H p,q (X) → (H n−p,n−q (X))∗ , [α] → X

is an isomorphism. We deduce that [α] = 0, so π ∗ is injective. We show similarly " C) is injective. that π ∗ : H j (X, C) → H j (X, p,q " By Corollary 1.4.13 for X, " we know If α ∈ H (X), then π ∗ α ∈ H p,q (X). ∗ ∗ ∗ that ∂π α = 0. Thus ∂α = 0 on X, as π ∂α = ∂π α = 0. This means dα = 0 for any α ∈ H p,q (X). But by (2.2.28)-(2.2.29), this implies that E1p,q (X) = Erp,q (X) for any r ≥ 1, together with (2.2.30), we get (a). To prove (b), we observe that H j (X, C) and H j (X, C) are C-anti-isomorphic, j so H (X, C) ∼ = ⊕p+q=j H p,q (X). Now, π ∗ (α) = π ∗ (α) for any α ∈ Ωp,q (X). There

  ∗ q,p " and π ∗ H q,p (X) ∩ H p1 ,q1 (X) = fore π H (X) = π ∗ (H q,p (X)) ⊂ H p,q (X) {0} if (q, p) = (q1 , p1 ). Since π ∗ is injective, we must have H q,p (X) ∼ = H p,q (X).
As an application of the Hodge decomposition, let us study the analytic cohomology classes. Definition 2.2.19. Given an n-dimensional compact complex manifold X, we call a 2k-cycle i ci Yi in X (ci ∈ Z) analytic if Yi are compact k-dimensional analytic sets. We call analytic 2-cycles curves. A homology class in H2k (X, Z) (k ∈ N) is called analytic, if it represented by an analytic 2k-cycle. We denote by A2k (X, Z) ⊂ H2k (X, Z) and the subgroup generated by analytic homology classes and set A2k (X, Q) = A2k (X, Z) ⊗Z Q and A2k (X, R) = A2k (X, Q) ⊗Q R.  Two curves C, C  are called numerically  equivalent, written C ≡ C , if D·C =  D·C for any divisor D on X. Here D·C = C c1 (OX (D)) is the intersection number of D with C. Two divisors D, D are called numerically equivalent, written D ≡ D if D · C = D · C for any irreducible curve C in X. Recall that there exists an isomorphism Ψ : H2k (X, Z) → H 2n−2k (X, Z) (by Poincar´e duality), which associates to each homology class [Y ] ∈ H2k (X, Z) of a

92

Chapter 2. Characterization of Moishezon Manifolds

2k-cycle Y a cohomolgy class [ηY ] ∈ H 2n−2k (X, Z) ⊂ H 2n−2k (X, R) of a closed (2n − 2k)-form ηY such that   α= α ∧ ηY , for any closed 2k-form α. (2.2.33) Y

X

ηY is called the fundamental class of Y . A cohomology class of H 2n−2k (X, Z) is called analytic if it is the fundamental class of an analytic 2k-cycle. For a form α ∈ Ω2k (X) we denote by αp,q ∈ Ωp,q (X) the component of α in the decomposition Ω2k (X) = ⊕p+q=2k Ωp,q (X). Assume that Y is an analytic 2k-cycle. By the Hodge decomposition theorem 2.2.18 (i), we can choose a representative ηY in the fundamental class of Y such that ηYp,q are d- and ∂-closed for any p, q. Y being analytic also entails   α= αk,k . (2.2.34) Y

Y

 If (p, q) = (n − k, n − k) we deduce from (2.2.33) and (2.2.34) that X α ∧ ηYp,q = 0 for any d-closed α ∈ Ω• (X). By another form of the Poincar´ e duality, the pairing  ψ : H • (X, C) × H • (X, C) → C, given by ψ(α, β) = X α ∧ β, is non-degenerate (which can also be seen from the Hodge theory for de Rham cohomology). Hence [ηYp,q ] = 0 ∈ H • (X, C). (Using the spectral sequence we see that actually [ηYp,q ] = 0 ∈ H p,q (X).) Therefore, [ηY ] = [ηYn−k,n−k ] ∈ H 2n−2k (X, C) and the de Rham cohomology class of ηY in H 2n−2k (X, C) (i.e., the fundamental class of Y ) lies in H n−k,n−k (X). For (1, 1)-classes on Moishezon manifolds the converse is also true, namely the Lefschetz theorem on (1, 1)-classes holds. Theorem 2.2.20. Let X be a Moishezon manifold. Then for any α ∈ H 1,1 (X) ∩ H 2 (X, Z), there exists a divisor D ∈ Div(X) with α = c1 (OX (D)). Hence, α is the fundamental class of D, i.e. α = ηD . Proof. Remark that the last assertion follows from the equality c1 (OX (D)) = ηD ,

(2.2.35)

valid for any divisor D, so it suffices to find D ∈ Div(X) with α = c1 (OX (D)). Let (F, hF ) be a holomorphic Hermitian line bundle on X such that c1 (F, hF ) = α (cf. Lemma 2.3.5). We assume first that X is projective. Consider a positive line bundle (L, hL ) on X. By the Kodaira-Serre vanishing theorem 1.5.6 and the Riemann-RochHirzebruch theorem 1.4.6, √ 

−1 L n n 1 0 p R dim H (X, L ⊗ F ) = p + O(pn−1 ), n! X 2π

2.2. The Siu–Demailly criterion

93

for p ∈ N sufficiently large. Since the coefficient of pn is positive, H 0 (X, Lp ⊗F ) = 0 for some fixed large p (cf. also (1.7.1) with q = 1). Hence Lp ⊗ F ∼ = OX (D) where D is the divisor defined by a non-trivial s ∈ H 0 (X, Lp ⊗ F ). Applying this argument for F trivial we see that Lp ∼ = OX (D1 ) for D1 ∈ Div(X) and finally F ∼ = OX (D − D1 ). " −→ X as in Theorem 2.2.16. For general X, consider a modification π : X " " " " Let Since X is projective, there exists D ∈ Div(X) such that π ∗ (F ) ∼ = OX" (D). −1 " A ⊂ X be an analytic set of codimension  2 such that π : X
π (A) → X
A is biholomorphic (note that the centers of blow-up have codimension  2). " in X
A By the Remmert-Stein extension theorem B.1.7, the image of D ∼ extends over A to a divisor D on X. Then F = OX (D) on X
A, which means that there exists a holomorphic section s of F ⊗ OX (−D) over X
A which has no zeros. In trivialization patches s is given by a holomorphic function which extends holomorphically over A (by Riemann’s second extension theorem B.1.6). Hence s extends to a holomorphic section s of F ⊗ OX (−D) over X. But s cannot have zeros (they would form an analytic set of codimension 1), therefore F ⊗ OX (−D) is trivial over X, i.e., F ∼
= OX (D) on X. Our next goal is to prove that a Moishezon K¨ ahler manifold is projective and prepare for this purpose some lemmas. Lemma 2.2.21. Let X be a Moishezon manifold. Then for any β ∈ H n−1,n−1 (X) ∩ 2n−2 (X, Z), there is a curve i ni Ci , ni ∈ Z whose fundamental class is β.The H assertion still hold if we consider cohomology with values in Q and curves with rational coefficients. Proof. Let us first assume that X is projective and we fix an embeding in a projective space C Pm . By the hard Lefschetz theorem we have a commutative diagram Φ

H 1,1 (X) ∩ H 2 (X, Z) −−−−→ H n−1,n−1 (X) ∩ H 2n−2 (X, Z) ⏐ ⏐ ⏐Ψ ⏐ Ψ1 ( ( 2 A2n−2 (X, Z)

Ξ

−−−−→

(2.2.36)

H2 (X, Z)

where Φ is the isomorphism given by the multiplication with ωFn−1 S (ωF S being the restriction of the Fubini-Study form to X) and Ψ1 , Ψ2 are induced by the Poincar´e duality. Finally, Ξ is the intersection with n − 1 hyperplanes in C Pm , which means that the image of Ψ2 is exactly A2 (X, Z). By Theorem 2.2.20, Ψ1 is an isomorphism. Since Φ is an isomorphism and Ξ injective, we have an isomorphism Ψ2 : H n−1,n−1 (X) ∩ H 2n−2 (X, Z) → A2 (X, Z). " −→ Assume now that X is just Moishezon and consider a modification π : X ∗ X as in Theorem 2.2.16. The class π (β) is represented by a curve i∈I ni Ci , " Then β is represented by where Ci ⊂ X. j∈J nj π(Cj ) , where the sum is restricted to the indices j ∈ I such that π(Cj ) is a curve in X (note that by Rem-

94

Chapter 2. Characterization of Moishezon Manifolds

mert’s proper image theorem B.1.12, the image of analytic sets through proper holomorphic maps are analytic sets). Indeed, for any ∂-closed form η ∈ Ω2 (X) we have  
 β∧η = π ∗ (β) ∧ π ∗ (η) = ni π ∗ (η) " X C X i i∈I (2.2.37)

 ∗ = ni π (η) = ni η. i∈J

Ci

i∈J

π(Ci )

Since H • (X, Q) = H • (X, Z) ⊗Z Q, we also get the Q-coefficient version.




Lemma 2.2.22. The numerical and homological equivalence for curves and divisors in a Moishezon manifold coincide. Proof. Let D ∈ Div(D). We have to show that c1 (OX (D)) = 0 in H 2 (X, Q) if and only if D · C = 0 for all curves C ⊂ X. The “only if” part is trivial. We prove the “if” part. Assume that D · C = 0 for all curves C ⊂ X. We have to show that  α∧ψ =0 (2.2.38) X

for any d-closed form ψ ∈ Ω (X), where α is a (1, 1)-form representing c1 (OX (D)). By the Hodge decomposition 2n−2

H 2n−2 (X, C) = H n−1,n−1 (X) ⊕ H n,n−2 (X) ⊕ H n−2,n (X) and since α is of bidegree (1, 1) it suffices to show (2.2.38) for ψ ∈ Ωn−1,n−1 (X). Moreover, since c1 (OX (D)) ∈ H 1,1 (X) ∩ H 2 (X, Q), it’s enough to consider forms ψ ∈ Ωn−1,n−1 (X) with [ψ] ∈ H n−1,n−1 (X) ∩ H 2n−2 (X, Q). Using Lemma 2.2.21 we know that [ψ] is represented by a curve with rational coefficients. But then (2.2.38) follows from the hypothesis. The proof that for a curve C, [C] = 0 if and only if D · C = 0 for any D ∈ Div(X), is analogous.
Lemma 2.2.23. Let X be a Moishezon manifold. Then the canonical morphism Φ : A2n−2 (X, Q) → A2 (X, Q)∗ given by Φ([D])([C]) = D · C is an isomorphism. Proof. Consider a linear map ϕ : A2 (X, Q) → Q. We identify now A2 (X, Q) to ϕR : A2 (X, R) = H n−1,n−1 (X) ∩ H 2n−2 (X, Q) by Lemma 2.2.21. The extension  H n−1,n−1 (X) → R is given by a ∂-closed form ψ: ϕR (γ) = X γ ∧ ψ, where γ is a ∂-closed (n − 1, n − 1)-form. The trivial extension ϕ " : H 2n−2 (X, Q) → Q of ϕ is 2 given by an element of H (X, Q). Therefore [ψ] ∈ H 1,1 (X) ∩ H 2 (X, Q). Theorem 2.2.20 provides D ∈ Div(X) and k ∈ N∗ such that [ψ] =

1 c1 (OX (D)), k

so

ϕ([C]) =

1 D · C. k

By Lemma 2.2.22, the class of k1 D in A2n−2 (X, Q) is uniquely determined by this property.


2.2. The Siu–Demailly criterion

95

For a compact curve C ⊂ X we define its multiplicity as m(C) = sup mx (C), x∈C

where mx (C) is the multiplicity at x (see Theorem and Definition B.1.3). Lemma 2.2.24 (Seshadri’s ampleness criterion). Let X be a Moishezon manifold and D ∈ Div(X). If there exists ε > 0 such that D · C > εm(C) for all irreducible curves C ⊂ X, then OX (D) is positive. Lemma 2.2.25. Let X be a Moishezon manifold. Assume that there exists ε > 0 and ϕ ∈ A2 (X, Q)∗ such that ϕ([C])  εm(C) for any irreducible curve C ⊂ X. Then X is projective. Proof. By Lemma 2.2.23 there exist k ∈ N∗ , D ∈ A2n−2 (X, Z) such that ϕ([C]) = 1 k D · C for any curve C ⊂ X. By Seshadri’s criterion we deduce that OX (D) is positive and X is projective by the Kodaira embedding theorem 5.1.12.
Theorem 2.2.26 (Moishezon). If X is a Moishezon K¨ ahler manifold, then X is projective. Proof. Let ωbe a K¨ ahler form on X. We consider the linear map ϕ : A2 (X; R) → R, ϕ([C]) = C ω. Lemma 2.2.23 entails that there exists D ∈ A2n−2 (X, R), with k ϕ([C]) = D · C for all curves C ⊂ X. Set D = i=1 mi Di , where mi ∈ R, Di ∈ Div(X). We consider sequences of rational numbers miν → mi and set k Dν = i=1 miν Di . We define αν =

  √−1 ROX (Di ) , (mi − miν ) 2π i=1

k


(2.2.39)

where ROX (Di ) is the curvature of a Chern connection on OX (Di ). Then αν → 0 uniformly, as ν → ∞. Thus, there exists ν0 such that ω − αν is strictly positive for  = ω − αν . Then ω  is K¨ahler and  ν  ν0ν . Let ν  ν0 be fixed and set ω ω  = D · C. This shows that by substituting ω by ω  and D by Dν we can C assume D ∈ Div(X). x → X the blow-up of X with center x and by Ex We denote by πx : X the exceptional divisor. Due to the compactness of X we can find ε > 0 and a √ −1 OX ∗ "x = πx (ω) − ε 2π R &x (Ex ) is positive Hermitian metric on OX x (Ex ) such that ω for any x ∈ X (cf. the proof of Lemma 2.1.8). Consider a curve C ⊂ X and a " be the strict transform of C in X x . Then ϕ([C]) = D · C = point x ∈ C. Let C   ∗ " " πx (ω). It is easy to see that mx (C) = Ex · C. Thus Cω = C   " ω − εmx (C) = ω − εEx · C C C   √  (2.2.40) −1 OX ∗ &x (Ex ) = πx (ω) − ε = ω "x > 0. 2π R " C

" C

" C

96

Chapter 2. Characterization of Moishezon Manifolds

Since ε does not depend on C and x, we can apply Lemma 2.2.25 and obtain that X is projective.
We come now to a characterization `a la Kodaira of the Moishezon manifolds. In view of Theorem 2.2.16, there exists a projective proper modification π : X  −→ X if X is Moishezon. By taking the push-forward of the positive line bundle on X  , we obtain in general a sheaf on X, which outside a proper analytic set A is locally free and has a smooth metric of positive curvature. Such sheaves are called quasipositive. The question is, whether this property characterizes Moishezon manifolds. Since the Moishezon property is bimeromorphically invariant, we can blow up X in order to obtain a manifold X  possessing a line bundle with semi-positive curvature everywhere and positive outside a proper analytic set. If we show that X  is Moishezon, it follows that X is Moishezon too. Grauert-Riemenschneider conjecture. If X possesses a smooth Hermitian line bundle which is semi-positive everywhere and positive on an open dense set, then X is Moishezon. Therefore, X is Moishezon if and only if X carries a quasi-positive sheaf. The following theorem gives a positive answer to the above conjecture. Theorem 2.2.27 (Siu–Demailly). Let X be a compact connected complex manifold of dimension n and (L, hL ) be a holomorphic Hermitian line bundle over X. Then X is Moishezon if one of the following conditions is verified (i) (Siu’s criterion) (L, hL ) is semi-positive on X and positive at one point. (ii) (Demailly’s criterion) (L, hL ) satisfies  √ n −1 L > 0. (2.2.41) 2π R X(
1)

Proof. We apply Theorem 1.7.1 for q = 1 and obtain  √ n pn −1 L 0 p dim H (X, L )  R + o(pn ). n! X(
1) 2π

(2.2.42)

√ First observe that if (L, hL ) is semi-positive, then −1RL has no negative eigenvalue, i.e., X(1) = ∅. If (L, hL ) is moreover positive at at least one point, n √  −1 L then X(0) = ∅ and of course X(0) 2π R > 0. Thus condition (i) implies (ii), so it suffices to prove that (ii) implies that X is Moishezon. Siegel’s lemma 2.2.6, (2.2.41) and (2.2.42) show that there exist C1 , C2 > 0 and p0 ∈ N such that for p > p0 , C2 pp  dim H 0 (X, Lp )  C1 pn .

(2.2.43)

Therefore p = n for p  p0 , so κ(L) = n and L is big. By Theorem 2.2.15, we conclude that X is Moishezon.


2.3. The Shiffman–Ji–Bonavero–Takayama criterion

97

2.3 The Shiffman–Ji–Bonavero–Takayama criterion This section is dedicated to the proof of Shiffman’s conjecture that Moishezon manifolds can be characterized in terms of integral K¨ ahler currents. The conjecture was first solved by Ji–Shiffman. We will present a proof using the singular holomorphic Morse inequalities of Bonavero in Section 2.3.2. Later, in Section 6.2, we prove the criterion using the holomorphic Morse inequalities on non-compact manifolds derived from the expansion of the Bergman kernel. This section is organized as follows. In Section 2.3.1, we introduce singular Hermitian metrics on line bundles. In Section 2.3.2, we establish the singular holomorphic Morse inequalities of Bonavero from the holomorphic Morse inequalities, then prove Shiffman’s conjecture. In Section 2.3.3, we study the volume of big line bundles. In Section 2.3.4, we present an example of a non-projective Moishezon manifold.

2.3.1 Singular Hermitian metrics on line bundles The proof of the Grauert–Riemenschneider criterion from Section 2.2 deals with smooth Hermitian line bundles. However, smooth Hermitian metrics with semipositive curvature do not characterize Moishezon manifolds, since there exist examples of Moishezon manifolds which do not possess a line bundle satisfying (2.2.41) for a smooth metric, cf. Section 2.3.4. Actually, the solution of the Grauert–Riemenschneider conjecture just shows that Moishezon manifolds can be characterized in terms of quasi-positive analytic sheaves. Nevertheless, one can conjecture another characterization, in terms of currents. Since the pioneering work of P. Lelong, currents have had a deep impact on complex analysis. While intrinsically linked to analytic objects, they are much more flexible. Let X be a complex manifold. Let L be a holomorphic line bundle over X. Definition 2.3.1. A singular Hermitian metric hL on L is a choice of a sesquilinear, Hermitian-symmetric form hL x : Lx × Lx → C ∪ {∞} on each fiber Lx , such that, for any local holomorphic frame eL of L on U , with local weights ϕ ∈ L1loc (U ), we have |eL |2hL = hL (eL , eL ) = e−2ϕ ∈ [0, ∞]. (2.3.1) Here L1loc (U ) is the space of local integrable functions on U . If the local weights ϕ ∈ C ∞ (U ), we obtain the usual definition of a Hermitian metric. If s is an arbitrary holomorphic section of L on U , s = f eL with f ∈ OX (U ) we cannot set |s|2hL = |f |2 |eL |2hL everywhere (e.g., if s(x) = 0 and |eL |hL (x) = ∞). Of course, the equality holds a.e. (almost everywhere). Moreover log |s|2hL = log |f |2 − 2ϕ

in L1loc (U ) .

(2.3.2)

In terms of an open covering {Uα } and holomorphic frames eα of L on Uα , the metric is given by a family {hα } of functions hα = |eα |2hL : Uα → [0, ∞] with local

98

Chapter 2. Characterization of Moishezon Manifolds

∗ weights ϕα = −(1/2) log hα ∈ L1loc (Uα ). If (gαβ ∈ OX (Uα ∩ Uβ )) is the cocycle of 2 L, then hβ = |gαβ | hα on Uα ∩ Uβ , so

log hβ = log |gαβ |2 + log hα

in L1loc (Uα ∩ Uβ ).

(2.3.3)

Since ∂∂ log |gαβ |2 = 0, (2.3.3) allows us to define the generalized curvature. Definition 2.3.2. The curvature current and the Chern–Weil representative of the first Chern class of a singular Hermitian line bundle (L, hL ) are the global (1, 1)L currents RL = R(L,h ) , c1 (L, hL ) defined locally by √ −1 (L,hL ) (L,hL ) L R = −∂∂ log hα = 2∂∂ϕα , c1 (L, h ) = . (2.3.4) R 2π c1 (L, hL ) is obviously closed and integral, representing the Chern class of L in the de Rham cohomology group H 2 (X, R) (which as we saw in Appendix B.2, can be also calculated using currents). Another way to define a singular Hermitian metric hL on L is to give a smooth 1 L L −2ϕ . From Hermitian metric hL 0 and a function ϕ ∈ Lloc (X) and then set h = h0 e (1.5.9), the curvature current is then √ L L −1 ∂∂ϕ. ) + (2.3.5) R(L,h ) = R(L,h0 ) + 2∂∂ϕ, c1 (L, hL ) = c1 (L, hL 0 π Theorem 2.3.3 (Poincar´e–Lelong formula). Let s be a meromorphic section of L on X and hL be a singular Hermitian metric on L. Then √ −1 2π ∂∂

log |s|2hL = [Div(s)] − c1 (L, hL ).

(2.3.6)

Proof. Indeed, writing locally s = fα eα and plugging in (2.3.2), (2.3.6) follows from the Poincar´e–Lelong formula (B.2.18).
Example 2.3.4. Consider a non-trivial meromorphic section s of L on X. We define a singular Hermitian metric hL on L (associated to s) by |s(x)|hL = 1

for x ∈ X
Div(s).

(2.3.7)

We conclude from (2.3.6) that the curvature of the singular Hermitian metric (2.3.7) is given by (2.3.8) c1 (L, hL ) = [Div(s)]. Under the notation in (2.1.5), the canonical section sD of OX (D) is defined by {fα } on Uα . The singular Hermitian metric hOX (D) associated to sD is defined by (2.3.7). (2.3.8) reads c1 (OX (D), hOX (D) ) = [D] .

(2.3.9)

2.3. The Shiffman–Ji–Bonavero–Takayama criterion

99

The following result is quite useful. For another proof in the case ω is smooth, see Proposition 1.5.3. Lemma 2.3.5. Let ω be a real, closed, (1, 1)-current of order 0. If the cohomology class of ω in H 2 (X, R) is integral, there exists a holomorphic line bundle L, endowed with a singular Hermitian metric hL , such that c1 (L, hL ) = ω in the sense of currents. Proof. Let U = {Uα } be a covering of X with geodesically convex open sets; hence all intersections of sets in U are contractible. We can also assume that Uα are biholomorphic to open balls in the Euclidean space. For a sheaf F on X, we denote by C q (U , F ), Z q (U , F ) and B q (U , F ) the spaces of q-cochains, q-cocycles and q-coboundaries of F with respect to U . Let δ : C q (U , F ) → C q+1 (U , F ) be the ˇ coboundary operator in Cech cohomology. √ We start by solving ω = 2 −1 ∂∂uα on Uα with uα ∈ L1loc (U, R), using Lemma B.2.20. Define hα := exp(−4πuα ). We will √ construct a line bundle L such that hα is a singular metric on L. Writing dc := −1(∂ − ∂), we have ω = ddc uα . Let (2.3.10) uαβ = uβ − uα ∈ L1loc (Uα ∩ Uβ ) . Then ddc uαβ = 0, so uαβ is pluriharmonic and there exists vαβ ∈ C ∞ (Uα ∩ Uβ , R) with (2.3.11) dvαβ = dc uαβ . Set cαβγ := vαβ + vβγ + vγα .

(2.3.12)

Since dcαβγ = dc (uαβ + uβγ + uγα ) = dc 0 = 0, cαβγ are constant, so (cαβγ ) ∈ Z 2 (U , R). The de Rham isomorphism and Leray Theorem for acyclic coverings (note that U is acyclic for the constant sheaf R) induce a canonical isomorphism ˇ H 2 (U , R) cohomology groups. Let between the de Rham H 2 (X, R) and the Cech us describe the image of the class [ω] through this isomorphism. If ϕα := dc uα , we have dϕα = ω and (δϕ)αβ = ϕβ − ϕα = dc uαβ = dvαβ . Then δ(vαβ ) = (cαβγ ) is the image of [ω] under the de Rham isomorphism. Hence (cαβγ ) is integral; i.e., there exists a 1-cochain {bαβ } ∈ C 1 (U , R) such that for all α, β, γ, mαβγ := cαβγ + bαβ + bβγ + bγα ∈ Z. Let fαβ = uαβ +

√ −1(vαβ + bαβ ) ,

ραβ = exp(−2πfαβ ) .

(2.3.13)

(2.3.14)

By (2.3.11), the fαβ are holomorphic functions, and by (2.3.10), (2.3.12) and (2.3.13), √ (2.3.15) fαβ + fβγ + fγα = −1mαβγ . √ ∗ By (2.3.15), ραβ ρβγ ραγ = exp(−2π −1mαβγ ) = 1, and thus (ραβ ) ∈ Z 1 (U , OX ) is the cocycle of a holomorphic line bundle L. Moreover, (hα ) defines indeed a

100

Chapter 2. Characterization of Moishezon Manifolds

Hermitian metric hL on L: 2 hβ h−1 α = exp(−4πuαβ ) = | exp(−4πfαβ )| = |ραβ | .

(2.3.16)

Finally, 1 ω = ddc uα = − 4π ddc log hα = −

√ −1 2π ∂∂

log hα = c1 (L, hL ) .

(2.3.17)


This achieves the proof.

Recall that a K¨ ahler current ω on X is a closed, strictly positive (1, 1)-current ω. Back to Moishezon manifolds we have the following. Lemma 2.3.6. Any compact Moishezon manifold X of dimension n carries an integral K¨ ahler current ω whose singular support is contained in a proper analytic set. Therefore, there exists a holomorphic line bundle L, endowed with a singular Hermitian metric hL , smooth outside an analytic set, such that c1 (L, hL ) = ω in the sense of currents. " and a Proof. By Moishezon’s theorem 2.2.16, there exists a projective manifold X " L " " proper modification π : X → X. Let (L, h ) be a positive line bundle with smooth " " set ω " hL" ). The push-forward ω := π∗ ω metric hL on X, " = c1 (L, " is a (1, 1)-current defined by  ∗ " , ϕ) := (" ω , π ϕ) = ω " ∧ π ∗ ϕ, (2.3.18) (π∗ ω " X

for any (n − 1, n − 1)-form ϕ on X. We show that ω " forms an integral K¨ ahler current on X. It is clear that the current π∗ ω " is smooth outside the analytic set where π is not biholomorphic. Let θ be an arbitrary positive smooth (1, 1)-form " Hence there exists on X. Then π ∗ θ is a semi-positive smooth (1, 1)-form on X. C > 0 such that ω "  Cπ ∗ θ. We deduce that for any positive (n − 1, n − 1)-form ϕ on X    (π∗ ω " , ϕ) = ω " ∧ π∗ ϕ  Cπ ∗ θ ∧ π ∗ ϕ = Cθ ∧ ϕ = (Cθ, ϕ) > 0, " X

" X

X

thus π∗ ω " is positive. It is also clear that dπ∗ ω " = π∗ d" ω = 0. Let A ⊂ X be a "
π −1 (A) → X
A is biholomorphic. nowhere dense analytic set such that π : X Since π is a composition of blow-ups in codimension  2, A is of codimension  2. " As " such that ω We consider a divisor D " represents the first Chern class of OX" (D). "
π −1 (A)) extends by the Remmertshown in the proof of Theorem 2.2.20, π(D Stein theorem B.1.7 to a divisor D on X. Then ω represents the first Chern class
of OX (D) and thus [ω] ∈ H 2 (X, Z). The converse is also true: Theorem 2.3.7. A compact connected complex manifold is Moishezon if and only if it admits an integral K¨ ahler current with singular support contained in a proper analytic set.

2.3. The Shiffman–Ji–Bonavero–Takayama criterion

101

There are two strategies to prove Theorem 2.3.7, both based on holomorphic Morse inequalities for dim H 0 (X, Lp ) for a suitable line bundle L. One can work directly with the given current and apply the Morse inequalities on the complement of the analytic set which contains the singular support. We develop this method in Section 6.2. Another possibility is to apply Demailly’s approximation Theorem 2.3.10 and suppose that the current has analytic singularities (cf. Definition 2.3.9) which allows us to apply Bonavero’s holomorphic Morse inequalities (see Section 2.3.2). With the help of Theorem 2.3.10, Theorem 2.3.7 has its equivalent version, Theorem 2.3.8 which was conjectured by Shiffman. Note that in the proof of Theorem 2.3.7 in Section 6.2, we do not use Demailly’s approximation theorem 2.3.10. We will prove Theorems 2.3.7 and 2.3.8 in the next section. Theorem 2.3.8 (Ji–Shiffman). A compact connected complex manifold is Moishezon if and only if it admits a strictly positive singular polarization (L, hL ).

2.3.2 Bonavero’s singular holomorphic Morse inequalities Bonavero gave a proof of Theorem 2.3.8 in terms of his singular holomorphic Morse inequalities. One crucial ingredient of the proof is the approximation result of Demailly which permits us to apply the singular holomorphic Morse inequalities, Theorem 2.3.18 (or Theorem 2.3.7). We start by recalling the notion of singular Hermitian metric with analytic singularities and state the approximation Theorem 2.3.10. After introducing the notion of Nadel multiplier sheaf, we prove the singular holomorphic Morse inequalities in Theorem 2.3.18. This permits us to show the characterization of Moishezon manifolds in Theorem 2.3.28; it contains also Theorems 2.3.7 and 2.3.8. Let X be a complex manifold of dimension n, and let L be a holomorphic line bundle on X. Definition 2.3.9. A real function with analytic singularities is a locally integrable function ϕ on X which has locally the form  
c (2.3.19) |fj |2 + ψ, ϕ = log 2 j∈J

where J is at most countable, fj are non-vanishing holomorphic functions and ψ is smooth and c is a Q+ -valued, locally constant function on X . In particular the singular support of ϕ and ∂∂ϕ is an analytic set. By a Hermitian metric with analytic singularities we understand a singular Hermitian metric hL on L with −2ϕ hL = hL , where hL 0 e 0 is a smooth Hermitian metric on L and ϕ : X → R has analytic √ singularities. A (1, 1)-current ω is called current with analytic singularities if ω = 2 −1∂∂ϕ + ω0 where ω0 is a smooth representative of the de Rham class of ω and ϕ : X → R has analytic singularities.

102

Chapter 2. Characterization of Moishezon Manifolds

Note that a function with analytic singularities is quasi-plurisubharmonic (cf. Definition B.2.16). With this notion we can state the approximation theorem. Theorem 2.3.10 (Demailly). √Let X be a compact complex manifold, and consider a closed (1, 1)-current ω = 2 −1∂∂ϕ + ω0 on X, where ω0 is a smooth (1, 1)-form in the same de Rham cohomology class as ω and ϕ is a quasi-plurisubharmonic function. Let θ be a smooth (1, 1)-form such that ω  θ, and let Θ be a Hermitian 1 form on X as in (1.2.49). Then there exists a sequence {ϕm }∞ m=1 ⊂ Lloc (X) of real functions with analytic singularities such that for m → ∞, the following assertions hold: (i) ϕm is decreasing and converges pointwise and in L1loc (X) to ϕ ; √ (ii) ωm := 2 −1∂∂ϕm + ω0 is in the same de Rham cohomology class as ω, and converges weakly to ω ; (iii) ωm  θ − εm Θ, where εm > 0 converges to 0. Remark 2.3.11. It was shown by Boucksom that we can choose ωm such that ωm,ac → ωac as m → ∞. Remark 2.3.12. To explain the form (2.3.19) of the approximating functions, let us consider the local part of the proof of Theorem 2.3.10. Let ψ be a plurisubharmonic function defined in the unit ball Bn ⊂ Cn . Let {fim }∞ i=1 be an orthonormal basis of the Hilbert space {f ∈ OCn (Bn ) : Bn |f |2 e−2m ψ dZ < ∞}, where dZ is the canon 1 log( i |fim |2 ) → ψ pointwise and in ical Euclidean volume form on Cn . Then 2m L1loc on Bn as m → ∞. By carefully using a partition of unity, Demailly shows that one can glue the local approximations and obtain a global approximation ϕm having analytic singularities. The new element in Bonavero’s approach is the introduction of the Nadel multiplier sheaf in the holomorphic Morse inequalities. Definition 2.3.13. Let ϕ ∈ L1loc (X, R). The Nadel multiplier ideal sheaf I (ϕ) is the ideal subsheaf of germs of holomorphic functions f ∈ OX,x such that |f |2 e−2ϕ is integrable with respect to the Lebesgue measure in local coordinates near x. Let −2ϕ hL = hL be a singular Hermitian metric on L where hL 0e 0 is smooth and ϕ ∈ 1 Lloc (X, R). The Nadel multiplier ideal sheaf of hL is defined by I (hL ) = I (ϕ), which does not depend on the choice of ϕ. Remark 2.3.14. Let E be a holomorphic vector bundle on X. Since I (ϕ) is a subsheaf of OX , we have H 0 (X, E ⊗ I (ϕ)) ⊂ H 0 (X, E).

(2.3.20)

An important property of the multiplier ideal sheaves is the coherence. Theorem 2.3.15 (Nadel). Let hL be a singular Hermitian metric on L. Assume that c1 (L, hL )  CΘ for some real constant C and a smooth Hermitian form Θ on X as in (1.2.49). Then I (hL ) is coherent.

2.3. The Shiffman–Ji–Bonavero–Takayama criterion

103

The proof is an application of Theorem B.4.6 for a small coordinate ball B X (x, ε) with the trivial bundle endowed with the singular metric hL · |z − x|−(2n+2j)

for j ∈ N.

Example 2.3.16. (1) Let ϕ be bounded from below near x ∈ X. Then I (ϕ)x = OX,x . (2) Let X = Cn and for c1 , . . . , ck ∈ R∗+ , p ∈ N∗ , set ϕp (z) =

p log(|z1 |2c1 + · · · + |zk |2ck ) , 2

z ∈ Cn .

Set P (0, r) = {z ∈ Cn : |zj | < r for any 1  j  n}, r > 0, and let dZ be the functions z β (β ∈ Nn ) are Euclidean volume form on Cn . Then the holomorphic  2 orthogonal with respect to the norm f ϕp = P (0,r) |f |2 e−2ϕp dZ, and z β ∈ OCn , 0 is an element of I (ϕp )0 if and only if z β 2ϕp < ∞. Passing to polar coordinates, we see that z β 2ϕp < ∞ if and only if 

((2β1 +2)/c1 )−1

((2β +2)/ck )−1

· · · uk k 2 (u1 + · · · + u2k )p

u1

[0,r]k

du1 · · · duk < ∞ .

Using the homogeneity, this is the case if and only if  0

r

t2

k

j=1 ((βj +1)/cj )−k

t2p

tk−1 dt < ∞ ,

k β +1 that is, if and only if 2 j=1 jcj − k − 2p + (k − 1) > −1. Hence I (ϕp )0 is k #k β generated by j=1 zj j over OCn ,0 , with βj ∈ N and j=1 (βj + 1)/cj > p . If c1 = · · · = ck , the condition on β1 , . . . , βk is kj=1 βj  pc1 − k + 1. (3) If Dj = gj−1 (0) are smooth divisors with transversal intersections with gj ∈ k OX (X), and c1 , . . . , ck ∈ R∗+ . Set ϕ = j=1 cj log |gj |. Then I (ϕ) = OX (− j cj Dj ). (2.3.21) Indeed f ∈ I (ϕ)x if and only if |f |2 |g1 |−2c1 · · · |gk |−2ck ∈ L1loc . Since gj are functionally independent, this means that 2dj − 2cj > −2 where dj is the order of vanishing of f along Dj . Thus dj  cj and we know that the sections of the line bundle OX (− j cj Dj ) can be identified with holomorphic functions vanishing along Dj to order  cj . Definition 2.3.17. Let hL be a Hermitian metric with analytic singularities on L with curvature RL . We define the set X(q) = X(q, c1 (L, hL )) of points of index q to (1,0) be the open set of points x ∈ X such that hL is smooth at x, R˙ xL ∈ End(Tx X) is invertible and has exactly q negative eigenvalues. We set as usual X( q) = X(0) ∪ · · · ∪ X(q).

104

Chapter 2. Characterization of Moishezon Manifolds

Theorem 2.3.18 (Bonavero). Let X be a compact complex manifold of dimension n and let L be a holomorphic line bundle on X equipped with a Hermitian metric hL with analytic singularities. Let (E, hE ) be a holomorphic Hermitian vector bundle on X. Then for 0  q  n, we have p

dim H q (X, Lp ⊗ E ⊗ I (hL ))  pn  rk(E) · (−1)q c1 (L, hL )n + o(pn ), n! X(q) q


(2.3.22)

p

(−1)q−j dim H j (X, Lp ⊗ E ⊗ I (hL ))

j=0

pn  rk(E) · n!

(2.3.23)

 q

L n

n

(−1) c1 (L, h ) + o(p ) . X(
q)

Proof. We will suppose in the sequel that X is connected. Thus c from (2.3.19) takes a constant positive rational value on X.  −→ X such that The idea is to construct a proper modification π  : X  L ∗ ∗ L  π L, π  h ) has singularities along a codimension 1 analytic set, which (L, h ) = (  means that I (hL ) is invertible. Then we apply the holomorphic Morse inequalities  and relate the cohomology spaces on X  with those on X. from Theorem 1.7.1 on X First step: blowing up the singularities of hL . Let us consider ϕ, c and fj as in Definition 2.3.9. We first make sense of the “ideal generated by the functions fj ”. The functions fj are given just locally and we wish to have a globally defined ideal. We introduce the ideal I ⊂ OX by Ix = {f ∈ OX,x : there exists C > 0 with |f |  Ceϕ/c in a neighborhood of x}, (2.3.24) for any x ∈ X. This is a globally defined sheaf which coincides with the integral closure of the ideal generated by the fj ’s on each open set where ϕ has the form (2.3.19) (Brian¸con–Skoda theorem). We now blow X up along the ideal I .  −→ X (a composition Lemma 2.3.19. There exists a proper modification π  : X of finitely many blow-ups with smooth centers) such that the local weight ϕ  of   = π the metric hL = π ∗ hL on L ∗ L has the form ϕ  = c j cj log |gj | + ψ in appropriate local holomorphic coordinates centered in a given point x , where ψ is and define a divisor with normal smooth, cj ∈ N∗ and gj are irreducible in OX,  x crossings. Proof. By Proposition 2.1.16, there exists a blow-up σ : X  −→ X along I such that the pull-back σ −1 I · OX
is an invertible sheaf. Theorem 2.1.25 shows that

2.3. The Shiffman–Ji–Bonavero–Takayama criterion

105

 −→ X of finitely many blow-ups with smooth there exists a composition π  :X  centers which dominates σ : X −→ X:  X ~~ ~ ~ π  ~~ ~~~  /X X ρ

σ

The pull-back π −1 I · OX is also invertible. Let us denote by g the local generator of the ideal generated by fj ◦ π . Then fj ◦ π  = g · hj , where hj have no common zeros. In view of (2.3.19), the local weight of π ∗ hL has the form  
 ϕ  = 2c log |fj ◦ π |2 + ψ ◦ π j

=

c 2

log |g|2 +

c 2

log




  |hj |2 + ψ ◦ π

(2.3.25)

j

= log |g| + ψ ,   2 +ψ◦π  is smooth. where ψ := 2c log j |hj | # c We consider the decomposition g = gj j of g in irreducible factors in OX,  x.   We introduce global divisors Dj given locally by Dj = {gj = 0}. By Theorem 2.1.13, we can further blow up to make the divisor Dj defined by gj with normal crossings.
2

c 2

 −→ X, c, cj and D  j as in Lemma 2.3.19. For any Hermitian vector Let π :X   F  bundle (F, h ) on X, we set (F , hF ) := ( π ∗ F, π ∗ hF ). By Example 2.3.16 we have 
 p j . I (hL ) = OX − (2.3.26) c cj p D j

 We take advantage of the fact Second step: holomorphic Morse inequalities on X. p p L p  that I (h ) is invertible and we write L ⊗ I (hL ) as a tensor power of a fixed line bundle, in order to apply the holomorphic Morse inequalities in the smooth case. We fix r ∈ N, m ∈ N∗ such that c = r/m. Set  = r cj D &j , L "=L  m ⊗ O  (−D).  D (2.3.27) j X   defined in Example Let hOX& (−D) be the singular Hermitian metric on OX (−D) m   "  " induced by hL m and hOX& (−D) . 2.3.4. Let hL = hL ⊗ hOX& (−D) be the metric on L "  and Lemma 2.3.20. The metric hL is smooth on X, "

R(L,h

" L

)



= m R(L,h

 L

)

= mπ ∗ R(L,h

L

)


supp(D).  on X

(2.3.28)

106

Chapter 2. Characterization of Moishezon Manifolds

"  Proof. The local weight of hL is m ϕ−r  j cj log |gj | = m ψ which is smooth. Since " " L)  we have R(L,h 
supp(D)  . ∂∂ log |gj | = 0 outside D, = 2m∂∂ ψ = 2m ∂∂ ϕ  on X The proof of Lemma 2.3.20 is completed.
Lemma 2.3.21. As p → ∞, the following inequality holds: q


 L p ⊗ E  ⊗ I (hL p )) (−1)q−j dim H j (X,

j=0

pn  rk(E) n!

 (−1)q c1 (L, hL )n + o(pn ). (2.3.29) X(
q)

Proof. Observe first that by (2.3.26) and (2.3.27), we have for p ∈ N:  mp


I (hL




" p
= L  mp
⊗ I (hL mp ). L

 ) = OX (−p D),

(2.3.30)

By Theorem 1.7.1, Lemma 2.3.20 and (2.3.30) with p = mp , we obtain q
 L p ⊗ E  ⊗ I (hL p )) (−1)q−j dim H j (X, j=0

 pn " hL" )n + o(p n ) (−1)q c1 (L, n! X(
q)   pn  hL )n + o(pn ) = rk(E) (−1)q c1 (L, n! X(
q)supp(   D)  pn = rk(E) (−1)q c1 (L, hL )n + o(pn ) . n! X(
q)  rk(E)

(2.3.31)

Now we show that the inequality holds for any p. We write p = p m + m (where c = r/m as in (2.3.27), 0  m < m, p , m ∈ N); then c cj p = rcj p + c cj m .  m
⊗ E  ⊗ O  (− c cj m D  j ). We infer from (2.3.27) m
= L Set E j X p ⊗ E  ⊗ O  (− L X




j) = L " p
⊗ E m
. c cj p D

(2.3.32)

j

We can now argue as in the first part, by taking a smooth Hermitian metric on m
and the smooth metric hL" on L " as in Lemma 2.3.20. E
Third step: relation to the cohomology on X. Lemma 2.3.22 (Skoda). Let U ⊂ Cn be open, and let A ⊂ U be an analytic subset. If f ∈ L2loc (U ) is holomorphic on U
A, then f is holomorphic on U . Proof. As in Definition B.1.1, we decompose the analytic set A = Areg ∪ Asing where Areg is the set of regular points of A. By Lemma B.1.2, Areg , Asing are analytic subsets of X and dim Asing < dim Areg .

2.3. The Shiffman–Ji–Bonavero–Takayama criterion

107

Arguing by induction on the dimension of A, we see that it suffices to prove the assertion for U a neighborhood of a regular point x ∈ A. We can thus assume that x = 0, A ∩ U ⊂ {z1 = 0}. We will show that ∂f = 0 in the sense of ∗ distributions on U . Then ∂ ∂f = 0 and the regularity theorem A.3.4 implies that f is smooth and hence holomorphic on U . We have to show that  f ∂s = 0 for any s ∈ Ωn,n−1 (U ). (2.3.33) 0 U ∞

Consider χ ∈ C (R) such that χ(t) = 0 for t  1/2, χ(t) = 1 for t  1. Set χε : Cn −→  R, χε (z) = χ(|z1 |/ε). Since ∂f = 0 on U
A and supp(χε s) ⊂ U
A, we have U f ∂(χε s) = 0 and thus   χε f ∂s = − f ∂χε ∧ s. (2.3.34) U

U

By the Cauchy–Schwarz inequality      f ∂χε ∧ s   U



|z1 |
ε

|f s|2 dZ

|∂χε |2 dZ , supp(s)

n where dZ is the Euclidean  volume 2form on C . 2 Since f ∈ Lloc (U ), |z1 |
ε |f s| dZ −→ 0, for ε → 0. On the other hand, there exist constants C, C  > 0 such that 

 C |∂χε |2 dZ  2 vol supp(s) ∩ {|z1 |  ε}  C  . ε supp(s)

   Thus U f ∂χε ∧s −→ 0, for ε → 0. Since f ∈ L2loc (U ), U χε f ∂s −→ U f ∂s as ε → 0. From (2.3.34), we get (2.3.33). The proof of Lemma 2.3.22 is completed.
Lemma 2.3.23. Let π : X  −→ X be a proper modification. Assume ϕ is a quasiplurisubharmonic function on X. Then π∗ (KX
⊗ I (ϕ ◦ π)) = KX ⊗ I (ϕ) .

(2.3.35)

Proof. Let A ⊂ X be a proper analytic set such that π : X 
π −1 (A) −→ X
A is biholomorphic. If V ⊂ X is open and f is a section of KX ⊗ I (ϕ) over V , then f √ 2 is a holomorphic (n, 0)–form with ( −1)n f ∧ f e−2ϕ ∈ L1loc (V ) with n = dim X. Since ϕ is bounded from above, it follows that f ∈ L2loc (V ). By the change of variable formula,   √ √ 2 2 ( −1)n f ∧ f e−2ϕ = ( −1)n π ∗ f ∧ π ∗ f e−2ϕ◦π , V A

π −1 (V )π −1 (A)

thus f is a section of KX ⊗ I (ϕ) over V if and only if π ∗ f is a section of KX
⊗ I (ϕ ◦ π) over π −1 (V ) from Lemma 2.3.22.


108

Chapter 2. Characterization of Moishezon Manifolds

Let π : X  −→ X be the blow-up of X with smooth center Y ⊂ X. Let A by a sheaf of abelian groups on X  and q ∈ N. Recall that the qth direct image of A by π is the sheaf on X associated to the pre-sheaf U −→ H q (π −1 (U ), A ), U ⊂ X open set, and is denoted by Rq π∗ A . Theorem 2.3.24 (Leray). If Rq π∗ A = 0 for q  1, then there is a canonical isomorphism H l (X, π∗ A )  H l (X  , A ) , for any l  0. (2.3.36) Proposition 2.3.25. Assume that in the neighborhood of any point of the exceptional divisor D of π, a local weight of the metric hL satisfies ϕ ◦ π = c log |f | + ψ ,

(2.3.37)

for some c > 0, f is a local definition function of D and ψ is quasi-plurisubharmonic. Then, for any p > 1/c, q  0,





p  p  H q X  , KX
⊗π ∗ (Lp ⊗E)⊗I π ∗ hL  H q X, KX ⊗Lp ⊗E⊗I hL . (2.3.38) Proof. Recall that a locally integrable function with analytic singularities is quasiplurisubharmonic. For any holomorphic line bundle F , endowed with a singular Hermitian metric hF with analytic singularities on X, by (2.3.35), π∗ (KX
⊗ π ∗ F ⊗ I (π ∗ hF )) = KX ⊗ F ⊗ I (hF ).

(2.3.39)

In order to show (2.3.38) by applying Leray’s theorem 2.3.24, we have to verify that the higher direct image sheaves vanish: Rq π∗ (KX
⊗ π ∗ (Lp ⊗ E) ⊗ I (π ∗ hL )) = 0 , p

q  1.

(2.3.40)

The sheaf Rq π∗ (KX
⊗ π ∗ (Lp ⊗ E)) ⊗ I (π ∗ hL )) is supported on Y and its fiber over y ∈ Y is p

Fp,y = lim H q (π −1 (U ), KX
⊗ π ∗ (Lp ⊗ E)) ⊗ I (π ∗ hL )), −→ p

(2.3.41)

Uy

where U runs over the neighborhoods of y in X. Since the question is local, we can assume that L and E are trivial, when proving (2.3.40). Let U be a Stein neighborhood (cf. Definition B.3.3) of y (on which L and E are trivial) and let ϕ be a local weight of hL over U . From (2.3.37), we infer

 (2.3.42) I (p ϕ ◦ π) = OX
(−pc D) ⊗ I pψ + (pc − pc ) log |f | , hence H q (π −1 (U ), KX
⊗ I (π ∗ hL ))

 = H q π −1 (U ), KX
⊗ OX
(−pc D) ⊗ I (pψ + (pc − pc ) log |f |) . p

(2.3.43)

2.3. The Shiffman–Ji–Bonavero–Takayama criterion

109

Note that we can assume ψ to be plurisubharmonic; by hypothesis, ψ = ψ1 + ψ2 , with ψ1 plurisubharmonic and ψ2 smooth, so ψ2 has no influence on the Nadel multiplier ideal sheaves from (2.3.43), i.e., I (pψ + (pc − pc ) log |f |) = I (pψ1 + (pc − pc ) log |f |). We endow F = OX
(−pc D) with a smooth Hermitian metric hF with positive curvature on π −1 (U ). For this we argue as in Propositions 2.1.8 and 2.1.11; the role of the positive line bundle therein is played now by (U × C, e−η ), where η is a strictly plurisubharmonic function in the neighborhood of U . Then hF e−pψ−(pc−pc) log |f | has positive curvature on π −1 (U ) (since ψ and log |f | are plurisubharmonic). We remark further that π −1 (U ) is a weakly 1-complete manifold. Indeed, if ρ : U → R is a strictly plurisubharmonic exhaustion function, then ρ ◦ π : π −1 (U ) → R is a plurisubharmonic exhaustion function which is strictly plurisubharmonic outside the exceptional divisor D. Then, for λ : R → R increasing fast enough and strictly convex (i.e., λ > 0), √ (2.3.44) Θ = −1(RF + ∂∂λ(ρ ◦ π)), delivers the associated (1, 1)-form of a complete K¨ ahler metric on π −1 (U ). MoreF −pψ−(pc−pc) log |f | −λ(ρ◦π) e on π −1 (U ) dominates this over, the curvature of h e −1 metric on π (U ). As a consequence of Theorem B.4.7, we obtain that the righthand side of (2.3.43) vanishes for q  1. Thus (2.3.38) follows from (2.3.43) together with the Leray theorem and (2.3.39). The proof of Proposition 2.3.25 is completed.
We apply now Proposition 2.3.25 at each step of the blowing-up of I per∗ . formed in Lemma 2.3.19. In doing so, we replace at the first step E by E ⊗ KX The hypothesis (2.3.37) is satisfied, since in Lemma 2.3.19 the local weight ϕ has analytic singularities and the centers of the blow-ups are included in the singular locus of the metric. Thus at each step (2.3.38) holds and for all q  0 and p large enough:

p 

p L ∗  L p ⊗ E  ⊗K ⊗K * (2.3.45) H q X, Lp ⊗ E ⊗ I hL  H q (X, X ⊗ I (h )). X Applying Lemma 2.3.21 to the right-hand side term of (2.3.45), we obtain (2.3.23). Finally, (2.3.22) follows by subtracting inequalities (2.3.23) for q and q + 1. We conclude thus the proof of Theorem 2.3.18.
Corollary 2.3.26. In the same conditions as in Theorem 2.3.18,  p pn dim H 0 (X, Lp ⊗ E ⊗ I (hL ))  rk(E) · c1 (L, hL )n + o(pn ) . (2.3.46) n! X(
1) Proof. By (2.3.23) for q = 1, we get immediately (2.3.46).




Remark 2.3.27. We can reformulate Theorem 2.3.18 using the absolute continuous component of the curvature current (cf. Definition B.2.12). Namely let ω ∈ c1 (L)

110

Chapter 2. Characterization of Moishezon Manifolds √

be a current with analytic singularities. Then ω = ω0 + π−1 ∂∂ϕ, where ω0 ∈ c1 (L) is smooth, and ϕ has analytic singularities. By Lemma 2.3.5, there exists a smooth L L L −2ϕ Hermitian metric hL , we obtain 0 on L with c1 (L, h0 ) = ω0 . Setting h = h0 e a Hermitian metric with analytic singularities on L such that c1 (L, hL ) = ω. In view of Definition 2.3.17, we define + p I (pω) := I (hL ), X(q, ω) = X(q, c1 (L, hL )), X(≤ q, ω) = X(i, ω). i≤q

(2.3.47) Note that X(q, ω) ⊂ X
sing supp(ω). Outside the singular support sing supp(ω) of ω, we have ωac = ω, so (2.3.23) takes the form:  pn 0 p dim H (X, L ⊗ I (pω))  ω n + o(pn ) , p −→ ∞ . (2.3.48) n! X(
1,ω) ac We summarize now the different characterizations of Moishezon manifolds (including Theorems 2.3.7 and 2.3.8) in terms of singular Hermitian metrics. Theorem 2.3.28. Let X be a compact connected complex manifold of dimension n. Then the following are equivalent: (1) X is Moishezon. (2) X admits an integral K¨ ahler current with singular support contained in a proper analytic set. (3) (Ji–Shiffman) X admits an integral K¨ ahler current. (4) X admits an integral K¨ ahler current with analytic singularities. (5) (Bonavero) a closed integral current ω with analytic singularities  X admits n such that X(
1,ω) ωac > 0. (6) (Takayama) X admits a closed integral current ω with singular support contained in  a propern analytic set Σ such that ω is positive in a neighborhood of > 0. Σ and X(
1,ω) ωac Proof. (1) ⇒ (2) follows from Lemma 2.3.6. (2) ⇒ (3) is trivial. (3) ⇒ (4). Let ω be an integral K¨ ahler current. Let Θ be a smooth positive (1, 1)-form on X such that ω  Θ. Theorem 2.3.10 shows the existence of currents ωε , 1 > ε > 0, in the same de Rham cohomology class as ω and having analytic singularities, such that ωε  (1 − ε)Θ. (4) ⇒ (5). If ω is a K¨ahler current with analytic singularities  with Σ :=  n = XΣ ω n > 0. singsupp(ω), then X(1,ω) = ∅, X(0,ω) = X
Σ, thus X(
1,ω) ωac (5) ⇒ (1). Corollary 2.3.26 (or rather (2.3.48)) and (2.3.20) show that there exists C > 0 such that for p ∈ N∗ , dim H 0 (X, Lp )  dim H 0 (X, Lp ⊗ I (pω))  C pn , and therefore L is big and X is Moishezon.

(2.3.49)

2.3. The Shiffman–Ji–Bonavero–Takayama criterion

111

(2) ⇒ (6) is trivial. (6) ⇒ (5) follows from Theorem 2.3.10. This achieves the proof. We remark that (6) ⇒ (1) is proved independently by Problem 6.2.


2.3.3 Volume of big line bundles Let X be a compact connected complex manifold of dimension n. Let L be a holomorphic line bundle on X. We characterize now the big line bundles in terms of singular Hermitian metrics, first on projective and then on general manifolds. Theorem 2.3.29. If X is projective, then the following conditions are equivalent: (1) L is big. (2) For p ∈ N large enough, there exists a decomposition of divisors Lp = OX (A+ D) with D effective and OX (A) positive on X. (3) L has a singular Hermitian metric hL with strictly positive curvature current and with analytic singularities. Proof. (1) ⇒ (2). As X is a submanifold of C PN , we denote it by φ : X → C PN , then the base locus of the linear system |H 0 (X, φ∗ OC PN (1))| is the empty set. By Bertini’s theorem 2.2.2, there exists s ∈ H 0 (X, φ∗ OC PN (1)) such that A = Div(s) is a smooth submanifold of X. Then we have the exact sequence of sheaves 0 → Lp ⊗ OX (−A) −→ Lp −→ Lp |A → 0 which induces the exact sequence 0 −→ H 0 (X, Lp ⊗ OX (−A)) −→ H 0 (X, Lp ) −→ H 0 (A, Lp |A ).

(2.3.50)

Since dim A = n − 1, by (1.7.2) or (2.2.13), we have dim H 0 (A, Lp |A )  c1 pn−1 , and by assumption dim H 0 (X, Lp )  c2 pn for some c1 , c2 > 0 and p large enough. Therefore dim H 0 (X, Lp ⊗ OX (−A)) > 0 for p large enough, so there exists an effective divisor D with Lp ⊗ OX (−A) = OX (D). √ (2) ⇒ (3). Let hOX (A) be a smooth metric on OX (A) with −1ROX (A) > 0. We endow OX (D) with the metric hOX (D) constructed in Example 2.3.4 and by (2.3.9), we have c1 (OX (D), hOX (D) ) = [D]  0. The induced metric hL = (hOX (A) ⊗ hOX (D) )1/p has curvature √ −1RL =

√

 −1  OX (A) R + ROX (D) > 0 . p

(3) ⇒ (1). This follows from Corollary 2.3.26 as in (2.3.49).




About the characterization of big line bundles, we have a parallel result to Theorem 2.3.28.

112

Chapter 2. Characterization of Moishezon Manifolds

Theorem 2.3.30. The following are equivalent: (1) L is big. (2) L admits a singular Hermitian metric hL , smooth outside a proper analytic set and whose curvature is a strictly positive current. (3) (Ji–Shiffman) L admits a singular Hermitian metric hL , whose curvature is a strictly positive current. (4) L admits a singular Hermitian metric hL with analytic singularities, whose curvature is a strictly positive current. (5) (Bonavero) L admits a singular Hermitian metric hL with analytic singular ities such that X(
1) c1 (L, hL )n > 0. (6) (Takayama) L admits a singular Hermitian metric hL , whose curvature is smooth outside a proper analytic set Σ, positive in a neighborhood of Σ, and  c (L, hL )n > 0. X(
1) 1 Proof. The proof is parallel to the proof of Theorem 2.3.28. As an example, we prove (1) ⇒ (3). " −→ X be a If L is big, by Theorem 2.2.15, X is Moishezon. Let π : X " projective (Theorem 2.2.16). Set L " = π ∗ L. Since π ∗ : proper modification with X " L " p ) is injective, thus lim supp→∞ p−n dim H 0 (X, " L " p ) > 0, H 0 (X, Lp ) −→ H 0 (X, " is big (by Theorem 2.2.7). By (1) ⇒ (3) in Theorem 2.3.29, there exists so L " " with c1 (L, " hL" ) > 0. We consider hL = a singular Hermitian metric hL on L " " " hL" ) is also strictly positive (see the argument π∗ hL . Then c1 (L, π∗ hL ) = π∗ c1 (L, in Lemma 2.3.6).
After characterizing big line bundles, we wish to find a measure of the bigness. Definition 2.3.31. The volume of the holomorphic line bundle L is defined by vol(L) := lim sup p→∞

n! dim H 0 (X, Lp ). pn

(2.3.51)

Thus, L is big if and only if vol(L) > 0. If L is positive on X, by the Kodaira– Serre vanishing theorem 1.5.6 and by the asymptotic Riemann–Roch–Hirzebruch formula (1.7.1) (for q = n), we have  vol(L) = c1 (L)n (2.3.52) X

which gives a nice description of the volume in terms of the first Chern class. If L is semi-positive, then for any q  1, since X(q) = ∅, by (1.7.2), for p → ∞, dim H q (X, Lp ) = o(pn ). Thus (1.7.1) for q = n shows again that (2.3.52) holds. We consider next numerically effective line bundles, which are the counterpart of semi-positive line bundles in the algebraic geometry.

2.3. The Shiffman–Ji–Bonavero–Takayama criterion

113

Definition 2.3.32. A holomorphic line bundle F over a projective manifold M is  said to be numerically effective, nef for short, if F · C = C c1 (F )  0 for every curve C ⊂ M . It is easily seen that a semi-positive line bundle is nef, but the converse is not true. Proposition 2.3.33 (Kleiman). If X is projective  and L is nef, then for any qdimensional subvariety Y ⊂ X, with the integral Y taken over Yreg as in (B.2.16),  Lq · Y := c1 (L)q  0. (2.3.53) Y

Let us recall the Nakai–Moishezon ampleness criterion: Theorem 2.3.34 (Nakai–Moishezon). If X is projective, then L is positive if and only if Lq · Y > 0 for every q-dimensional subvariety Y ⊂ X. From this, we easily infer: Proposition 2.3.35. If X is a projective manifold on which a positive line bundle F and a positive (1, 1)-form Θ are given. The following properties are equivalent: (a) L is nef; (b) For any integer p  1, the line bundle Lp ⊗ F is positive; L (c) For every ε > 0, there is a smooth metric hL ε on L such that c1 (L, hε )  −εΘ. Proof. (a) ⇒ (b). If L is nef and F is positive, then clearly Lp ⊗ F satisfies the Nakai–Moishezon criterion, hence Lp ⊗ F is positive. (b) ⇒ (c). Let hF be a Hermitian metric on F such that c1 (F, hF ) > 0. Condition (c) is independent of the choice of the positive (1, 1)-form Θ; thus we p set Θ = c1 (F, hF ). If Lp ⊗ F is positive, there exists a metric hL ⊗F of positive p ∗ curvature. Then for the curvature RL of (L, hL = (hL ⊗F ⊗ hF )1/p ), √ √ p 1 √ 1√ 2π −1RL = ( −1RL ⊗F − −1RF )  − −1RF = − Θ . p p p

(2.3.54)

When p is large enough in (2.3.54), we get (c).   (c) ⇒ (a). Under hypothesis (c), we get L · C = C c1 (L, hL ε )  −ε C Θ for every curve C and ε > 0, hence L · C  0 and L is nef.
Since there need not exist any curve in an arbitrary compact complex manifold X, Proposition 2.2.35 (c) will be taken as a definition of nefness: Definition 2.3.36. L is said to be nef if for every ε > 0, there is a smooth Hermitian L metric hL ε on L such that c1 (L, hε )  −εΘ. Proposition 2.3.37. If (X, ω) is a compact K¨ ahler manifold and L is a nef line bundle. Then for any q  1, dim H q (X, Lp ) = o(pn ) ,

as p → ∞ .

(2.3.55)

114

Chapter 2. Characterization of Moishezon Manifolds

L L Proof. We set ωε = c1 (L, hL ε ) for (L, hε ) with hε as in Proposition 2.3.35 (c). We apply the weak holomorphic Morse inequalities (1.7.2) as p → ∞, we get  pn (−1)q ωεn + o(pn ) . (2.3.56) dim H q (X, Lp )  n! X(q,ωε )

The characteristic function 1X(q,ωε ) is defined by 1 on X(q, ωε ), 0 otherwise. Let us remark that 0

1 1 (−1)q n ωε 1X(q,ωε )  (εω)q ∧ (ωε + εω)n−q n! q! (n − q)!

on X .

(2.3.57)

For a closed form ϑ, we will denote by [ϑ] its cohomology class. Then [ωε ] = c1 (L). As ω is closed, from (2.3.56) and (2.3.57), we get  pn ε q q p [ω]q · (c1 (L) + ε[ω])n−q + o(pn ). (2.3.58) dim H (X, L )  q!(n − q)! X When we take ε → 0, we get (2.3.55).




Corollary 2.3.38. If (X, ω) is a compact K¨ ahler manifold  of dimension n and L is a nef line bundle on X.Then L is big if and only if X c1 (L)n > 0. If this is the case, we have vol(L) = X c1 (L)n . We continue our train of thought and wish to compute the volume of big line bundles which are not necessarily nef. For this we need to introduce additional notions. Definition 2.3.39. A Q-divisor on a compact connected complex manifold X, dim X = n, is an element D of the vector space DivQ (X) := Div(X) ⊗Z Q. Then D is a finite linear combination D = i ci Vi , where Vi are irreducible analytic hypersurfaces of X and ci ∈ Q. Obviously Div(X) ⊂ DivQ (X) and elements of Div(X) are called integral divisors. D ∈ DivQ (X) is called effective if ci  0 for all i. A ample (resp. big) Q-divisor is a Q-divisor D such that there exists m ∈ Z with mD ∈ Div(X) and OX (mD) is positive (resp. big). The volume of a divisor D ∈ Div(X) is defined by vol(D) := vol(OX (D)). The volume of a Q-divisor is defined by vol(D) := vol(mD)/mn where m ∈ Z satisfies mD ∈ Div(X). The definition is independent of the choice of m with this property. A divisor D ∈ Div(X) is said to have a Zariski decomposition if there exist a nef Q-divisor N and an effective Q-divisor E on X such that (i) D = N + E as Q-divisors. (ii) The canonical inclusion of H 0 (X, OX (mN )) in H 0 (X, OX (mD)) is surjective for every multiple m of the denominator of N . If a Zariski decomposition exists, then vol(D) = vol(N ), so the computation of the volume of L reduces to that of the nef part N . The following result, due to Fujita, states that we can recover most of the volume of D from the volume of a positive Q-divisor on a modification:

2.3. The Shiffman–Ji–Bonavero–Takayama criterion

115

Theorem 2.3.40 (Fujita). If X is a projective manifold and D ∈ Div(X) is a big divisor on X, then, for every ε > 0, there exists a proper modification µε : Xε −→ X, an ample Q-divisor Fε and an effective Q-divisor Eε on Xε such that: (i) µ∗ε D = Fε + Eε as Q-divisors, (ii) | vol(Fε ) − vol(D)| < ε. A decomposition as in (i) is called an approximate Zariski decomposition. Approximate Zariski decompositions enable us to compute the volume of big line bundles on K¨ ahler manifolds. We can actually give a formula for a larger class of bundles. Recall from Theorem 2.3.30 that the first Chern class of a big line bundle contains a strictly positive current. Definition 2.3.41. A holomorphic line bundle F on X is called pseudo-effective if there exists a positive (1, 1)-current ω such that [ω] = c1 (F ). A divisor D is called pseudo-effective if OX (D) is pseudo-effective. From Lemma 2.3.5, we deduce that F is pseudo-effective exactly when F has a singular Hermitian metric whose curvature is a positive current. If D is an effective divisor and we endow OX (D) with the singular metric from Example 2.3.4, we see by (2.3.9) that OX (D) has a singular metric whose curvature current is positive. Thus every effective divisor is pseudo-effective. To approximate vol(D), the following two results are useful. Theorem 2.3.42 (Demailly). Under the same hypotheses as in Theorem 2.3.10, there exist closed currents ωm , m  1, such that √ (i) ωm = 2 −1∂∂ϕm + ω0 , where ϕm are smooth real functions, and ω0 is a C ∞ representative of ω, (ii) ωm converge weakly to ω, for m → ∞, and pointwise on the set where the Lelong number of ω vanishes. (iii) ωm  θ − Cλm Θ, where C > 0 depends only on (X, Θ), λm are continuous and λm → ν(ω, x) for all x ∈ X as m → ∞, where ν(ω, x) is the Lelong number of ω at x, cf. (B.2.19). Lemma 2.3.43. If (X, Θ) is a compact K¨ ahler manifold and if ω is a closed positive k (1, 1)-current on X, then X Θn−k ∧ ωac are finite and can be bounded in terms of Θ and the cohomology class of ω only. Proof. We first observe that the Lelong numbers ν(ω, x) in (B.2.19) are bounded. In fact, taking r > 0 small enough, we have  Θn−1 ∧ ω, (2.3.59) ν(ω, x) < ν(ω, x, r) < C X

and the last integral depends only on the cohomology class of ω.

116

Chapter 2. Characterization of Moishezon Manifolds

Using Theorem 2.3.42, we find smooth forms ωε approximating ω and such that ωε  −CΘ, where C > 0 depends only on [ω] and Θ. Hence   (ωε + CΘ)k ∧ Θn−k = [ω + CΘ]k · [Θ]n−k , (2.3.60) X

X

does not depend on ε. Since ωε + CΘ converges to ωac + CΘ almost everywhere, the result follows from Fatou’s lemma.
We need the following application of the holomorphic Morse inequalities. Lemma 2.3.44. If (X, Θ) is a compact K¨ ahler manifold and L is a pseudo-effective line bundle on X and ω ∈ c1 (L) is a closed positive current, one has:  n vol(L)  ωac . (2.3.61) X

Proof. We choose a sequence ωk of currents with analytic singularities as in Remark 2.3.11, i.e., in such a way that ωk,ac → ωac . We denote by λ1  · · ·  λn (k) (k) (resp. λ1  · · ·  λn ) the eigenvalues of ωac (resp. ωk,ac ) with respect to Θ. (k) We have by assumption and Remark B.2.13 that λ1  0, λ1  −εk , εk → 0 and (k) λj (x) → λj (x) almost everywhere.  n We may assume that X ωac > 0, which means that the set A := {λ1 > 0} has positive measure. For each small δ > 0, Egoroff’s lemma delivers some Bδ ⊂ A (k) δ. such that λ1 → λ1 uniformly on Bδ and also A − Bδ has measure  less than n Thus Bδ ⊂ X(0, ωk ) for k big enough, and consequently lim sup X(0,ωk ) ωk,ac    n n Bδ lim inf ωk,ac = Bδ ωac , using Fatou’s lemma. Letting now δ → 0, we get    n n n lim sup ωk,ac  ωac = ωac . k→∞



X(0,ωk )

A



X

n  vol(L) − X(1,ωk ) ωk,ac for every k, we Since by (2.3.48) we have  n can conclude our argument if we can show that − X(1,ωk ) ωk,ac → 0. But n X(0,ωk ) ωk,ac

n 0  −ωk,ac 1X(1,ωk )  nεk Θ ∧ (ωk,ac + εk Θ)n−1 ,



hence



0−

n ωk,ac X(1,ωk )

 nεk

Θ ∧ (ωk,ac + εk Θ)n−1 .

(2.3.62)

X

From Lemma 2.3.43, it follows that the integrals in (2.3.62) are bounded, which ends the proof of Lemma 2.3.44.
Theorem 2.3.45 (Boucksom). If (X, Θ) is a compact K¨ ahler manifold and L is pseudo-effective, then   n ωac : ω ∈ c1 (L) closed positive current . (2.3.63) vol(L) = sup X

2.3. The Shiffman–Ji–Bonavero–Takayama criterion

117

Proof. By Lemma 2.3.44, we need just to construct a family of positive currents  n → vol(L), for ε → 0. If vol(L) = 0, (2.3.63) is clear ωε , ε > 0, such that X ωε,ac from (2.3.61). So we can assume vol(L) > 0. Then L is big, hence X is Moishezon. But as X is K¨ahler, from Theorem 2.2.26, X is projective. Observe that H 0 (X, Lm ) = 0 for some m so Lm  OX (D) for the effective divisor D = {s = 0}, with a non-trivial s ∈ H 0 (X, Lm ). Since vol(L) = vol(D)/mn and c1 (L) = c1 (OX (D))/m, it is enough to prove (2.3.63) for D. We apply thus Fujita’s theorem 2.3.40 for D; for any ε > 0, there exists a proper modification µε : Xε −→ X and a decomposition µ∗ε D = Fε + Eε , where Fε is an ample Q-divisor, Eε is an effective Q-divisor, and | vol(D) − vol(Fε )| < ε. Let ϑε ∈ c1 (OXε (Fε )) be a K¨ ahler form. Consider the push-forward ωε = µε,∗ (ϑε + [Eε ]) (where [Eε ] is the current of integration on Eε ). Then ωε ∈ c1 (OX (D)) is a positive current. Moreover,     n ωε,ac = (ϑε + [Eε ])nac = ϑnε = c1 (OXε (Fε ))n = vol(Fε ). (2.3.64) X

Xε

Hence | vol(D) −

Xε

 X

Xε

n ωε,ac | < ε, which achieves the proof of Theorem 2.3.45.




Therefore, one has a Grauert–Riemenschneider-type criterion: Corollary 2.3.46 (Boucksom). If (X, Θ) is K¨ ahler, L is pseudo-effective and  and n > 0, then vol(L) > 0 and its Chern class c1 (L) contains a current ω with X ωac L is big. Especially, X is projective. Actually, we can make Theorem 2.3.45 precise as follows.

 Theorem 2.3.47. Under the assumption in Theorem 2.3.45, assume that X Θn = 1. Then there exists a singular Hermitian metric hL on L with positive curvature n current ω := c1 (L, hL ) and ωac = vol(L)Θn almost everywhere. In particular, ω realizes the maximum in (2.3.63). The proof is based on the well-known 2.3.48 (Calabi–Yau). Let (X, Θ) be a compact K¨ ahler manifold with Theorem n 1,1 Θ = 1. Then for any K¨ a hler cohomology class θ ∈ H (X, R), there exists a X  unique K¨ ahler form ω in θ such that ω n = ( X θn )Θn on X.

2.3.4 Some examples of Moishezon manifolds We already observed that connected projective manifolds are Moishezon. Are there other Moishezon manifolds than projective ones? If X is a compact complex manifold of dimension 1, then X is a projective manifold (Problem 5.6). From Theorem 2.2.16, we infer easily that a smooth connected complex surface is Moishezon if and only if it is projective (Problem 2.5). Thus we should look for an example of non-projective Moishezon manifold starting with dimension 3.

118

Chapter 2. Characterization of Moishezon Manifolds

In this section, we present an example of Koll´ar, used by Koll´ ar and Bonavero, to show that there exist Moishezon manifolds without line bundles satisfying the conditions from the Siu–Demailly criterion. Therefore for this class of manifolds only the Shiffman–Ji–Bonavero–Takayama criterion applies. Let Q = {[z] ∈ C P3 : z0 z1 = z2 z3 } be a smooth quadric. The map ϕ : C P1 ×CP1 −→ Q,

ϕ([u0 , u1 ], [v0 , v1 ]) = [u0 v0 , u1 v1 , u0 v1 , u1 v0 ]

(2.3.65)

is an isomorphism and consequently H2 (Q, Z)  Z2 is generated by L1 = ϕ([0, 1] × C P1 ) = {z0 = z2 = 0}, L2 = ϕ(C P1 ×[0, 1]) = {z0 = z3 = 0}.

(2.3.66)

We use the notation in (2.3.53) now. The intersection numbers of these curves are L1 · L1 = L2 · L2 = 0,

L1 · L2 = 1.

(2.3.67)

Any divisor D ⊂ Q is characterized by the pair (a, b) = (D · L1 , D · L2 ) ∈ Z2 called the type of D. For example the canonical divisor (line bundle) KQ is of type (−2, −2). We have KC P3 = OC P3 (−4),

OC P3 (Q)|Q = NQ/C P3 ,

KQ = KC P3 ⊗ OC P3 (Q)|Q = OC P3 (−2)|Q .

(2.3.68)

Let q1 , q2 be the projections from C P1 ×C P1 to the first and second factor. For any l, m ∈ N∗ , consider a section of the line bundle q1∗ OC P1 (l) × q2∗ OC P1 (m) on C P1 ×C P1  Q. Its zero set is Cl,m . From Theorem 1.4.6, as KCl,m = OQ (Cl,m ) ⊗ KQ |Cl,m , the genus gl,m = dim H 1 (Cl,m , OCl,m ) of Cl,m is given by 2(gl,m − 1) = Cl,m · (Cl,m + KQ ), Cl,m · Cl,m = 2lm, Cl,m · KQ = −2(l + m).

(2.3.69)

Thus by Bertini’s theorem 2.2.2, there exists a smooth curve Cl,m ⊂ Q of type (l, m), genus (l − 1)(m − 1) and degree l + m, where the degree of Cl,m is  c1 (OC P3 (1)). Cl,m

 −→ C P3 , with exceptional Let us blow up C P3 along Cl,m and obtain π1 : X divisor El,m  P(NCl,m /C P3 ), the projectivization of the normal bundle of Cl,m in C P3 . By Proposition 2.1.11 (b),  = Pic(C P3 ) ⊕ Z · O  (El,m ) = Z · OC P3 (1) ⊕ Z · O  (El,m ). Pic(X) X X  L 1 , L  2 the strict transforms of Q, L1 , L2 . Denote by Q,

(2.3.70)

2.3. The Shiffman–Ji–Bonavero–Takayama criterion

119

  Proposition 2.3.49. We have NQ/  X  · L1 = 2 − l, NQ/  X  · L2 = 2 − m. Proof. By Proposition 2.1.11 (a), KX = π1∗ KC P3 ⊗ OX (El,m ). From the exact  −→ T X|   −→ N   −→ 0, we deduce N   = K  ⊗ K ∗ |  . sequence 0 −→ T Q  Q Q Q/X Q/X Q X Hence for i = 1, 2, ∗     NQ/  X  · Li = KQ  · Li − π1 KC P3 · Li − OX  (El,m ) · Li

= KQ · Li − KC P3 · Li − Cl,m · Li . Since KQ = OC P3 (−2)|Q and KC P3 = OC P3 (−4), we conclude our proof.

(2.3.71)


 We shall take in the sequel l = 3. Therefore NQ/  X  · L1 = −1 and the restric tion of NQ/  X  to L1 is isomorphic to OC P1 (−1). We invoke now the Fujiki–Nakano criterion for exceptional divisors. Theorem 2.3.50 (Fujiki–Nakano). Let X be a compact complex manifold and D ⊂ X a smooth divisor isomorphic to the projectivization P(F ) of a holomorphic vector bundle F over a compact complex manifold Y . Let σ : P(F ) −→ Y be the projection. Assume that ND/X  OP(F ) (−1). Then there exists a complex manifold X  containing Y as a submanifold and a map π : X −→ X  such that π is the blow-up of X  along Y and π|D = σ.  2 and L 1, L  2 are   L 1 × L From (2.3.65) and (2.3.66), we know that Q 1  biholomorphic to C P . As for x ∈ L2 , NQ/  X  |L  1 ×{x}  OL  1 (−1), applying Theorem   2.3.50 to σ : Q → L2 , there exists a complex manifold Xm and a map π2 :  −→ Xm such that π2 is the blow-up of rational curve Cm with NC /X  X m m  OC P1 (2 − m) ⊗ C2 and such that the exceptional divisor of π2 is exactly Q. Since Xm is bimeromorphically equivalent to C P3 , Xm is Moishezon. Moreover, the Picard group of Xm is isomorphic to Z, by Prop. 2.1.11 and (2.3.70). Theorem 2.3.51 (Koll´ar–Bonavero). (a) If m > 3, there exists no nef and big holomorphic line bundle L on Xm . (b) If m > no holomorphic Hermitian line bundle (L, hL ) on  5, there exists L 3 Xm such that Xm (
1) c1 (L, h ) > 0. Corollary 2.3.52. Xm is a non-projective Moishezon manifold for m > 3. Proof of Theorem 2.3.51. (a) Let L be a non-trivial holomorphic line bundle on Xm . Then by (2.3.70), there exist k, r ∈ N such that π2∗ L = π1∗ OC P3 (r) ⊗ OX (−kE3,m ).  1 is a fiber of π2 , we have π ∗ L · L  1 = 0, but E3,m · L1 = 3, OC P3 (1) · L1 = 1, Since L 2 and hence r = 3k and then π2∗ L = π1∗ OC P3 (3k) ⊗ OX (−kE3,m ).

(2.3.72)

120

Chapter 2. Characterization of Moishezon Manifolds

 2 = k(3 − m) and π2∗ L · F = k (cf. For any fiber F of π1 , we have π2∗ L · L ∗ Prop. 2.1.11 (a)). Thus, for m > 3, π2 L cannot be nef since its intersection with  2 is negative. L (b) Denote by L the generator of the Picard group of Xm which satisfies π2∗ L = π1∗ OC P3 (3) ⊗ OX (−E3,m ). Let us observe that H 0 (Xm , L q ) = 0 for any q < 0.

(2.3.73)

Indeed, by Proposition 2.1.4, H 0 (Xm , L q ) can be identified with the space of sections of OC P3 (3q) having a pole of order  |q| along C3,m . Any such sections extends to a holomorphic section of OC P3 (3q). But for q < 0, OC P3 (3q) has only the trivial global section 0. From (2.3.73), the Kodaira–Iitaka dimension of L ∗ is −∞. Since Xm is Moishezon with Picard group Z, we deduce that L is big. Next, we prove that KXm = L −2 . (2.3.74)  by construction. If F is a fiber of π1 , π2∗ KXm · F = We have KX = π2∗ KXm ⊗OX (Q) ∗  −2, and π2 L · F = 1, hence (2.3.74). By Serre duality and (2.3.74), we obtain H 3 (Xm , L p )  H 0 (Xm , L −p ⊗ KXm )∗ = 0 ,

for p > −2 .

(2.3.75)

Assume now that L has a Hermitian metric hL as in (b). (1.7.2) implies then that hp := dim H 0 (Xm , L p ) − dim H 1 (Xm , L p )  Cp3 + o(p3 )

(2.3.76)



√−1 L 3 for C = 16 Xm (
1) 2π R > 0. On the other hand, we know from the Riemann–Roch–Hirzebruch theorem 1.4.6 and (2.3.75) that hp + dim H 2 (Xm , L p ) = c1 (L )3 ·

p3 + o(p3 ) . 6

(2.3.77)

We will show that c1 (L )3  0 for m > 5, which is a contradiction to the existence of the Hermitian metric hL as in (b) from (2.3.76) and (2.3.77). We compute now:

3 c1 (L )3 =c1 π1∗ OC P3 (3) ⊗ OX (−E3,m ) 



3 2 =c1 OC P3 (3) − 3 c1 π1∗ OC P3 (3)  +3

 X

E3,m

(2.3.78)



 2 3 c1 π1∗ OC P3 (3) ∧ c1 OX (E3,m ) − E3,m .

We calculate next the terms of the right-hand side of (2.3.78). It is well known



3  2 that c1 OC P3 (3) = 27 and E3,m c1 π1∗ OC P3 (3) = 0.

2.4. Algebraic Morse inequalities

121

 



Remark that c1 OX (E3,m ) |E3,m = −h, where h = c1 OP(NC /C P3 ) (1) , by 3,m Prop. 2.1.11. Hence  

∗  2

 c1 π1 OC P3 (3) ∧ c1 (OX (E3,m ) = − π1∗ c1 OC P3 (3) ∧ h  X

E3,m

 =− C3,m

 c1 OC P3 (3) = −3(3 + m),

(2.3.79)

3 since the degree of C3,m in C P3 is 3 + m. To compute E3,m , we use the formula

h2 + π1∗ c1 (NC3,m /C P3 )h + π1∗ c2 (NC3,m /C P3 ) = 0,

(2.3.80)

which in our case becomes h2 + π1∗ c1 (NC3,m /C P3 )h = 0, as dim C3,m = 1, thus c2 (NC3,m /C P3 ) = 0. Thus    3 = h2 = π1∗ c1 (NC∗ 3,m /C P3 )h = c1 (NC∗ 3,m /C P3 ). (2.3.81) E3,m E3,m

E3,m

C3,m

For the latter integral, we use the exact sequence 0 → T (1,0) C3,m → T (1,0) C P3 |C3,m → NC3,m /C P3 → 0, which gives   c1 (NC∗ 3,m /C P3 ) = C3,m

C3,m

c1 (OC P3 (−4)) − 2g3,m + 2 = −6 − 8m .

(2.3.82)

Finally we get c1 (L )3 = 27 − 27 − 9m + 6 + 8m = 6 − m  0 for m > 5 . This concludes the proof of Theorem 2.3.51.

(2.3.83)


2.4 Algebraic Morse inequalities The curvature integrals appearing in the holomorphic Morse inequalities (1.7.1) and (1.7.2) are neither topological nor algebraic invariants. That is why it is interesting to have an algebraic reformulation. Theorem 2.4.1 (Demailly). Let L = F ⊗ G∗ where F, G are holomorphic nef line bundles over a compact connected K¨ ahler manifold X of dimension n. Then for any q = 0, 1, . . . , n, there is a strong holomorphic Morse inequality for p → ∞:   q q
pn
q−j j p q−j n (−1) dim H (X, L )  (−1) c1 (F )n−j ∧ c1 (G)j + o(pn ) . j n! X j=0 j=0 (2.4.1)

122

Chapter 2. Characterization of Moishezon Manifolds

Proof. Let ω be a K¨ ahler form on X. For ε > 0 consider Hermitian metrics hF ε , G G ) + εω and Θ (G) := c (G, h ) + εω hε on F and G such that Θε (F ) := c1 (F, hF ε 1 ε ε are positive forms on X. We denote by λε1  · · ·  λεn > 0 the eigenvalues of F G −1 Θε (G) with respect to Θε (F ). Then hL is a Hermitian metric on ε = hε (hε ) L F G L and c1 (L, hε ) = c1 (F, hε ) − c1 (G, hε ) = Θε (F ) − Θε (G). The eigenvalues of L ε ε L n c# 1 (L, hε ) with respect to Θε (F ) are 1 − λ1  · · ·  1 − λn . Hence c1 (L, hε ) = n ε n L ε j=1 (1−λj )Θε (F ) and X( q, c1 (L, hε )) = {x ∈ X : λq+1 (x) < 1}. By applying (1.7.1), we obtain for p → ∞:  q n
 pn q−j j p (−1) dim H (X, L )  (−1)q (1 − λεj )Θε (F )n + o(pn ) . ε n! {λ <1} q+1 j=0 j=1 (2.4.2)

 One calculates that nj Θε (F )n−j ∧ Θε (G)j = σnj (λε )Θε (F )n where σnj (λε ) is the jth elementary function in λε1 , . . . , λεn . As ω is closed, we have as in (2.3.58)   n−j j c1 (F ) ∧ c1 (G) = lim (c1 (F ) + ε[ω])n−j ∧ (c1 (G) + ε[ω])j ε→0 X X   j! (n − j)! lim = lim Θε (F )n−j ∧ Θε (G)j = σnj (λε )Θε (F )n . (2.4.3) ε→0 X ε→0 X n! In order to achieve the proof, we need to show that q n
 (−1)q−j σnj (λε ) − 1{λεq+1 <1} (−1)q (1 − λεj )  0 , j=0

(2.4.4)

j=1




which follows by induction on n.

Corollary 2.4.2 (Trapani, Siu). In the same hypothesis as in Theorem 2.4.1, we have for p → ∞:  

pn 0 p 1 p dim H (X, L ) − dim H (X, L )  c1 (F )n − nc1 (F )n−1 ∧ c1 (G) − o(pn ) . n! X (2.4.5) Corollary 2.4.3. Assume that F and G are nef line bundles over a compact connected K¨ ahler manifold X of dimension n, and F is moreover big. Then some tensor power of F m ⊗ G∗ has a section if −1    c1 (F )n−1 ∧ c1 (G) c1 (F )n . (2.4.6) m>n X

X

Finally let us mention the following algebraic version of the singular holomorphic Morse inequalities, which is obtained with the same proof as of Theorem 2.4.1, by using (2.3.22) instead of (1.7.1). Let D = ci Di be a rational effective irreducible divisors. If gi are the local gendivisor, where ci ∈ Q+ and Di are erators of Di we set I (D) := I ( ci log |gi |). If Di intersect transversely, then I (D) = OX (−D ), where D := ci Di .

2.4. Algebraic Morse inequalities

123

Theorem 2.4.4 (Bonavero). Let X be a compact connected K¨ ahler manifold and F, G two holomorphic line bundles over X. Assume that G is nef and there exists m ∈ N∗ , a nef line bundle H and an effective divisor D such that F m = H ⊗ OX (D). Then for q = 0, 1, . . . , n = dim X, we have as p → ∞, q
j=0

 p (−1)q−j dim H j X, (F ⊗ G∗ )p ⊗ I ( m D)    q pn
q−j n −n+j (−1) c1 (H)n−j ∧ c1 (G)j + o(pn ) . m  j n! j=0 X

(2.4.7)

Problems Problem 2.1 ([2]). Let X be a complex manifold and f ∈ MX (X). (1) Show that there exists an open cover {Ui } of X and ri ∈ OX (Ui ), si ∈ SX (Ui ) such that the germs ri,x , si,x ∈ OX,x are coprime for any x ∈ Ui and f |Ui = rsii . (Hint: OX,x is a unique factorization domain and if rx , sx ∈ OX,x are coprime, then ry , sy ∈ OX,y are coprime for y in a neighborhood of x, cf. [120, p.10]) ∗ (2) Show that there exists gij ∈ OX (Ui ∩ Uj ) with ri = gij rj and si = gij sj on Ui ∩ Uj . (Hint: ri sj = rj si on Ui ∩ Uj .) (3) Show that gij gjk gki = 1 on Ui ∩ Uj ∩ Uk = ∅. Deduce that there exists a holomorphic line bundle L on X and S0 , S1 ∈ H 0 (X, L) with S0 ≡ 0 and f = S1 /S0 . In fact, L = OX (P ol(f )) where Pol (f ) is the divisor on X defined by
ordV (f ) V. Pol (f ) = − V, ordV (f )<0

Problem 2.2 ([2]). Let X be a complex manifold and f1 . . . , fk ∈ MX (X). Show that there exists a holomorphic line bundle L on X and S0 , S1 , . . . , Sk ∈ H 0 (X, L) with S0 ≡ 0 and fi = Si /S0 for i = 1, . . . , k. (Hint: Apply Problem 2.1 and (i) (i) (i) find line bundles Li and holomorphic sections S0 , S1 such that S0 ≡ 0 and (i) (i) (1) (k) (1) fi = S1 /S0 . Take L = L1 ⊗ · · · ⊗ Lk and S0 = S0 ⊗ · · · ⊗ S0 , Si = S0 ⊗ (i) (k) · · · ⊗ S1 ⊗ · · · ⊗ S0 for i = 1, . . . , k.) Problem 2.3 (Schwarz inequality). For z ∈ Cn , set |z|0 = sup{|zj | : 1  j  n}. Denote P (0, r) = {z ∈ Cn : |z|0 < r} for r > 0. Let f ∈ C 0 (P (0, r)) ∩ OX (P (0, r)) which vanishes up to order m ∈ N at 0. (a) Let z ∈ P (0, r), z = 0 fixed. Show that g : {t ∈ C : |t|  r} −→ C, ) is continuous and holomorphic in {t ∈ C : |t| < r}. g(t) = t−m f (tz/|z|0 ∞ (Hint: Write f (z) = k=m Pk (z) with homogeneous polynomials Pk of degree k.) (b) Show that |g(t)|  f ∞ r−m for |t| = r.

124

Chapter 2. Characterization of Moishezon Manifolds

(c) Deduce the Schwarz inequality: |f (z)|  f ∞

|z|m 0 , rm

z ∈ P (0, r) .

Problem 2.4. Let X be a projective manifold, dim X = n, let L be a nef line bundle and E be a holomorphic vector bundle over X. Show that for q  0, dim H q (X, Lp ⊗ E) = O(pn−q ) ,

p → ∞.

(Hint: Induction on n. Prove the result for n = 1 and for q = 0 and n arbitrary. For the induction step consider D a smooth positive divisor. Using Proposition 2.3.35 (b) and the Kodaira–Serre vanishing theorem 1.5.6, show that for k  1, H q (X, OX (kD) ⊗ Lp ⊗ E) = 0, for all p  0, q  1. Fix a large k and conclude by using the exact cohomology sequence of the sheaf exact sequence 0 −→ Lp −→ Lp ⊗ OX (kD) −→ (Lp ⊗ OX (kD))|D
−→ 0 and the induction hypothesis (note that dim D = n − 1).) Here D is a smooth divisor of X which is the zero-set of a section of OX (kD). Problem 2.5.  → X be the blow-up of a complex manifold X, dim X = n, with (a) Let π : X  is projective if and only if X is projective. center x ∈ X. Show that X (b) Prove the following theorem of Chow and Kodaira: A complex connected surface is Moishezon if and only if X is projective. (Hint: For (a), show that X
{x} is quasiprojective and apply [139, Cor. 3, p. 328]. For (b), use the fact that the center of a blow-up of a surface can be only zero dimensional; apply Theorem 2.2.16.)

2.5 Bibliographic notes The material from Section 2.1 is classical; good sources are [120, 79]. For Proposition 2.1.11 (c), one may consult [1, p. 269], [124, Ch. II. Prob. 8.5 (a)]. Bertini’s Theorem 2.2.2 can be read at [120, p. 137]. For more details on the blow up with center a submanifold see [79, VII, §12], [107, p. 162]. Hironaka’s famous resolution of singularities Theorem 2.1.13 stems from [128, Main Th. I]. An elementary and constructive proof is given by Bierstone and Milman [22]. For the definition and properties of the blow up of a coherent ideal (Prop. 2.1.14) and for the notion of ample line bundle (invertible sheaf) on a complex space, we refer to [124, Ch. II, §7]. The proof of Theorem 2.1.18 is in [107, p. 186]. A constructive proof of the Hironaka–Rossi theorem is given in [114]. The notion of meromorphic map we used here is due to Remmert [204]. For properties of flat morphisms, see [107, Ch. 3]. The flattening theorem, the elimination of indeterminacies of meromorphic maps and the Chow lemma can be found in Hironaka [128, 129].

2.5. Bibliographic notes

125

Section 2.2. Siegel’s lemma 2.2.6 stems from Siegel [223]. We have followed here the account of Andreotti [2], where the method of Siegel is extended to pseudoconcave complex spaces. Theorem 2.2.11 of Siegel-Remmert-Thimm saying that MX (X) is an algebraic field was proved in [239, 223, 203]. In the case of complex tori, the result was communicated in 1860 by Riemann to Hermite and it was stated by Weierstrass in 1869. A proof of Hurwitz Theorem is contained in [107, §4.7], [120, p. 168]. For Theorem 2.2.13, we refer to [216, I.§5.4]. Moishezon’s theorems 2.2.16, 2.2.26 appeared in [177]. We have followed [195, 196] in the proof of Theorem 2.2.26. Another proof follows from the techniques of [81]. The Seshadri criterion can be found in [123]; the proof given there holds for Moishezon manifolds without change. For the hard Lefschetz theorem we refer to [120, p. 122]. Formula (2.2.35) can be found in [120, p. 141]. The Poincar´e duality and the spectral sequences are treated in [120]. The Grauert–Riemenschneider [118] conjecture was proved by Siu [224, 225] and Demailly [72]. Another notable application of the strong Morse inequalities is to the proof of the effective Matsusaka theorem by Siu [226, 78]. Section 2.3. The characterization of Moishezon manifolds Theorem 2.3.28 given by Ji–Shiffman [134], Bonavero [41] and Takayama [234], has many applications. An important one is the projectivity criterion for hyperk¨ ahler manifolds given by Huybrechts [133]. For the singular holomorphic Morse inequalities of Bonavero, we followed closely [41]. For the Brian¸con–Skoda theorem, see [79, VIII.10.4]. For the Leray and de Rham isomorphism we refer to [79, IV.6], [120]. The systematic use of singular Hermitian metrics was initiated by Nadel [180] and Demailly [76] (building on previous work of Bombieri [39] and Skoda [229]). For the Nadel coherence theorem 2.3.15 see [180], [78, Prop. 5.7]. Demailly’s approximation theorems 2.3.10, 2.3.42 stems from [75, Th. 1.1]. The refinement from Remark 2.3.11 has been shown by Boucksom in [50, Th. 2.4]. Proposition 2.3.33 is proved in [139]. The Nakai-Moishezon criterion (Theorem 2.3.34) can be found in Kleiman [139]. The Definition 2.3.36 of nefness on arbitrary compact manifolds is introduced by Demailly-Peternell-Schneider [82]. On a surface, it is known that the Zariski decomposition exists, but this is no longer true in general by a counter-example of Cutkovsky. The Fujita theorem 2.3.40 stems from [109], cf. also [80]. Formula (2.3.63) for the volume of a pseudoeffective line bundle was proved by Boucksom [50]. The Calabi-Yau theorem 2.3.48 can be found in [257]. Using techniques from the proof of the Nakai-Moishezon criterion of Demailly-Paun [81], Boucksom extends the Grauert-Riemenschneider criterion from Corollary 2.3.46 to non-necessarily rational pseudo-effective classes. For a general reference about the volume of a line bundle, we refer to Lazarsfeld [149, §2.2.C, §11.4]. It seems that the first example of non-projective Moishezon spaces appeared as folklore in Russia during the 50’s and was named in Princeton the ”algebraic

126

Chapter 2. Characterization of Moishezon Manifolds

Sputnik”. Hironaka was the first to write down an example, and we refer to the books of Hartshorne [124, Ex. 3.4.2] and Shafarevich [216, VIII. §3.3]. Grauert constructed an example of a two-dimensional non-projective normal Moishezon space with a singular point in [116, §4]. Koll´ar’s example is given in [146]. The material of Section 2.3.4 is taken from [41]. For (2.3.80) see [120, p. 606]. Section 2.4. The algebraic Morse inequalities were obtained by Demailly [78]. The case q = 1 (Corollary 2.4.2) was proved before by Siu [226] and Trapani [245]. Theorem 2.4.4 appeared in [40]. Angelini gave an algebraic version in [8]. Using a mean value estimate of eigensections of the Kodaira Laplacian, Berndtsson [20] shows that the assertion of Problem 2.4 holds for a compact complex manifold X and a semi-positive line bundle L (and E trivial). Generalizing this method, Berman [16] reproved the holomorphic Morse inequalities (1.7.1).

Chapter 3

Holomorphic Morse Inequalities on Non-compact Manifolds We start by the L2 Hodge theory on non-compact Hermitian manifolds in Section 3.1. In Section 3.2, we prove holomorphic Morse inequalities for the L2 -cohomology in a quite general context, namely, when the fundamental estimate (3.2.2) holds. This gives a fairly general method which may be applied in many situations. The main idea, going back to Witten, is to show that the spectral spaces of the Laplacian, corresponding to small eigenvalues, inject in the spectral spaces of the Laplacian with Dirichlet boundary conditions on a smooth relatively domain, The asymptotic of the latter operator is calculated in Theorem 3.2.9. For a compact manifold we recover of course Theorem 1.7.1. In Section 3.3, we specialize the abstract Morse inequalities to a geometric situation and we assume that we have a line bundle which is uniformly positive on a complete Hermitian manifold. In Section 3.4, we establish Siu–Demailly’s type criterion for complex compact spaces with isolated singularities. Section 3.5 treats the q-convex and weakly 1-complete manifolds, while in Section 3.6 we treat the case of covering manifolds. We will revisit some of these topics in Chapter 6 and treat them from the point of view of Bergman kernels.

3.1 L2-cohomology and Hodge theory In this section, we establish the L2 Hodge theory on non-compact Hermitian manifolds and introduce the fundamental estimate (Poincar´e inequality at infinity), which entails the strong Hodge decomposition. Let (X, J, Θ) be a complex Hermitian manifold and let (E, hE ) be a holomorphic Hermitian vector bundle on X. Let dvX be the Riemannian volume form of X. The boundary of X may be non-empty.

128

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

We start with a review of Hilbertian extension of differential operators. Let E 2 R : Ω0,• 0 (X, E) → L0,• (X, E) be a differential operator (e.g., ∂ ). Then R is preclosed, so there exists closed extensions of R. In general, at least two possibilities present themselves to extend R. The first choice is to consider the closure R whose graph is the closure of the graph of R. This is called the minimal extension of R denoted also Rmin . We denote by R∗ the formal adjoint of R. This is a differential operator 2 satisfying Rs1 , s2  = s1 , R∗ s2  for all s1 , s2 ∈ Ω0,• 0 (X, E). For s1 ∈ L0,• (X, E), we can still calculate Rs1 in the sense of currents: Rs1 is the current defined by Rs1 , s2  = s1 , R∗ s2 ,

for s2 ∈ Ω00,• (X, E).

(3.1.1)

Lemma 3.1.1. The operator Rmax defined by Dom(Rmax ) = {s ∈ L20,• (X, E) : Rs ∈ L20,• (X, E)}, for s ∈ Dom(Rmax ),

Rmax s = Rs,

(3.1.2)

is a densely defined, closed extension, called the maximal extension of R. We leave the proof as an exercise. In the sequel we will work mainly with the maximal extension of R, so we will simplify the notation and write R = Rmax . ∗ ∗ . Note that RH ⊂ (R∗ )max . The Hilbert space adjoint of Rmax is denoted by RH ∗ ∗ The two operators are in general not equal, but Rmax = (RH )H . E E,∗

E,∗

E

+∂ ∂ is a densely defined, posThe Kodaira Laplacian
E = ∂ ∂ itive operator, so we can consider its Friedrichs extension (cf. Definition C.1.7). However, we will describe now a self-adjoint extension of
E which is very useful in conjunction with the study of L2 cohomology. Consider the complex of closed, densely defined operators T =∂

E

S=∂

E

L20,q−1 (X, E) −→ L20,q (X, E) −→ L20,q+1 (X, E) ,

(3.1.3)

E

where T and S are the maximal extensions of ∂ . Let T ∗ , S ∗ be their Hilbert E,∗ E,∗ space adjoints, thus T ∗ = ∂ H , S ∗ = ∂ H . Note that Im(T ) ⊂ Ker(S), so ST = 0. Proposition 3.1.2 (Gaffney). The operator defined by Dom(
E ) = {s ∈ Dom(S) ∩ Dom(T ∗ ) : Ss ∈ Dom(S ∗ ), T ∗ s ∈ Dom(T )} ,
E s = S ∗ Ss + T T ∗ s ,

for s ∈ Dom(
E ) , (3.1.4)

is a positive self-adjoint extension of the Kodaira Laplacian, called the Gaffney extension. The quadratic form associated to
E is the form Q given by Dom(Q) := Dom(S) ∩ Dom(T ∗ ), Q(s1 , s2 ) =Ss1 , Ss2  + T ∗ s1 , T ∗ s2  ,

for s1 , s2 ∈ Dom(Q).

(3.1.5)

3.1. L2 -cohomology and Hodge theory

129

Proof. It is easy to show that
E is closed (since T and S are) and positive. We need to show that it is self-adjoint. Note that T = (T ∗ )∗H . By Lemma C.1.3, we know that (1 + T T ∗)−1 and (1 + S ∗ S)−1 are bounded self-adjoint operators on L20,q (X, E). We have Im(1 + T T ∗ )−1 = Dom(T T ∗),

Im(1 + S ∗ S)−1 = Dom(S ∗ S).

(3.1.6)

We define a bounded self-adjoint operator R by R = (1 + T T ∗ )−1 + (1 + S ∗ S)−1 − 1.

(3.1.7)

Since R = (1 + S ∗ S)−1 − T T ∗ (1 + T T ∗)−1 , ST = 0 and (3.1.6), we get Im(R) ⊂ Dom(S ∗ S),

S ∗ SR = S ∗ S(1 + S ∗ S)−1 .

(3.1.8)

In the same way, from T ∗ S ∗ = 0, (3.1.6) and (3.1.7), we have Im(R) ⊂ Dom(T T ∗ ),

T T ∗R = T T ∗(1 + T T ∗ )−1 .

(3.1.9)

Thus, from (3.1.6)–(3.1.9), we have Im(R) ⊂ Dom(
E ) and (1 +
E )R = R + S ∗ S(1 + S ∗ S)−1 + T T ∗(1 + T T ∗)−1 = 1.

(3.1.10)

Since 1 +
E is injective, we get R = (1 +
E )−1 . By Lemma C.1.2, 1 +
E and hence
E are self-adjoint. We prove now that Q is the associated quadratic form of
E in the sense of Proposition C.1.4. First remark that Q is a closed form (Problem 3.2). By Proposition C.1.5, there exists a self-adjoint, positive operator B with Q = QB . The domain of B is given by (C.1.6). For s ∈ Dom(
E ), it is clear that s ∈ Dom(QB ) and s satisfies (C.1.6) with v =
E s. Therefore s ∈ Dom(B) and Bs =
E s, i.e.,
E ⊂ B and, since both operators are self-adjoint, B =
E (for, in general, B1 ⊂ B implies B ∗ ⊂ B1∗ ).
Friedrichs’ lemma about the identity of weak and strong derivatives is very useful. Lemma 3.1.3 (Friedrichs). Let Rf = gk ∂f /∂xk + gf be a differential operator of order 1 on an open set U ⊂ Rm , with coefficients gk ∈ C 1 (U ), g ∈ C 0 (U ). Then for any s ∈ L2 (Rm ) with compact support in U , ρ ∈ C0∞ (Rm ), we have lim R(s ∗ ρε ) − (Rs) ∗ ρε L2 = 0.

ε−→0

Here we denote by dx the Euclidean volume form on Rm , and  1 s(y − x) ρε (x) dx. ρε (x) = m ρ(x/ε), (s ∗ ρε )(y) = ε Rm

(3.1.11)

(3.1.12)

130

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

The Gaffney and Friedrichs extensions are in general not equal, but they coincide in all the cases we consider (Kodaira Laplacian on complete manifolds or with ∂-Neumann boundary conditions). However, the quadratic forms of the two extensions are always equal for elements with compact support (Problem 3.3). This can be expressed also as the following regularity result for elements of the domain of the quadratic Q. Proposition 3.1.4. The domain of the quadratic form (3.1.5) satisfies Dom(Q) ⊂ H 1 (X, Λ• (T ∗(0,1) X) ⊗ E, loc). Proof. The statement follows from general arguments about the regularity of E E,∗ (cf. Theorem A.3.4), but we give solutions of the elliptic operator ∂ + ∂ here a direct argument. Let s ∈ Dom(Q) and ϕ ∈ C0∞ (X). We have to show that ϕs ∈ H 10 (K, Λ• (T ∗(0,1) X) ⊗ E), for K = supp(ϕ). It easy to see that E E,∗ ϕs ∈ Dom(∂ ) and ϕs ∈ Dom(∂ H ). Without loss of generality, we can replace s by ϕs and assume that s has compact support in K. After using a partition of unity in order to divide the support of s in small pieces contained in coordinate charts, by Lemma 3.1.3, we obtain a sequence E E E,∗ E,∗ 2 {sk } ⊂ Ω0,• 0 (X, E) such that sk −→ s, ∂ sk −→ ∂ s, ∂ H sk −→ ∂ H s in L . By G˚ arding’s inequality (A.3.5) applied to sk1 − sk2 , we obtain that {sk } is a Cauchy sequence in H 10 (K, Λ• (T ∗(0,1) X) ⊗ E) and therefore s ∈ H 10 (K, Λ• (T ∗(0,1) X) ⊗ E).
Definition 3.1.5. The space of harmonic forms H 0,• (X, E) is defined by H 0,• (X, E) := Ker(
E ) = {s ∈ Dom(
E ) :
E s = 0}.

(3.1.13)

The qth reduced L2 Dolbeault cohomology is defined by ,  E E 0,q H (2) (X, E) := Ker(∂ ) ∩ L20,q (X, E) Im(∂ ) ∩ L20,q (X, E) ,

(3.1.14)

where [V ] denotes the closure of the space V . The qth (non-reduced) L2 Dolbeault cohomology is defined by , E E 0,q H(2) (X, E) := Ker(∂ ) ∩ L20,q (X, E) Im(∂ ) ∩ L20,q (X, E) .

(3.1.15)

For q, r  0, we set: r,q 0,q q 0,q H(2) (X, E) :=H(2) (X, E ⊗ Λr (T ∗(1,0) X)) , H(2) (X, E) := H(2) (X, E) . (3.1.16)

By (3.1.4), we see that E

E,∗

H 0,• (X, E) = Ker(∂ ) ∩ Ker(∂ H ). Using the notation of (3.1.3), by (C.1.2), we have



 Im(T )⊥ = Ker(T ∗ ) = Ker(T ∗ ) ∩ Ker(S) ⊕ Ker(T ∗ ) ∩ Ker(S)⊥ .

(3.1.17)

(3.1.18)

3.1. L2 -cohomology and Hodge theory

131

But by (C.1.2), Ker(S)⊥ = [Im(S ∗ )] and by S ∗ T ∗ = 0, [Im(S ∗ )] ⊂ Ker(T ∗ ). Thus Im(T )⊥ = H 0,• (X, E) ⊕ [Im(S ∗ )]. (3.1.19)

 Likewise, Ker(S) = [Im(T )] ⊕ Ker(S) ∩ Im(T )⊥ and by Im(T )⊥ = Ker(T ∗ ), we get Ker(S) = H 0,• (X, E) ⊕ [Im(T )]. (3.1.20) Since L20,q (X, E) = Im(T )⊥ ⊕ [Im(T )], we deduce the weak Hodge decomposition:     E E,∗ L20,• (X, E) = H 0,• (X, E) ⊕ Im(∂ ) ⊕ Im(∂ H ) ,   E E Ker(∂ ) = H 0,• (X, E) ⊕ Im(∂ ) .

(3.1.21)

By (3.1.21), we have a canonical isomorphism 0,•

H (2) (X, E)  H 0,• (X, E) ,

(3.1.22)

which associates to each cohomology class its unique harmonic representative. E

However, from the point of view of solving the ∂ equation, the (non-reduced) 0,• L Dolbeault cohomology H(2) (X, E) is more interesting. A general condition for 2

the range of ∂ follows.

E

to be closed and for the finiteness of the L2 cohomology is as

Proposition 3.1.6. E

E,∗

(i) A necessary and sufficient condition for Im (∂ ) ∩ L20,q (X, E) and Im(∂ H ) ∩ L20,q (X, E) to be closed is that there exists C > 0 such that E

E,∗

s2L2  C(∂ s2L2 + ∂ H s2L2 ), E

E,∗

s ∈ Dom(∂ )∩ Dom(∂ H ) ∩ L20,q (X, E) , s ⊥ H 0,q (X, E). E

(3.1.23)

E,∗

(ii) Assume that from every sequence sk ∈ Dom(∂ ) ∩ Dom(∂ H ) ∩ L20,q (X, E) with sk L2 bounded and E

∂ sk −→ 0 in L20,q+1 (X, E),

E,∗

∂ H sk −→ 0 in L20,q−1 (X, E),

(3.1.24)

one can select a strongly convergent subsequence. Then both E

Im(∂ ) ∩ L20,q (X, E), are closed. Moreover, H

0,q

E,∗

Im(∂ H ) ∩ L20,q (X, E)

(X, E) is finite-dimensional and

0,q H(2) (X, E)  H 0,q (X, E).

(3.1.25)

132

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

Proof. We use the notation from (3.1.3). (i) Assume that (3.1.23) holds. Then, for any s ∈ Dom(T ∗ ) ∩ [Im(T )], we have Ss = 0, hence s2L2  CT ∗ s2L2 . Likewise, for any s ∈ Dom(S) ∩ [Im(S ∗ )], we have T ∗ s = 0, thus s2L2  CSs2L2 . By Lemma C.1.1, T and S have closed range. (In fact, set T1 = T on Dom(T ) ⊂ L0,q−1 (X, E), and T1 = 0 on L0,q (X, E), then we apply Lemma C.1.1 to T1 .) Conversely if T and S have closed range, by (3.1.21), any s ∈ Dom(S) ∩ Dom(T ∗ ), s ⊥ H 0,q (X, E) can be decomposed in orthogonal sum s = s1 + s2 , s1 ∈ Im(T ), s2 ∈ Im(S ∗ ), and (3.1.23) follows from Lemma C.1.1. (ii) The space H 0,q (X, E) is closed, so a Hilbert space. The hypothesis implies that the unit ball B = {s ∈ L20,q (X, E) : sL2  1 , Ss = 0 , T ∗ s = 0} ⊂ H 0,q (X, E) is compact. Therefore H 0,q (X, E) is finite-dimensional. Assume that (3.1.23) were false. Then there exists {sk } ⊂ Dom(T ∗ )∩Dom(S), sk ⊥ H 0,q (X, E) such that sk 2L2 > k(T ∗sk 2L2 + Ssk 2L2 ).

(3.1.26)

Setting wk = sk /sk L2 , we have T ∗ wk 2L2 + Swk 2L2 < 1/k. By hypothesis, we can extract a convergent subsequence (still denoted by {wk }). Put w = limk wk . Then wL2 = 1 and w ⊥ H 0,q (X, E). But wk → w, Swk → 0 implies w ∈ Ker(S) and likewise we get w ∈ Ker(T ∗ ), hence w ∈ H 0,q (X, E). This means w = 0 which contradicts wL2 = 1. Thus (3.1.23) holds true.
In the geometric situations we shall encounter, the hypotheses of Proposition 3.1.6 are implied by the following estimate. Definition 3.1.7. We say that the fundamental estimate holds in bidegree (0, q) if there exists a compact set K in the interior of X and C > 0 such that    E E,∗ 2 2 2 2 |s| dvX , sL2  C ∂ sL2 + ∂ H sL2 + K (3.1.27) E

E,∗

for s ∈ Dom(∂ ) ∩ Dom(∂ H ) ∩ L20,q (X, E) . Theorem 3.1.8. If the fundamental estimate (3.1.27) holds in bidegree (0, q), then E

(i) The operators ∂ on L20,q−1 (X, E) and
E on L20,q (X, E) have closed range and we have the strong Hodge decomposition: E E,∗

E,∗ E

L20,q (X, E) = H 0,q (X, E) ⊕ Im(∂ ∂ H ) ⊕ Im(∂ H ∂ ) ,

 E E Ker(∂ ) ∩ L20,q (X, E) = H 0,q (X, E) ⊕ Im(∂ ) ∩ L20,q (X, E) .

(3.1.28)

Moreover, H 0,q (X, E) is finite-dimensional. We have a canonical isomorphism 0,q (X, E), s → [s]. (3.1.29) H 0,q (X, E) → H(2)

3.1. L2 -cohomology and Hodge theory

133

(ii) There exists a bounded operator G on L20,q (X, E), called the Green operator, such that (3.1.30)
E G = G
E = Id −P, P G = GP = 0, where P is the orthogonal projection from L20,q (X, E) onto H 0,q (X, E). E

E,∗

Proof. Consider a sequence {sk } ⊂ Dom(∂ ) ∩ Dom(∂ H ) ∩ L20,q (X, E) with E,∗

E

{sk L2 } bounded and ∂ H sk L2 + ∂ sk L2 −→ 0, for k → ∞. Let ϕ ∈ C0∞ (X) such that ϕ = 1 on K; then E

E,∗

Q(ϕsk , ϕsk ) + ϕsk 2L2 = ∂ (ϕsk )2L2 + ∂ H (ϕsk )2L2 + ϕsk 2L2

(3.1.31)



is also bounded. Let X be a compact manifold with boundary containing supp(ϕ) in its interior. By Proposition 3.1.4, ϕsk ∈ H 10 (X  , Λq (T ∗(0,1) X)⊗E). By applying G˚ arding’s inequality (A.3.5) and (3.1.31), we obtain that (ϕsk 1 ) is bounded. By Rellich’s theorem A.3.1, 



1  q ∗(0,1) X) ⊗ E),  · 1 → L20,q (X, E),  · L2 (3.1.32) H 0 (X , Λ (T is compact. We can select therefore a convergent subsequence in L20,q (X, E), denoted also {ϕsk }. Since ϕ = 1 on K, it follows that {sk |K } converges in  · L2 . By estimate (3.1.27), this entails that {sk } converges in L20,q (X, E). Proposition 3.1.6 implies that (3.1.23) holds and H 0,q is finite-dimensional. From (3.1.23), we infer that sL2  C
E sL2 , Therefore


E

for s ∈ Dom(
E ), s ⊥ Ker(
E ) .

has closed range. Since


E

(3.1.33)

is self-adjoint, by (C.1.2), we have E

E,∗

E,∗ E

L20,q (X, E) = Im(
E ) ⊕ Ker(
E ) = Im(∂ ∂ H ) ⊕ Im(∂ H ∂ ) ⊕ H 0,q (X, E) . By (3.1.33), there exists a bounded inverse G of
E on Im(
E ). We extend G to L20,q (X, E) by setting G = 0 on H 0,q (X, E). We obtain thus a bounded operator G on L20,q (X, E) (bounded by 1/C, where C is the constant from (3.1.33)), satisfying Ker(G) = H 0,q (X, E) and Im(G) = Im(
E ). The proof of Theorem 3.1.8 is completed.
Corollary 3.1.9. Assume that the fundamental estimate (3.1.27) holds in bidegree (0, q). E

E

(i) If f ∈ Im(∂ ) ∩ L20,q (X, E), the unique solution s ⊥ Ker(∂ ) ∩ L20,q−1 (X, E) E

E,∗

of the equation ∂ s = f is given by s = ∂ H Gf . (ii) The operator G maps L20,q (X, E) ∩ Ω0,q (X, E) into itself. Assertion (i) is immediate from the preceding proof. Finally, assertion (ii) follows from the interior regularity for the elliptic operator
E (Theorem A.3.4). Remark 3.1.10. If X is compact, then the fundamental estimate (3.1.27) holds in bidegree (0, q) for any q ∈ N. Thus from Theorem 3.1.8, we get the Hodge decomposition on compact manifolds, Theorem 1.4.1.

134

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

3.2 Abstract Morse inequalities for the L2-cohomology We shall examine a general situation which permits us to prove asymptotic Morse inequalities for the L2 -Dolbeault cohomology groups of complex manifolds. We show in Section 3.2.1 that the fundamental estimate allows us to reduce the study of the spectral spaces of the Kodaira Laplacian on the whole manifold to the study of spectral spaces of the Kodaira Laplacian with Dirichlet boundary conditions on a smooth relatively compact domain, whose asymptotic we determine in Section 3.2.2. In Section 3.2.3, we apply then Theorem 3.2.9 in order to prove the abstract holomorphic Morse inequalities.

3.2.1 The fundamental estimate Let (X, J, Θ) be a complex Hermitian manifold of dimension n. Let (L, hL ) and (E, hE ) be holomorphic Hermitian vector bundles on X with rk(L) = 1. Let dvX be the Riemannian volume form on X associated to the metric g T X = Θ(·, J·) on T X. From (1.5.20) and (1.6.1), we will denote simply that Epj = Λj (T ∗(0,1) X) ⊗ Lp ⊗ E, E

∂ p := ∂

Lp ⊗E

,

E,∗

∂p

:= ∂

Ep = ⊕j Epj ,

Lp ⊗E,∗

,


p :=
L

p

⊗E

(3.2.1) .

We postulate a general estimate for the quadratic form of the Kodaira Laplacian
p acting on the bundle Lp ⊗ E, which implies estimates from above of the spectral function. Definition 3.2.1 (fundamental estimate). We say that the fundamental estimate holds in bidegree (0, q) for forms with values in Lp ⊗ E if there exists a compact K ⊂ X and C0 > 0 such that for sufficiently large p, we have  C0 E 2 E,∗ 2  2 sL2  ∂ p sL2 + ∂ p sL2 + C0 |s|2 dvX , p K (3.2.2) E E,∗ 2 p for s ∈ Dom(∂ p ) ∩ Dom(∂ p ) ∩ L0,q (X, L ⊗ E). K is called the exceptional compact set of the estimate. Let us consider the Gaffney extension
p of the Kodaira Laplacian acting on Lp ⊗ E which we normalize by p1
p . (3.2.2) is of course a variant with parameters of the fundamental estimate (3.1.27). It allows us to compare the spectral spaces of 1 1 p
p on X and of p
p with Dirichlet boundary conditions on a relatively compact domain U containing K. Definition 3.2.2. Let U ⊂ X be an open set. The Friedrichs extension (cf. Definition C.1.7) of the positive operator p1
p : C0∞ (U, Ep ) → L2 (U, Ep ) is called the operator p1
p with Dirichlet boundary conditions. We denote the Kodaira Laplacian with Dirichlet boundary conditions by
p,U and the quadratic form associated to p1
p,U by Qp,U .

3.2. Abstract Morse inequalities for the L2 -cohomology

135

Lemma 3.2.3. If U is relatively compact, then Dom(
p,U ) = H 20 (U, Ep ) (cf. Appendix A.3), and
p,U has discrete spectrum. Proof. It is easy to see that Dom(Qp,U ) = H 10 (U, Ep ). By G˚ arding’s inequality (A.3.5) and Rellich’s theorem A.3.1, we deduce that (Dom(Qp,U ), ·Qp,U ) → L2 (U, Ep ),

(3.2.3)

with sQp,U := (Qp,U (s, s) + s2L2 )1/2 ,

is a compact operator. By Proposition C.2.4,
p,U has discrete spectrum. Using Definition C.1.7, and Theorem A.3.4, we know Dom(
p,U ) = H 20 (U, Ep ). Note that
∗p = (
p )max .
According to Proposition 3.1.2, the quadratic form associated to p1
p is Qp (s, s) =

 1 E 2 E,∗ ∂ p sL2 + ∂ p s2L2 , p

s ∈ Dom(Qp ) =

E Dom(∂ p )

∩

(3.2.4)

E,∗ Dom(∂ p ) .

Let {Eλ ( 1p
p )}λ be the spectral resolution of p1
p and let E (λ, 1p
p ) = Im(Eλ ( p1
p )) be the corresponding spectral spaces. All these objects decompose in a direct sum according to the decomposition of forms after bidegree. Let us fix an open, relatively compact neighborhood U of K with smooth boundary. Let {Eλ ( 1p
p,U )}λ be the spectral resolution of p1
p,U and for q ∈ N, set: E q (λ, p1
p ) = Im(Eλ ( 1p
p )) ∩ L20,q (X, Lp ⊗ E), E q (λ, 1p
p,U ) = Im(Eλ ( 1p
p,U )) ∩ L20,q (X, Lp ⊗ E),

(3.2.5)

N q (λ, p1
p ) = dim E q (λ, 1p
p ), N q (λ, 1p
p,U ) = dim E q (λ, 1p
p,U ).

One of the tools for the proof of the holomorphic Morse inequalities is to estimate N • (λ, 1p
p ) from above and from below. Proposition 3.2.4 (decomposition principle). The operators have the same essential spectrum.

1 p
p

and

1 p
p,XU

Proof. Let ϕ ∈ C0∞ (X) be a nonnegative function which is equal to 1 on a neighborhood of U , the closure of U . Let U1 be a relatively compact neighborhood of the support of ϕ. Let {fk }k∈N be an orthonormal characteristic sequence for ( p1
p , λ) for some λ  0 as in Theorem C.3.3. Rellich’s theorem A.3.1 implies that {ϕfk }k∈N is compact in H 20 (U1 , Ep ). Thus, by passing to a subsequence of {fk }k∈N , if necessary, we may assume that {ϕfk }k∈N is convergent.

136

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

Then we set gk = f2k − f2k−1 (k  1). We see that {gk }k∈N is non-compact, limk−→∞ ( p1
p − λ Id)gk L2 = 0, and ϕgk → 0 in the Sobolev space H 20 (U1 , Ep ). Then limk→∞ ( p1
p,XU − λ Id)(1 − ϕ)gk L2 = 0, and consequently { gk }k∈N

with gk :=

(1 − ϕ)gk , (1 − ϕ)gk L2

is a characteristic sequence for ( p1
p,XU , λ). So σess ( 1p
p ) ⊂ σess ( p1
p,XU ). We trivially have σess ( p1
p,XU ) ⊂ σess ( p1
p ).
Proposition 3.2.4 and its consequence, Theorem 3.2.5(1) will not be used in the sequel (e.g., for the proof of Theorem 3.2.5(2)), but they provide a good description of the spectrum of 1p
p . Consider a smooth function ρ on X , 0  ρ  1 such that ρ = 1 on K and ρ = 0 on X
U . Set C1 = sup |dρ|2 . Theorem 3.2.5. If the fundamental estimate (3.2.2) holds in bidegree (0, q), then on L20,q (X, Lp ⊗ E) has only discrete spectrum in [0, 1/C0 ] for large enough p. (2) There exists a constant C2 depending only on C0 and C1 such that for λ < 1/(2C0 ), the following maps are injective for p sufficiently large,

(1)

1 p
p

E q (λ, 1p
p ) −→ E q (3C0 λ + C2 p−1 , p1
p,U ), s −→ E3C0 λ+C2 p−1 ( p1
p,U )(ρs).

(3.2.6)

In particular for any λ < 1/(2C0 ) , p  1, N q (λ, p1
p )  N q (3C0 λ + C2 p−1 , p1
p,U ) .

(3.2.7)

Proof. (1) By Proposition 3.2.4, p1
p has the same essential spectrum as 1p
p,XU . The fundamental estimate (3.2.2) shows then that Qp,XU (s)  C10 s2 , s ∈ Dom(Qp,XU ), since Dom(Qp,XU ) ⊂ Dom(Qp ). It follows that 1p
p,XU has no essential spectrum in [0, C10 ] and p1
p has the same property. E

E,∗

(2) For s ∈ Dom(∂ p ) ∩ Dom(∂ p ), we obtain by Leibniz’s formula E

E,∗

E,∗

E,∗

∂ p (ρs)2L2 + ∂ p (ρs)2L2 = ρ∂ p s + ∂ρ ∧ s2L2 + ρ∂ p s + i∂ρ s2L2 . Using the inequality (x + y)2  32 x2 + 3y 2 together with the triangle inequality, we obtain with C1 = sup |dρ|2 , E

E,∗

∂ p (ρs)2L2 + ∂ p (ρs)2L2 

3 E E,∗ (∂ p s2L2 + ∂ p s2L2 ) + 6C1 s2L2 . 2

(3.2.8)

3.2. Abstract Morse inequalities for the L2 -cohomology

137

Let s ∈ E q (λ, 1p
p ), λ < 1/(2C0 ). Then by (C.3.2) and (3.2.2), we get  1 E 2 E,∗ ∂ p sL2 + ∂ p s2L2  λs2L2 , p   2C0 |s|2 dvX .

Qp (s, s) = s2L2

(3.2.9)

K

By (3.2.8) and (3.2.9), we have 3 6C1 Qp (s, s) + s2L2 2 p  3 6C1   |s|2 dvX . λ+ · 2C0 2 p K

Qp,U (ρs, ρs) 

(3.2.10)

We set C2 = 12C0 C1 . Inequality (3.2.10) and E3C0 λ+C2 p−1 ( p1
p,U )(ρs) = 0 imply ρs = 0, by Problem 3.4. Since ρ = 1 on K, it follows that s = 0 on K and by (3.2.9), we infer s = 0. Thus (3.2.6) is injective.
As for a lower bound of the counting function N q (λ, 1p
p ), we have a general result which does not depend on the fundamental estimate. Lemma 3.2.6. The following estimate from below holds for any q ∈ N: N q (λ, p1
p )  N q (λ, p1
p,U ) .

(3.2.11)

Proof. Let us denote by λ0  λ1  · · · the spectrum of p1
p,U acting on (0, q)forms. Let {si }i be an orthonormal basis which consists of eigenforms corresponding to the eigenvalues {λi }i ; if we let si = 0 on X
U and si = si on U , then si ∈ Dom(Qp ) and Qp ( si , sj ) = δi,j λi . Let Vλ0 be the subspace spanned 2 by {si : λi  λ} in L0,q (U, Lp ⊗ E) and Vλ the closed subspace spanned by { si : λi  λ} in L20,q (X, Lp ⊗ E). Then dim Vλ = dim Vλ0 = N q (λ, p1
p,U ). If f is a linear combination of { si : λi  λ}, then Qp (f, f )  λf 2L2 and, as Dom(Qp ) is complete in the graph norm, we obtain Vλ ⊂ Dom(Qp ) and Qp (f, f )  λf 2L2 , for f ∈ Vλ . Glazman’s lemma C.3.1 implies now Lemma 3.2.6.

(3.2.12)


3.2.2 Asymptotic distribution of eigenvalues We use the notation and assumption in Section 1.6.1, except that X is a compact complex manifold with boundary ∂X. 2 As in Section 3.2.1, we denote by Dp,X = 2
p,X the operator Dp2 = 2
p with Dirichlet boundary condition (cf. (D.1.25)) on X.

138

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

For j = 0, 1, . . . , let λp,q be the eigenvalues (counted with multiplicities) j acting on (0, q)-forms with values in Lp ⊗ E. We define the counting of function N q (λ, 1p
p,X ) by (3.2.5), and consider the measure µqp on R+ , 1 p
p,X

µqp = p−n

∞
d q N (λ, p1
p,X ) = p−n δ(λp,q j ). dλ j=0

(3.2.13)

By integrating the function e−uλ against this measure, we obtain the trace of the u 2 operator exp(− 2p Dp,X ), so using (1.7.4), we get 

∞

0

e−uλ dµqp (λ) = p−n

 u 2  −n exp(− D ) . exp(−u λp,q ) = p Tr q j 2p p,X j=0

∞


Proposition 3.2.7. For any u > 0 fixed, the following relation holds:   ∞ det(R˙ L /(2π)) TrΛ0,q [exp(uωd )] −uλ q lim e dµp (λ) = rk(E) . p→∞ 0 det(1 − exp(−uR˙ L )) X

(3.2.14)

(3.2.15)

The convergence in (3.2.15) is uniform as u varies in any compact subset of R∗+ . Proof. Let N be the normal bundle of ∂X in X; we identify it as the orthogonal complement of T ∂X in T X. Let en be the inward pointing unit normal at any boundary point of X. For y0 ∈ ∂X, 0  xn  , by using Ny0  xn en → expX y0 (xn en ), we get a neighborhood of ∂X for  > 0 small enough, and we identify ∂X × [0, [ as a neighborhood of ∂X in this way. At first, when we choose {xi } in Section 1.6.2, we will take first xi ∈ ∂X for 0 1  i  k0 such that B X (xi , ε/2) covers ∂X, then we choose {xi }N i=k0 +1 such that N0 k0 N0 X X X {B (xi , ε/2)}i=k0 +1 covers X
∪i=1 B (xi , ε) and ∪i=k0 +1 B (xi , ε) ∩ ∂X = ∅. We still define s2H l (p) as in (1.6.5). By basic elliptic estimate, Theorem A.3.2, the Dirichlet boundary condition is an elliptic boundary problem for Dp2 , thus (1.6.8) still holds for s ∈ H 20 (X, Ep ) (cf. (3.2.1) for Ep ). Thus for s ∈ H 02m+2 (X, Ep ), p ∈ N, we have the analogue of (1.6.6),  4m+4 p sH 2m+2 (p)  Cm

m+1


p−4j Dp2j sL2 .

(3.2.16)

j=0

 u, H  u,ς are the holomorphic functions on C defined by  u, G Recall that F (1.6.26). By (3.2.16) as in (1.6.18), for any differential operators P, Q of order m, m with compact support in Uxi := B X (xi , ε), Uxj respectively, there exists C > 0 such that for p  1, u  u0 , s ∈ C0∞ (X, Ep ),  2 1−θ   u ,pθ (D2 )QsL2  Cp2(m+m
+2)(θ+1) exp − ε p sL2 . P H p,X p 16u

(3.2.17)

3.2. Abstract Morse inequalities for the L2 -cohomology

139

 u ( u D2 ) = H  u ,pθ (D2 ). On Ux × Ux , by using Sobolev By (1.6.14), G i j p,X p p,X p p1−θ inequality from Theorem A.1.6, we get the following generalization of (1.6.15) : For any m ∈ N, 0  θ < 1, u0 > 0, ε > 0, there exists C > 0 such that for any x, x ∈ X, p ∈ N∗ , u > u0 ,    ε2 p1−θ    u u 2 . G p1−θ ( Dp,X )(x, x ) m  Cp2(m+4n+4)(1+θ) exp − p 16u C

(3.2.18)

Using (1.6.13) and finite propagation speed, Theorem D.2.1 and (D.2.17), it  u/p1−θ ( u D2 )(x, x ) only depends on the restriction is clear that for x, x ∈ X, F p p,X of Dp2 to B X (x, εp−θ/2 ), and is zero if d(x, x )  εp−θ/2 .  (which need not be complex) and Now we embed X into a closed manifold X ∗(0,1)  X), L, E to X with smooth metrics and smooth we extend the bundles Λ(T Hermitian connections (cf. Problem 3.5), moreover, we extend the smooth section  thus we get a generalized Laplacian L  p on X  such that the ΦE in (1.6.20) to X, 2    p )(x, x ). u ( up L restriction of Lp on X is Dp . Certainly, (3.2.18) also holds for G 1−θ p

Thus from (1.6.14) and (3.2.18), we get uniformly for x0 ∈ X
(∂X × [0, εp−θ/2]), p  1, u > u0 ,   u  u 2   )(x0 , x0 ) − exp(− L  exp(− Dp,X p )(x0 , x0 ) 2p 2p

 ε2 p1−θ  . (3.2.19)  Cp8(n+1)(1+θ) exp − 16u

For y0 ∈ ∂X, we use the coordinate V × [0, [ of y0 in X such that V is a normal coordinate of y0 in ∂X. We still trivialize the bundles Λ(T ∗(0,1) X), L, 0,• E by the parallel transport with respect to the connections ∇B,Λ , ∇L , ∇E 2 along the curve [0, 1]  v → vZ. Now we extend the operator Dp,X to Lp,D,x0 on 2n 2n−1 ∗ R+ = R ×R+ by (1.6.23) with the Dirichlet boundary condition (cf. (D.1.25)). Then by the argument in Lemma 1.6.5 and (3.2.18), for 0  xn  ,   u u 2   )((y0 , xn ), (y0 , xn )) − exp(− Lp,D )((0, xn ), (0, xn ))  exp(− Dp,X 2p 2p ε2 p1−θ ). (3.2.20)  Cp8(n+1)(1+θ) exp(− 16u We fix the coordinate such that en = e2n . In our coordinate, for j = 2n, we have gj,2n (Z) = 0 in (1.6.31), thus we have Lt2 = − (∇t,e2n )2 −

2n−1


  g ij (tZ) ∇t,ei ∇t,ej − tΓkij (tZ)∇t,ek

i,j=1

+ ρ(|tZ|/ε)(−2ωd,tZ − τtZ + t2 ΦE,tZ ).

(3.2.21)

140

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

We denote by Lt2,D the corresponding operator Lt2 with the Dirichlet boundary ∞ 2n condition. Let H m 0,t be the Sobolev space which is the completion of C0 (R+ , Ex0 ) with respect to the norm  t,m . Then Theorems 1.6.7–1.6.9 still hold for Lt2,D t −1 and s, s ∈ C0∞ (R2n exists for + , Ex0 ) without any change. Especially, (λ − L2,D ) t ∈]0, 1], λ ∈ Γ. To prove the analogue of Theorem 1.6.10, we need to prove that for Q0 , . . . , 2n α Qm ∈ {∇t,ei }2n i=1 , Qm+1 , . . . , Qm+|α| ∈ {Zi }i=1 , and Qm+1 · · · Qm+|α| = Z , there ∞ 2n exist M ∈ N, Cα,m > 0 such that for t ∈]0, 1], λ ∈ Γ, s ∈ C (R+ , Ex0 ) with compact support,

Q0 · · · Qm+|α| (λ − Lt2,D )−1 st,0  Cα,m (1 + |λ|2 )M Z α st,m . (3.2.22) α

α 2n−1 preserves the Dirichlet boundary conThe main point here is that {∇t,ei }i=1 dition, but ∇t,e2n does not preserve the Dirichlet boundary condition, thus the following equation only holds for 1  i  2n − 1,

[∇t,ei , (λ − Lt2,D )−1 ] = (λ − Lt2,D )−1 [∇t,ei , Lt2 ](λ − Lt2,D )−1 .

(3.2.23)

2n−1 If Q0 , . . . , Qm ∈ {∇t,ei }i=1 , then the same proof of Theorem 1.6.10 gives 2n−1 (3.2.22), as {∇t,ei }i=1 preserves the Dirichlet boundary condition. If there are only one Qi such that Qi = ∇t,e2n , then by using (1.6.41), we can assume that Q0 = ∇t,e2n , and the same proof of Theorem 1.6.10 still gives (3.2.22). If there are at least two ∇t,e2n in Q0 , . . . , Qm , then by using (1.6.41) and [∇t,ei , Zj ] = δij , we move all ∇t,e2n to the right-hand side of Q0 · · · Qm+|α| , thus we express Q0 · · · Qm+|α| as a linear combination of operators of type

Qj1 · · · Qjl (∇t,e2n )k (λ − Lt2,D )−1 ;

(3.2.24)

2n−1 ∪ {Zi }2n with k  m + 1, l + k  m + |α| + 1 and Qji ∈ {∇t,ei }i=1 i=1 . If k < 2, we apply the previous argument to get the estimate (3.2.22). If k  2, then we replace (∇t,e2n )2 (λ − Lt2,D )−1 in (3.2.24) by

1 + (−λ + Lt2 + (∇t,e2n )2 )(λ − Lt2,D )−1 .

(3.2.25)

By (3.2.21), the degree of ∇t,e2n in Lt2 +(∇t,e2n )2 is at most 1, and we can continue the process to get (3.2.24) with k  1. Thus we have proved (3.2.22).  m By (3.2.22), as in (1.6.49), (1.6.51), for s ∈ C0∞ (R2n + , Ex0 ), Q, Q ∈ Q , Q(λ − Lt2,D )−m st,0  Cm (1 + |λ|2 )M st,0 , (λ − Lt2,D )−m Qst,0  Cm (1 + |λ|2 )M st,0 .

(3.2.26)

Thus u  Cm . QZ QZ
exp(− Lt2,D )0,0 t 2

(3.2.27)

3.2. Abstract Morse inequalities for the L2 -cohomology

141

Ex0 Let  · m be the usual Sobolev norm on C0∞ (R2n = + , Ex0 ) induced by h ∗(0,1)

hΛ(Tx0 X)⊗Ex0 and the volume form dvT X (Z) as in (1.6.32). Observe that by (1.6.31), (1.6.32), for m  0, there exists Cm > 0 such that for s ∈ C0∞ (R2n + , Ex0 ), supp(s) ⊂ B Tx0 X (0, q), 1 (1 + q)−m st,m  sm  Cm (1 + q)m st,m . Cm

(3.2.28)

Now (3.2.27), (3.2.28) together with Sobolev’s inequalities, Theorem A.2.2, implies that u | exp(− Lt2,D )(Z, Z  )|  C(1 + |Z| + |Z  |)2n+2 . 2

(3.2.29)

By (1.6.66), (3.2.29), we get for 0  xn  , | exp(−

u √ Lp,D )((0, xn ), (0, xn ))|  Cpn (1 + p|xn |)2n+2 . 2p

(3.2.30)

From (3.2.20) and (3.2.30), we get for 0  xn  εp−θ/2 , p

    u 2  exp(− 2p Dp,X )((y0 , xn ), (y0 , xn ))

−n 

 Cp8(n+1)(1+θ) exp(−

ε2 p1−θ ) + Cp(n+1)(1−θ) . (3.2.31) 16u

Now, we fix 0 < θ < 1 such that (n + 1)(1 − θ) < θ/2, then by Theorem 1.6.1 and (3.2.14), (3.2.19), (3.2.31) and dominated convergence, we get for u > 0 fixed,  ∞   u 2  e−uλ dµqp (λ) = lim p−n Trq e− 2p Dp,X (x, x) dvX (x) lim p→∞ 0 p→∞ X  u 2 = lim p−n Trq [exp(− Dp,X )(x, x)]dvX (x) p→∞ X(∂X×[0,εp−θ/2 [) 2p  (3.2.32) u 2 p−n Trq [exp(− Dp,X )(x, x)]dvX (x) + lim p→∞ ∂X×[0,εp−θ/2 [ 2p  L ˙ det(R /(2π)) TrΛ0,q [exp(u ωd )] = rk(E) . det(1 − exp(−uR˙ L )) X From Theorem 1.6.1 and (3.2.31), the convergence in (3.2.32) is uniform as u varies
in any compact subset of R∗+ . The proof of Proposition 3.2.7 is complete. We denote by D  (R) (resp. S  (R)) the space of distributions (resp. temperate distributions) on R. Consider g ∈ D  (R) supported in [0, ∞ [. Assume that there exists u0 ∈ R such that for any u > u0 , λ → e−λu g(λ) belongs to S  (R). The

142

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

Laplace transform of g is the holomorphic function L(g) : {Re(z) > u0 } → C defined by (3.2.33) L(g)(z) = (g(λ), e−λz ), where z = u + iv. Relation (3.2.14) gives the Laplace transform of the measure µqp and (3.2.15) describes the limit of the Laplace transforms as p → ∞. Since we wish to calculate the limit of the sequence (µqp )p1 , we need the following. Theorem 3.2.8. A holomorphic function G : {Re(z) > u0 } → C, u0 ∈ R, is the Laplace transform of a distribution g ∈ D  (R) with support in [0, ∞ [ if and only if there exists a polynomial Q ∈ R[t] and u1 > u0 such that |G(z)|  Q(|z|) for Re(z) > u1 . The inverse Laplace transform L−1 (G) of G is given for any u > u0 by  u+i∞ 1 G(z)eλz dz . (3.2.34) L−1 (G)(λ) = 2πi u−i∞ As in Section 3.2.1, we use the notation (1.5.14)–(1.5.20). Especially, ai (x) (1,0) are eigenvalues of R˙ L (x) ∈ End(Tx X). Let us denote by l = l(x) the rank of R˙ L (x) and order the eigenvalues so that |a1 (x)|  · · ·  |al (x)| > 0 = al+1 (x) = · · · = an (x). For I ⊂ {1, . . . , n}, we set
aI (x) = ai (x), I = {1, . . . , n}
I. (3.2.35) i∈I

Let H+ (t) = 0 for t < 0, and 1 for t  0 be the Heaviside function. We introduce the function νRL (x) (λ) =

(2π)−n |a1 (x) · · · al (x)| (n − l)! 
 × (tn−l H+ (t)) (α1 ,...,αl

)∈Nl

t=λ−

j

, 1 (αj + 2 )|aj (x)|

where we set t0 = 1 for any t ∈ R. To shorten the notation, we set


 q νR νRL (x) λ + 12 (aI (x) − aI (x)) . L (λ, x) =

(3.2.36)

(3.2.37)

|I|=q

Theorem 3.2.9. There exists an at most countable set A q ⊂ R such that we have for λ ∈ R+
A q :  q νR (3.2.38) lim p−n N q (λ, p1
p,X ) = I q (X, λ) := rk(E) L (λ, x) dvX . p→∞

X

Moreover, 1 rk(E) lim I (X, λ) = I (X, 0) = λ→+0 n! q



q

(−1)q X(q)

√

−1 L 2π R

n .

(3.2.39)

3.2. Abstract Morse inequalities for the L2 -cohomology

143

Proof. Taking into account (1.7.11), we can reformulate (3.2.15) as follows: for any u > 0,  ∞ e−uλ dµqp (λ) (3.2.40) lim p→∞

0

= rk(E)






l |aj | exp( u2 (aI − aI − j=1 |aj |)) dvX . · # (2π)n un−l lj=1 (1 − e−u|aj | )

#l

X |I|=q

j=1

Let us denote the integrand in (3.2.40) by F (x, u). As a function of u, F (x, u) can be extended to a holomorphic function F (x, z) in {z ∈ C : Re(z) > 0}. There exists C , 2}. C > 0 such that for u ∈ C and a ∈ R we have |(1 − e−u|a| )−1 | ≤ max{ Re(u)|a| Then for any u0 > 0 we have |F (x, u)| ≤ C uniformly on x ∈ X, Re(u) > u0 . We see from Theorem 3.2.8, that F (x, u) admits an inverse Laplace transform L−1 (F (x, ·)) for each fixed x. We have (cf. Problem 3.6),  λ q L−1 (F (x, ·))(λ0 ) dλ0 = νR (3.2.41) L (λ, x). 0

Lemma 3.2.10. Consider a sequence of measures µp on R and a function Ω : ]c, ∞[−→ R such that when p → ∞, L(µp )(λ) → Ω(λ) for λ ∈ R, λ > c. Then there exists a measure µ on R with support in [0, ∞[ such that L(µ)(λ) = Ω(λ) for λ ∈ R, λ > c, and µp → µ in the sense of distributions. By Lemma 3.2.10 the sequence of measures (µqp )p1 converges in the sense of distributions to the measure µq satisfying:  q µq ([0, λ]) = rk(E) νR (3.2.42) L (λ, x) dvX , X

especially the measure µ is zero on ] − ∞, 0[. In other words, for any ϕ ∈ C0∞ (R), we have (3.2.43) lim (µqp , ϕ) = (µq , ϕ). q

p→∞

Now, take ϕk : R → [0, 1] smooth, supp(ϕk ) ⊂ ] − 2, λ[, and ϕk ! 1 on [−1, λ[. Then from (3.2.43), we have for any k ∈ N, lim inf µqp ([0, λ])  lim (µqp , ϕk ) = (µq , ϕk ). p→∞

p→∞

(3.2.44)

Relations (3.2.13) and (3.2.44) yield by taking k → ∞: lim inf p−n N q (λ, 1p
p,X )  µq ([0, λ[). p→∞

(3.2.45)

In the same way, we choose ϕk ∈ C0∞ (R), such that ϕk = 1 on [−1, λ], and ϕk " 1[0,λ] on [0, ∞[. Then we get as in (3.2.45), 1 lim sup p−n N q (λ,
p,X )  lim µq ([0, λ0 ]) = µq ([0, λ]). λ0 →+λ p p→∞

(3.2.46)

144

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

Let us denote by A q the countable set where λ → µq ([0, λ]) is possibly discontinuous. Then by (3.2.45) and (3.2.46), p−n N q (λ, p1
p,X ) → µq ([0, λ]) as p → ∞, for all λ ∈ R+
A q and formula (3.2.38) holds true. We compute the behavior of I q (X, λ) for λ → +0. First it is clear that  q q q lim I (X, λ) = I (X, 0) = rk(E) νR (3.2.47) L (0, x) dvX . λ→+0

X

Since 12 (aI − aI ) − α ∈ Nl ,



j (αj

+ 12 )|aj |  0 for all α ∈ Nl , it is clear that for a given

  (tn−l H+ (t))

unless l = n (i.e.,

1 t= 2 (a
I −aI )−

j

=0

1 (αj + 2 )|aj |

√ −1RL (x) is non-degenerate) and 1 2 (aI

− aI ) −



j (αj

+ 12 )|aj | = 0.

The last equality holds if and only if α1 = · · · = αn = 0,  aj <√0 for j ∈ I and aj > 0 for j ∈ I. In particular, if νRL (x) 12 (aI − aI ) = 0, −1RL (x) is non-degenerate and has exactly q negative eigenvalues. For x ∈ X(q), set I(x) = {j : aj (x) < 0}. For |I| = q, it follows that νRL (x)

1

2 (aI

 − aI ) =

(2π)−n |a1 (x) . . . an (x)| 0

for I = I(x), for I =  I(x).

(3.2.48)


By (1.7.15) and (3.2.48), we get (3.2.39).

3.2.3 Morse inequalities for the L2 cohomology Let (X, Θ) be a complex Hermitian manifold of dimension n. Let (L, hL ) and (E, hE ) be holomorphic Hermitian vector bundles on X with rk(L) = 1.

E E Let Dom(∂ p )∩L20,• (X, Lp ⊗E), ∂ p be the L2 -Dolbeault complex of densely E

defined closed operators. Since ∂ p commutes with spectral projections, i.e., E

∂ p Eλ

1

p
p



= Eλ

1

p
p

1 p
p

, it commutes with the

 E ∂p .

(3.2.49)

We obtain therefore a sub-complex

• 1  E 

E E E λ, p
p , ∂ p −→ Dom(∂ p ) ∩ L20,• (X, Lp ⊗ E), ∂ p .



 E The cohomology of this complex is denoted by H • E • λ, p1
p , ∂ p .

(3.2.50)

3.2. Abstract Morse inequalities for the L2 -cohomology

145

Proposition 3.2.11. If the fundamental estimate (3.2.2) holds for (0, q)-forms with q  m, then for q  m and λ  0,



 E 0,q (X, Lp ⊗ E). H q E • λ, 1p
p , ∂ p  H 0,q (X, Lp ⊗ E)  H(2)

(3.2.51)

Proof. By Theorem 3.1.8, the strong Hodge decomposition holds in bidegrees (0, q), for any q  m. Let us restrict (3.1.28) to the complex (3.2.50):

 E q λ, 1p
p = H

0,q

 E (X, Lp ⊗ E) ⊕ Im( ∂ p ) ∩ E q (λ, 1p
p )



 E,∗ ⊕ Im( ∂ p ) ∩ E q λ, 1p
p . (3.2.52)

Due to (3.2.49), we see that

 E E Im( ∂ p ) ∩ E q λ, 1p
p = Im( ∂ p |E q (λ, p1
p )) ,



 E,∗ E,∗ Im(∂ p ) ∩ E q λ, 1p
p = Im(∂ p |E q λ, 1p
p ).

(3.2.53)

Therefore





E E Ker(∂ p |E q λ, 1p
p ) = H 0,q (X, Lp ⊗ E) ⊕ Im ∂ p |E q λ, 1p
p .

(3.2.54)


By (3.1.29) and (3.2.54) , we get (3.2.51).

The following algebraic Morse inequalities which were essentially established in (1.7.9), are very useful. Lemma 3.2.12. Let d0

d1

dn−1

0 −→ V 0 −−−−→ V 1 −−−−→ · · · −−−−→ V n −→ 0 be a complex of vector spaces. Let H i (V • ) = Ker(di )/ Im(dj−1 ) with Im(d−1 ) = 0. If dim V q < +∞ for any q  m, then q


(−1)q−j dim H j (V • ) 

j=0

q


(−1)q−j dim V j , for q  m.

(3.2.55)

j=0

Proof. Set nj = dim Ker(dj ) , mj = dim Im(dj ). Then dim H j (V • ) = nj − mj−1

dim V j = nj + mj ,

(with m−1 = 0)

and for q  m, q
j=0

q−j

(−1)

j

dim V = mq +

q


(−1)q−j dim H j (V • ).

(3.2.56)

j=0

The inequalities (3.2.55) follow from (3.2.56).




146

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

We are prepared to state the Morse inequalities for the L2 cohomology. Theorem 3.2.13. Assume that there exists m ∈ N such that the fundamental estimate (3.2.2) holds for (0, q)-forms with any q  m. Let U ⊂ X be a relatively compact open set with smooth boundary such that K  U . Then as p → ∞, the following strong Morse inequalities hold for any q  m:  q √ n
pn 0,j −1 L q−j p (−1) dim H(2) (X, L ⊗ E)  rk(E) (−1)q 2π R + o(pn ). n! U(
q) j=0 (3.2.57) In particular, we get the weak Morse inequalities for any q  m:  √ n pn 0,q −1 L p (−1)q 2π R + o(pn ). dim H(2) (X, L ⊗ E)  rk(E) n! U(q)

(3.2.58)

Proof. Lemma 3.2.12 applied to the complex (3.2.50) together with Proposition 3.2.11 delivers q q



 0,j (3.2.59) (−1)q−j dim H(2) (X, Lp ⊗ E)  (−1)q−j N j λ, p1
p . j=0

j=0

We estimate the right-hand side of (3.2.59). Theorem 3.2.5 and (3.2.11) show that N j (λ, 1p
p,U )  N j (λ, p1
p )  N j (3C0 λ + C2 p−1 , 1p
p,U ) .

(3.2.60)

Thus (3.2.38) and (3.2.60) imply for λ ∈ A j , N j (λ, p1
p )  pn I j (U, λ) + o(pn ) ,

for p → ∞ .

(3.2.61)

Let us consider λ < 1/(2C0 ) and δ > 0. For p > C2 /δ, we have N j (3C0 λ + C2 p−1 , p1
p,U )  N j (3C0 λ + δ, 1p
p,U ).

(3.2.62)

For 3C0 λ + δ ∈ A j , we get from (3.2.38) and (3.2.62), lim sup p−n N j (λ, p1
p )  lim p−n N j (3C0 λ + δ, p1
p,U ) = I j (U, 3C0 λ + δ) . p→∞

p→∞

Since νRj L is right-continuous in λ and bounded on U , we can use the Lebesgue dominated convergence theorem to let δ → 0. By (3.2.38) and the above equation, we get N j (λ, 1p
p )  pn I j (U, 3C0 λ) + o(pn ) ,

for p → ∞ .

(3.2.63)

By (3.2.59), (3.2.61) and (3.2.63), we deduce q
j=0

0,j (−1)q−j dim H(2) (X, Lp ⊗ E)  pn

q


 (−1)q−j I j U, λ(q − j) + o(pn ), j=0 (3.2.64)

where λ(r) = λ for r odd, λ(r) = 3C0 λ for all r even. By passing to the limit λ → +0 such that λ(r) ∈ ∪j
q A j , we obtain (3.2.57) invoking (3.2.39).


3.2. Abstract Morse inequalities for the L2 -cohomology

147

Remark 3.2.14. If X is compact, as in Remark 3.1.10, the hypotheses of Theorem 3.2.13 are trivially satisfied for m = n, so that Theorem 1.7.1 is a special case of Theorem 3.2.13. Remark 3.2.15. In a similar manner as in Theorem 3.2.13, if the fundamental estimate (3.2.2) holds in bi-degree (0, q) for any q  m, then for any r  m,  n √ n
pn 0,j −1 L j−r p (−1) dim H(2) (X, L ⊗ E)  rk(E) (−1)r 2π R + o(pn ). n! X(r) j=r (3.2.65) Back to the general situation, observe that as in (2.2.42), we can estimate from below the dimension of the space E

0 H(2) (X, Lp ⊗ E) = {s ∈ L20,0 (X, Lp ⊗ E) : ∂ p s = 0} ,

(3.2.66)

by applying (3.2.57) for q = 1, provided the fundamental estimate holds in bidegrees (0, 0) and (0, 1). We show in the sequel that we can get such an estimate if we only assume that the fundamental estimate (3.2.2) holds in bidegree (0, 1), although (3.2.57) for q = 1 might not hold. Theorem 3.2.16. Assume that the fundamental estimate (3.2.2) holds for (0, 1)forms. Let U be a relatively compact open set with smooth boundary such that K  U . Then, for p → ∞,  √ n pn −1 L 0 p R + o(pn ) . (3.2.67) dim H(2) (X, L ⊗ E)  rk(E) 2π n! U(
1) Note that the left side of (3.2.67) may be infinite. E

E

Proof. For λ > 0, let ∂ p,λ : E 0 (λ, 1p
p ) −→ E 1 (λ, p1
p ) be the restriction of ∂ p . E

By the definition of E 0 (λ, p1
p ), ∂ p,λ is a bounded operator, and by (3.2.5), E

0 (X, Lp ⊗ E), Ker(∂ p,λ ) = H(2) E

E

N 0 (λ, 1p
p ) = dim Ker(∂ p,λ ) + dim Im(∂ p,λ ).

(3.2.68)

E

Obviously dim Im(∂ p,λ )  N 1 (λ, p1
p ). For λ < 1/(2C0 ) and sufficiently large p, by Theorem 3.2.5, N 1 (λ, 1p
p ) is finite-dimensional, and by (3.2.68), 0 (X, Lp ⊗ E)  N 0 (λ, p1
p ) − N 1 (λ, 1p
p ) . dim H(2)

(3.2.69)

We repeat now the proof of Theorem 3.2.13 to estimate N 1 (λ, p1
p ) from above and N 0 (λ, p1
p ) from below, and observe that by the latter estimate, we applied Lemma 3.2.6 which does not require any hypothesis on the spectrum of
p on (0, 0)-forms.


148

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

3.3 Uniformly positive line bundles In this section, we apply the results from the previous one to the study of the L2 cohomology of complex manifolds satisfying certain curvature conditions. If X is a complete K¨ahler manifold and L a positive line bundle on X, the L2 estimates of Andreotti–Vesentini–H¨ ormander allow us to find a lot of sections of Lp ⊗ KX . We prove now a “compact perturbation” of this result. In this case the underlying complete metric is no more assumed to be K¨ahler, but we assume instead the existence of a uniformly positive line bundle outside a compact set. As an application we prove the Nadel–Tsuji theorem in Corollary 3.3.7 and the holomorphic Morse inequalities for hyperconcave manifolds in Theorem 3.4.9. Let (X, Θ) be a complex Hermitian manifold of dimension n and (E, hE ) be a holomorphic Hermitian vector bundle on X. We start by proving the Andreotti–Vesentini density lemma. For a linear operator R between Hilbert spaces, the graph-norm is defined by Dom(R)  s −→ s + Rs. E

Lemma 3.3.1 (Andreotti–Vesentini). Assume that (X, Θ) is complete. Let ∂ and E,∗ ∂ be the maximal extensions as in Lemma 3.1.1. Then Ω0,• 0 (X, E) is dense in E E,∗ E E,∗ E E,∗ and Dom(∂ ), Dom(∂ ), Dom(∂ ) ∩ Dom(∂ ) in the graph norms of ∂ , ∂ E E,∗ ∂ + ∂ , respectively. Proof. We first reduce the proof to the case of a compactly supported form s. The completeness of the metric implies the existence of a sequence {ϕk }k ⊂ C0∞ (X), such that 0  ϕk  1, ϕk+1 = 1 on supp(ϕk ), |dϕk |  1/2k for every k  1 and {supp(ϕk )}k exhaust X. To construct this sequence, we first construct an exhaustive function ϕ : X −→ R with |dϕ| < 1. This is done by smoothing the distance to a point (we can assume that X is connected). Next, consider a smooth function ρ : R −→ [0, 1] such that ρ = 0 on ] − ∞, −2], ρ = 1 on [−1, ∞[ and 0  ρ   2. Then ϕk = ρ(−ϕ/2k+1 ) satisfies the conditions above. E

Thus there exists C > 0 such that for s ∈ Dom(∂ ) ∩ Dom(∂ E E,∗ ϕk s ∈ Dom(∂ ) ∩ Dom(∂ ) and E

E,∗

(ϕk s) − ϕk ∂

), we have

E

∂ (ϕk s) − ϕk ∂ sL2  CsL2 /2k , ∂

E,∗

E,∗

sL2  CsL2 /2k .

(3.3.1)

Hence {ϕk s} converges to s in the graph norm. So to prove the assertion, we can start with a form s having compact support in X. But then the approximation in the graph norm follows from the Friedrichs lemma 3.1.3.
As a by-product of Friedrichs lemma, we obtain also:

3.3. Uniformly positive line bundles

149

Corollary 3.3.2. Assume that E

s ∈ L20,q (X, E), w ∈ L20,q+1 (X, E), ∂ s ∈ L20,q+1 (X, E) and ∂

E,∗

w ∈ L20,q (X, E).

E

Suppose also that s and w have compact support. Then ∂ s, w = s, ∂

E,∗

w.

Proof. We may assume that s, w have support in a trivialization patch diffeomorphic to R2n . We denote wε = w ∗ ρε as in Lemma 3.1.3. We have: E

E

E

∂ s, w = lim (∂ s)ε , wε  = lim ∂ sε , wε  = lim sε , ∂ ε→0

ε→0

= lim sε , (∂

E,∗

ε→0

w)ε  = s, ∂

E,∗

ε→0

E,∗

wε  (3.3.2)

w.


From (3.3.2), we get Lemma 3.3.2. E,∗

E,∗

Corollary 3.3.3. If (X, Θ) is complete, then ∂ H = ∂ , that is the Hilbert space E adjoint and the maximal extension of the formal adjoint of ∂ coincide. E,∗

E,∗

E,∗

Proof. It is clear that ∂ H ⊂ ∂ . Conversely if s ∈ Dom(∂ ), from Lemma E,∗ E,∗ sk −→ ∂ s. Then by 3.3.1, there exist sk ∈ Ω0,• 0 (X, E) with sk −→ s, ∂ E definition of ∂ , E

∂ w, sk  = w, ∂

E,∗

sk  ,

E

for w ∈ Dom(∂ ).

(3.3.3)

The limit of this equality for k → ∞ gives E

∂ w, s = w, ∂ E,∗

E,∗

E,∗

Thus s ∈ Dom(∂ H ) and ∂ H s = ∂

s ,

E,∗

E

for w ∈ Dom(∂ ).

s.




Corollary 3.3.4. If (X, Θ) is complete, then
E =
E |Ω0,• (X,E) is essentially selfadjoint. In particular the Gaffney and Friedrichs extensions coincide. E Proof. We will show that
E max is self-adjoint. Since
|Ω0,• (X,E) is symmetric, E E ∗ E E
max = (
)H . This implies the closure
of
satisfies ∗ E E ∗
E = ((
E )∗H )∗H = (
E max )H =
max = (
)H .

(3.3.4)

It follows that the closure
E is self-adjoint and
E is essentially self-adjoint. 2 E ∗(0,1) X) ⊗ Let s ∈ Dom(
E max ). Since
is elliptic, we have s ∈ H (X, Λ(T E,∗ E 2 E, loc) by Theorem A.3.4. Thus ∂ s, ∂ s ∈ L0,• (X, E, loc) and by Corollary

150

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

3.3.2, we can integrate by parts if s is multiplied with a smooth compactly supported function. Let ϕk be the family of functions defined in the proof of Lemma 3.3.1. We obtain: E

ϕk ∂ s2L2 + ϕk ∂ =

ϕ2k s,
E s

E,∗

E

E

E

s2L2 = ϕ2k ∂ s, ∂ s + s, ∂ (ϕ2k ∂ E

E,∗

− 2∂ϕk ∧ s, ϕk ∂ s + 2s, ∂ϕk ∧ (ϕk ∂ E

−k (2ϕk ∂ sL2 sL2 + 2ϕk ∂  ϕ2k s,
E max s + 2 E

−k  ϕ2k s,
E (ϕk ∂ s2L2 + ϕk ∂ max s + 2

E,∗

s)

E,∗

E,∗

s)

sL2 sL2 )

(3.3.5)

s2L2 + 2s2L2 ).

We get therefore the Stampacchia type inequality E

ϕk ∂ s2L2 + ϕk ∂

E,∗

s2L2 

1 1−k (ϕ2k s,
E s2L2 ). max s + 2 1 − 2−k

E

By letting k → ∞, we obtain ∂ s2L2 + ∂ E

∂ s, ∂

E,∗

E,∗

s2L2  s,
E max s, in particular

s are in L20,• (X, E). This implies E

E

s,
E max s1  = ∂ s, ∂ s1  + ∂

E,∗

s, ∂

E,∗

s1 

s, s1 ∈ Dom(
E max ),

(3.3.6) E

because the equality holds for ϕk s and s1 , and since we have ϕk s → s, ∂ (ϕk s) → E E,∗ E,∗ ∂ s and ∂ (ϕk s) → ∂ s in L2 by the proof of Lemma 3.3.1. An analogous calculation shows that the right-hand side of (3.3.6) equals 
E max s, s1 . Thus E s,
E max s1  = 
max s, s1  ,

s, s1 ∈ Dom(
E max )

(3.3.7)

E ∗ E E which means that
E max ⊂ (
max ) . But
max is the maximal extension of
so E E ∗ that
max = (
max ) .


Theorem 3.3.5. Let (L, hL ) be a holomorphic Hermitian line bundle on a complex Hermitian manifold √ (X, Θ). If (X, Θ) is complete and there exist K  X and C0 > 0 such that −1RL  C0 Θ on X
K, (i) then, for p → ∞, n,0 (X, Lp ) dim H(2)

pn  n!

 X(
1)

√

−1 L 2π R

n

+ o(pn ) ,

(3.3.8)

n,0 where H(2) (X, Lp ) is the space of (n, 0)-forms with values in Lp which are L2 with respect to any metric on X and the metric hL on L. (ii) Assume moreover that the Hermitian torsion T = [i(Θ), ∂Θ] is bounded and ∗ Rdet = RKX is bounded from below with respect to Θ. Then, for p → ∞, √ 

−1 L n pn 0 dim H(2) R (X, Lp )  + o(pn ), (3.3.9) n! X(
1) 2π 0 (X, Lp ) is the space of holomorphic sections in Lp which are L2 where H(2) with respect to the Hermitian form Θ on X and hL on L.

3.3. Uniformly positive line bundles

151

Remark 3.3.6. We can state Theorem 3.3.5 (i) without reference to the auxiliary L metric √ Θ, byL saying that (L, h ) is positive outside a compact set K and the curvature −1R defines a complete metric on X (by extending it to a metric over K). √

−1 L Proof. (i) Let us endow X with a Hermitian form ω0 such that ω0 = 2π R outside K, which is complete, for ω0  C0 Θ on X
K. Since ω0 is K¨ahler outside K, the Bochner–Kodaira–Nakano formula (1.4.44) gives √

p p 
L s, s  p [ −1RL , i(ω0 )]s, s , for s ∈ Ωn,1 (3.3.10) 0 (X
K, L ). √

By (1.4.61) (cf. also Problem 1.10), we know that [ −1RL , i(ω0 )]s, s  a1 (x)|s|2 , √ where a1  · · ·  an are the eigenvalues of −1RL with respect to ω0 . In our case, 0,1 p p a1 = ···= an = 2π outside K. Thus for s ∈ Ωn,1 0 (X
K,L ) = Ω0 (X
K,L ⊗ KX ),  1  Lp 2 Lp ,∗ 2 (3.3.11) ∂ sL2 + ∂ s2L2  sL2 . p Let U be any open set with smooth boundary such that K  U  X. Choose ρ ∈ C0∞ (X) such that ρ = 1 on a neighborhood of K and supp(ρ) ⊂ U . Applying (3.3.11) for (1 − ρ)s and using (3.2.8), we obtain the fundamental estimate (3.2.2) (with a slightly larger K) in bidegree (0, 1) for all s ∈ Ω00,1 (X, Lp ⊗ KX ). By Lemma 3.3.1, we infer that (3.2.2) holds true in bidegree (0, 1) with KX as E therein. We conclude by Theorem 3.2.16 that as p → ∞, √ 

−1 L n n,0 dim H(2) R (X, Lp )0  pn + o(pn ) . (3.3.12) 2π U(
1) n,0 Here H(2) (X, Lp )0 is the L2 -cohomology group with respect to the metric ω0 on X. For any (n, 0)-form s with values in L, and any metrics g T X , g1T X on X, with Riemannian volume forms dvX , dvX,1 , respectively, we have pointwise |s|2gT X dvX = |s|2gT X dvX,1 . Thus the L2 condition for (n, 0)-forms does not depend 1 on the metric on X, so n,0 n,0 H(2) (X, Lp )0 = H(2) (X, Lp )

(3.3.13)

where in the latter group, the L2 condition is with respect to an arbitrary metric on X. √ ∗ (ii) Due to the bound −1RL  C0 Θ on X
K and since RKX is bounded from below, there exists C1 > 0 such that for s ∈ Ω0,1 (X
K, Lp ), we have pointwise on X
K,  p ∗  RL ⊗KX (wj , w k )w k ∧ iwj s, s  (pC0 − C1 )|s|2 . (3.3.14) As T = [i(Θ),∂Θ] is bounded, from the Bochner–Kodaira–Nakano formula (1.4.64) and (3.3.14), there exists C > 0 such that for s ∈ Ω00,1 (X
K, Lp ), p

3
L s, s  (2pC0 − C)s2L2 . We can thus proceed just as in the proof of (i).

(3.3.15)


152

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

Corollary 3.3.7 (Nadel–Tsuji). Let (X, Θ) be a complete K¨ ahler manifold with Rdet  −Θ. Then we have the following estimate: 

√−1 KX n pn p 0 dim H(2) (X, KX )  R + o(pn ), (3.3.16) n! X 2π where RKX = −Rdet is the curvature of the canonical bundle KX equipped with the metric induced from Θ. Proof. At first, T = 0 as Θ is K¨ahler. Since (3.3.14) holds for L = KX , (3.3.15) still holds. We get (3.3.16) by the same argument in Theorem 3.3.5 (ii).


3.4 Siu–Demailly criterion for isolated singularities In this section, we will prove Theorems 3.4.10, 3.4.14 (generalizations of the Siu– Demailly criterion for complex compact spaces with isolated singularities) by applying the holomorphic Morse inequalities and Theorems 3.3.5 and 3.4.9. It is useful to consider a more general class of manifolds which include as a special case the regular locus of compact complex spaces with isolated singularities. Definition 3.4.1. A complex manifold X is called hyperconcave or hyper 1-concave if there exists a smooth function ϕ : X −→] − ∞, u ] where u ∈ R, such that Xc := {ϕ > c}  X for all c ∈] − ∞, u ] and ϕ is strictly plurisubharmonic outside a compact set. Example 3.4.2. (i) Let Y be a compact complex manifold, S a complete pluripolar set. By definition, S is complete pluripolar, if there exists a neighborhood W of S and a plurisubharmonic function ψ : W −→ [−∞, ∞[ such that S = ψ −1 (−∞). If ψ is strictly plurisubharmonic, then X = Y
S is hyperconcave. Conversely, it is shown in Theorem 6.3.9 that any hyperconcave manifold M is biholomorphic to a complement of a pluripolar set in a compact manifold. (ii) Let X be a compact complex space with isolated singularities. Then the regular locus Xreg is hyperconcave. Indeed, let {Uα } be pairwise disjoint neighborhoods of the singular points {xα } and let ια : Uα −→ CNα be holomorphic embeddings. We may assume that the singular points are mapped to the origin, ια (xα ) = 0. The function z −→ log |z|2 is strictly plurisubharmonic on CNα . By taking its pull-back to each Uα through ια , we obtain a strictly plurisubharmonic function on Xreg ∩(∪α Uα ) which tends to −∞ at the singular points. By extending this function to Xreg by means of a partition of unity, we get a function as in the definition. (iii) If X is a complete K¨ahler manifold of finite volume and bounded negative sectional curvature, X is hyperconcave as shown by Siu–Yau in Theorem 6.3.8. Definition 3.4.3. We will call a connected complex manifold X Andreotti pseudoconcave if there exists a non-empty open set M  X with smooth boundary ∂M

3.4. Siu–Demailly criterion for isolated singularities

153

such that the Levi form of M restricted to the analytic tangent space T (1,0) ∂M has at least one negative eigenvalue at each point of ∂M . Immediate examples of Andreotti pseudoconcave manifolds are connected qconcave manifolds, q  n − 1. Indeed, let X be a connected q-concave manifold with associated function ϕ : X −→]u, v] as in Definition B.3.2. The definition function of Xc = {ϕ > c}  X is c − ϕ and for c sufficiently close to u, the Levi form of c − ϕ has at least n − q + 1 negative eigenvalues in a neighborhood of ∂Xc . Thus, the restriction of the Levi form on the analytic tangent space T (1,0) ∂Xc has at least n − q  1 negative eigenvalues. Lemma 3.4.4. In Definition 3.4.3, for each point x ∈ M , we can choose holomorphic coordinates (Ux , x) and a coordinate polydisc P (x, r) ⊂ Ux centered at x such that the Silov boundary S(P (x, r)) = {y ∈ Ux : |yi | = r} ⊂ M . Proof. Let ρ be the defining function of M near x = 0 as in Definition B.3.7. After a change of coordinates, we may assume that Lρ,0 (v, v) < 0 for v = (1, 0, . . . , 0) and ∂ρ(0) = dzn . The Taylor expansion of ρ gives:

 ρ(z) = ρ(0) + 2 Re ∂ρ(0) · z + Qρ (z, z) + Lρ,0 (z, z) + o(|z|2 ) , (3.4.1) where Qρ (z, z) =



∂2 ρ j,k ∂zj ∂zk (0)zj zk .

Therefore

∂2ρ (0)|z1 |2 ∂z1 ∂z 1 + O(|z1 ||zn |) + O(|zn |2 ) + o(|z1 |2 + |zn |2 ) ,

ρ(z1 , 0, . . . , 0, zn ) = 2 Re(h(z)) +

(3.4.2)

∂h (0) = 0, the set where h(z) = zn + Qρ (z, z) with z = (z1 , 0, . . . , 0, zn ). Since ∂z n n U = {z ∈ C : h(z) = 0, z2 = · · · = zn−1 = 0} is a complex submanifold of dimension one in the neighborhood of 0. Let B(0, ε) be the ball with center 0 2 ρ and radius ε. Since ∂z∂1 ∂z (0) < 0, we deduce that there exists ε > 0 such that 1 U ∩ B(0, ε)
{0} ⊂ M . In the holomorphic coordinates w1 (z) = h(z), w2 = z2 , . . . , wn = zn , U is a complex plane through 0. Moreover, for η > 0 small enough {w1 = . . . = wn−1 = 0, |wn | < η} ∩ B(0, ε)
{0} ⊂ M . The lemma follows now easily by a continuity argument.


Theorem 3.4.5. Let X be an Andreotti-pseudoconcave manifold. Then for any holomorphic line bundle L on X, we have dim H 0 (X, L) < ∞. Moreover, there exists C > 0 such that for p  1, (3.4.3) dim H 0 (X, Lp )  Cp p , where p is the maximal rank of the Kodaira map Φp : X
Bl|H 0 (X,Lp )| → P(H 0 (X, Lp )∗ ). The field of meromorphic functions MX (X) is an algebraic field of transcendence degree a(X)  dim X. Proof. First we choose an open set M  X as in Definition 3.4.3. As in the proof of Lemma 2.2.6, the set of points where the Kodaira map Φp has rank less than p

154

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

is a proper analytic set of X, so {x ∈ X, rkx Φp = p for any p  1} is dense in X. By Lemma 3.4.4, there exist x1 , . . . , xm ∈ M as in Lemma 2.2.1 such that Φp has −1 rank p at each xj for any p  1, M ⊂ ∪m ) , L is trivial on coordinate i=1 P (xi , ri e polydiscs P (xi , 2ri ) and the Silov boundary S(P (xi , ri )) ⊂ M , for i = 1, . . . , m. Then the proof goes through as in the compact case by observing that there exists q ∈ {1, . . . , m} such that for some y ∈ S(P (xq , rq )) with |sq (y)| = s. Thus y ∈ M and we can find j = q such that y ∈ P (xj , rj e−1 ). Now the same argument as in the proof of Lemma 2.2.6 gives (3.4.3). The proof of the last part is completely analogous to that of Theorem 2.2.11.
Remark 3.4.6. Under the hypothesis of Theorem 3.4.5 we observe that Siegel’s lemma 2.2.6 holds for the adjoint bundles, that is there exists C > 0 such that for and p ∈ N∗ , dim H 0 (X, Lp ⊗ KX )  Cpp , (3.4.4) where p is the maximal rank of the Kodaira map Φp associated to H 0 (X, Lp ⊗ KX ). Theorem 3.4.5 allows us to extend the notion of Moishezon manifold for the case of Andreotti-pseudoconcave manifolds. Thus, an Andreotti-pseudoconcave manifold is called Moishezon if a(X) = dim X. Theorem 3.4.7. Let X be an Andreotti-pseudoconcave manifold and L be a holomorphic line bundle over X. If there exists p ∈ N∗ such that rk Φp = dim X, where Φp is the Kodaira map associated to H 0 (X, Lp ) or H 0 (X, Lp ⊗ KX ), then X is Moishezon. Proof. By using Theorem 3.4.5, the proof is the same as that of Theorem 2.2.15.
Theorem 3.4.8 (Levi’s removable singularity theorem). Let X be a reduced and irreducible complex space with an analytic subset A ⊂ X of codimension at least 2 at every point. Then every meromorphic function f ∈ MX (X
A) has a unique extension f˜ ∈ MX (X). After this study of the function theory on pseudoconcave manifolds, we return to the holomorphic Morse inequalities. As in Section 1.7, we will use the notation X(q), X( q), X( q); when we need to make precise the metric hL as in Definition 2.3.17, (2.3.47), we denote them by X(q, c1 (L, hL )), X( q, c1 (L, hL )), X( q, c1 (L, hL )), respectively. Theorem 3.4.9. Let X be a hyperconcave manifold of dimension n carrying a holomorphic line bundle (L, hL ) which is semi-positive outside a compact set. Then, for p → ∞  √ n pn n,0 −1 L (X, Lp )  R + o(pn ) , (3.4.5) dim H(2) 2π n! X(
1) where the L2 condition is with respect to hL on L and any metric on X.

3.4. Siu–Demailly criterion for isolated singularities

155

Proof. Let us consider a proper smooth function ϕ : X −→] − ∞, 0 [ which is strictly plurisubharmonic outside a compact set K  X. The fact that ϕ goes to −∞ to the ideal boundary of X allows us to construct a complete metric g T X on X. Let χ be the smooth function on X defined by χ = − log (−ϕ).

(3.4.6)

Note that ∂∂χ =

∂∂ϕ ∂ϕ ∧ ∂ϕ + , −ϕ ϕ2

and

∂ϕ ∧ ∂ϕ = ∂χ ∧ ∂χ. ϕ2

(3.4.7)

√ √ √ −1∂χ ∧ ∂χ is semi-positive on X
K, thus −1∂∂χ As −1 ∂∂ϕ −ϕ is positive and is positive on X
K. We can now patch ∂∂χ and an arbitrary positive (1, 1)-form on X by using a smooth partition of unity, to get a positive (1, 1)-form Θ on X such that √ √ Θ = −1∂∂χ = − −1∂∂ log (−ϕ), on X
K1 with K  K1  X. (3.4.8) The metric associated to Θ is the desired g T X . In fact, from (3.4.7) and (3.4.8), we get |dχ|gT X  2 on X
K1 . (3.4.9) Since χ : X −→ R is proper, (3.4.9) ensures that g T X is complete. Indeed, (3.4.9) entails that χ is Lipschitz with respect to the geodesic distance induced by g T X , so any geodesic ball must be relatively compact. √ Note that g T X is obviously K¨ ahler on X
K1 . Let us assume −1RL  0 on L X
K1 (we stretch K1 if necessary). We equip L with the metric hL ε = h exp(−εχ) L and the curvature relative to the new metric satisfies c1 (L, hε )  ε Θ on X
K1 . L L (L, hL ε ) therefore in the conditions of Theorem 3.3.5. Since hε  Ch for some C > 0, there is an injective morphism n,0 n,0 p L H(2) (X, Lp , Θ, hL ε ) −→ H(2) (X, L , Θ, h ).

(3.4.10)

n,0 By Theorem 3.3.5 for the space H(2) (X, Lp , Θ, hL ε ) and (3.4.10), for any K1  L U  X, as c1 (L, hε )) is positive on X
U , we get  1 n,0 n (X, Lp , Θ, hL )  c1 (L, hL (3.4.11) lim inf p−n dim H(2) ε) . p→∞ n! U(
1,c1 (L,hLε ))

We let now ε → 0 in (3.4.11); since hL ε converges uniformly together with its L derivatives to hL on compact sets, we see that we can replace hL in ε with h the right-hand side of (3.4.11). Now, X( 1) = X(0) ∪ X(1). By hypothesis X(1) ⊂ K and on X(0) the integrand is positive. Hence we can let U exhaust X to get (3.4.5).


156

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

Theorem 3.4.9 implies the first main result of this section. Let us remark that Definition 2.2.8 and Theorem 2.2.11 carry over compact irreducible spaces. Such a space is called Moishezon if its algebraic and complex dimensions are equal. Theorem 3.4.10. Let X be a compact irreducible complex space of dimension n  2 with at most isolated singularities, and let (L, hL ) be a holomorphic Hermitian line bundle on Xreg which is semi-positive in a deleted neighborhood of the singular locus Xsing and satisfies Demailly’s condition:  c1 (L, hL )n > 0. (3.4.12) Xreg (
1)

Then X is Moishezon. Proof. By Example 3.4.2 (ii), Xreg here is a hyperconcave manifold. We deduce from Theorem 3.4.9 and (3.4.12) that dim H 0 (Xreg , Lp ⊗ KX )  Cpn for some C > 0 and p sufficiently large. By Theorem 3.4.5 and Remark 3.4.6, we deduce that the rank of the Kodaira map of H 0 (Xreg , Lp ⊗ KX ) is maximal. Theorem 3.4.7 entails that there exists dim X independent meromorphic functions on Xreg . By Theorem 3.4.8, we conclude that these functions extend to dim X independent meromorphic functions on X. Thus X is Moishezon.
Let us define the “adjoint” volume of a line bundle L over a complex manifold M of dimension n by vol∗ (L) = lim sup n! p−n dim H 0 (M, Lp ⊗ KM ).

(3.4.13)

p→∞

Corollary 3.4.11. Let L be a holomorphic line bundle over Xreg , where X is a compact complex space with only isolated singularities, dim X = n. (i) If L is semi-positive outside a compact set, then   ∗ L n 0 c1 (L, h )  vol (L) − c1 (L, hL )n < +∞. Xreg (0)

(3.4.14)

Xreg (1)

(ii) If ψ : Xreg −→ R is a smooth function which is plurisubharmonic outside a compact set, then   n n

√

√ −1∂∂ψ  − −1∂∂ψ < +∞, (3.4.15) Xreg (0)

Xreg (1)

where Xreg (0) is the open set where ψ is strictly plurisubharmonic. Proof. By Example 3.4.2 (ii) and Remark 3.4.6, vol∗ (L) < +∞. Under the condition (i), since Xreg (1) is relatively compact by the hypothesis on the semi-positivity of L, we have −∞ < Xreg (1) c1 (L, hL )n  0. Now (3.4.14) is from (3.4.5). To prove (ii), we apply (i) to the trivial bundle L endowed with the metric exp(−ψ) and we use the obvious fact that vol∗ (L) = 0, if L is trivial.


3.4. Siu–Demailly criterion for isolated singularities

157

We recall now the notion of Hermitian form (metric) on a singular space. Definition 3.4.12. Let us consider a covering {Uα } of a complex space X and embeddings ια : Uα → CNα . A Hermitian form on X is a Hermitian form Θ (i.e., Θ is a positive (1, 1)-form) on Xreg which on every open set Uα as above is the pullback of a Hermitian form on the ambient space CNα , Θ = ι∗α Θα . The Hermitian form is called distinguished , if in the neighborhood of the singular points, Θα is the Euclidean K¨ ahler form. A Hermitian form on a singular space is constructed as usual by a partition of unity argument. If the singularities are isolated, we can assume that the Hermitian form is distinguished. Definition 3.4.13. Let L be a holomorphic line bundle on a complex space X. Assume that L|Uα is the inverse image by ια of the trivial line bundle Cα on CNα . Consider Hermitian metrics hα = e−2ϕα on Cα such that ι∗α hα = ι∗β hβ on Uα ∩ Uβ ∩ Xreg . The system hL = {ι∗α hα } is called a Hermitian metric on L. It clearly induces a Hermitian metric on L|Xreg . We shall allow our metrics to be 1 Nα singular at the singular points, that ) and ϕα √is smooth outside α ∈ Lloc (C √ is, ϕ L ια (Xsing ). The curvature current −1R is given in Uα by ι∗α (2 −1∂∂ϕα ), which on Xreg agrees with the curvature of the induced metric. The following Theorem 3.4.14 is the second main result of this section. Theorems 3.4.10 and 3.4.14 show that Demailly’s criterion generalizes to singular spaces with at most isolated singularities under mild growth conditions of the curvature near the singular set. Theorem 3.4.14. Let X be a compact irreducible complex space of dimension n  2 and with isolated singularities. Assume that a holomorphic line bundle L is defined over all X, the Hermitian metric hL on L may be singular at Xsing , but the curvature current RL is dominated by the Euclidean metric near Xsing (cf. (3.4.18)) and moreover condition (3.4.12) is fulfilled on Xreg . Then X is Moishezon. Proof. In order to perform analysis on Xreg , we introduce first a good exhaustion function and a complete metric. Let us consider a coherent ideal I ⊂ OX  → X of the ideal I with supp(OX /I ) = Xsing such that the blow-up π : X is a resolution of singularities of X. Let E be the exceptional divisor and Ei its ∗ smooth irreducible components such that E = i ci Ei , where ci ∈ N . Then −1 (I ) equals the invertible sheaf OX (F ), where i Ei has normal crossings and π F = OX (−E). Lemma 3.4.15. There exists a Hermitian metric hF on F which has positive cur of E. vature in a neighborhood U Proof. Let us consider a small neighborhood Uy of y ∈ Xsing which does not contain any other singular point and ι : Uy −→ CN as an analytic an embedding N N 2 2 subset of a ball Bε = {z ∈ C : j |zj | < ε }. By definition of the blow-up,

158

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

 , F ) which define a map Φ : U  −→ C Pk , there exist sections s0 , . . . , sk ∈ H 0 (U k N  −→ Bε × C P is an embedding. Φ(x) = [s0 (x), . . . , sk (x)], such that π × Φ : U Moreover, F = Φ∗ OC Pk (1) = (π × Φ)∗ σ ∗ OC Pk (1), with σ : BεN × C Pk −→ C Pk is the projection. We can endow σ ∗ OC Pk (1) with a metric with positive curvature on BεN × C Pk and pulling back this metric by π × Φ, we obtain the desired hF .
Let us consider a canonical section s of F = OX (−E), and denote by |s|hF the pointwise norm of s. By the Poincar´e–Lelong formula (2.3.6), √ −1 2π ∂∂

log |s|2hF = [Div(s)] − c1 (F, hF ).

(3.4.16)


E and converges Hence the function log |s|2hF is strictly plurisubharmonic on U  with compact support in U  which to −∞ on E. By using a smooth function on X 
E  Xreg such that equals one near E, we construct a smooth function χ on X 
E. χ = − log(− log |s|2hF ) on U Let Θ be a distinguished Hermitian form on X, in particular Θ is K¨ ahler near Xsing . We consider then the positive (1, 1)-form on Xreg , √ −1 ∂∂χ, ω0 = A1 Θ + 2π

(3.4.17)

where A1 > 0 is chosen sufficiently large (to ensure that ω0 is positive away from the open set where ∂∂χ is positive definite, cf. (3.4.7)). The metric g0T X associated to ω0 is complete by the same argument as in the proof of Theorem 3.4.9 (see (3.4.9)). Lemma 3.4.16. The metric ω0 has finite volume. √  → X as above. Observe first that the integral of ( −1∂∂χ)n Proof. Let π : X √  is finite by Corollary 3.4.11 (ii). Since −1∂∂χ is a strictly positive current on U √  there exists C > 0 such that π ∗ (Θ)  −1 C∂∂χ on and π ∗ (Θ) is smooth on U, √ 
E. Therefore, ω n is dominated on U 
E by a multiple of ( −1∂∂χ)n . The U 0 integral of ω0n on Xreg must be finite.
We shall suppose in the sequel that the curvature current of the Hermitian ahler form, i.e., for ϕα in Definition metric hL on L is dominated by the Euclidean K¨ 3.4.13, there exists C > 0 with −CωE 

√ −1∂∂ϕα  CωE ,

ωE =

√ −1 j dzj 2

∧ dz j .

(3.4.18)

We assume that U is a small enough neighborhood√of the singular set so that on U , there are well-defined potentials ρ, ρ1 for 2πΘ, −1RL from ambient spaces, i.e., √ √ √ −1 ∂∂ρ, −1RL = −1∂∂ρ1 . (3.4.19) Θ= 2π

3.4. Siu–Demailly criterion for isolated singularities

159

By suitably cutting-off, we may define a function ψ ∈ C ∞ (Xreg ) such that near Xsing , ψ = χ − ρ1 + A1 ρ. (3.4.20) √ Remark that, since −1RL is bounded above by a continuous (1, 1)-form near Xsing , the potential −ρ1 is bounded above near the singular set. This holds true for ρ too (it is smooth) so that ψ tends to −∞ at the singular set Xsing . Let us consider a smooth function γ : R −→ R such that 0 if u  0 , γ(u) = u if u  −1 , and the functions γν : R −→ R given by γν (u) = γ(u + ν) for ν ∈ N.

 L L The curvature R(L,hν ) of the metric hL ν = h exp − γν (ψ) is L

R(L,hν ) = R(L,h

L

)

+ γν (ψ)∂∂ψ + γν (ψ)∂ψ ∧ ∂ψ .

(3.4.21)

Since ψ goes to −∞ when we approach the singular set, we may choose ν0 ∈ N such that for ν  ν0 we have {ψ  −ν − 1} ⊂ U . On the set {ψ  −ν − 1}, we have γν (ψ) = ψ + ν, so that γν (ψ) = 1 and γν (ψ) = 0, by (3.4.17) and (3.4.19), √ √ −1 (L,hLν) −1 L L c1 (L, hν ) = R ∂∂ψ = ω0 . = c1 (L, h ) + (3.4.22) 2π 2π Set Uν = {ψ  −ν}, Uν = {−ν − 2  ψ  −ν}, Uν = Uν ∪ Uν .

(3.4.23)

Then Uν = {ψ  −ν − 2} is a compact set. By Theorem 3.3.5 and the same argument in (3.4.11), as c1 (L, hL ν ) is positive on Xreg
Uν , we get for p → ∞,  pn n n dim H n,0 (Xreg , Lp , hL )  c1 (L, hL (3.4.24) ν ) + o(p ) . n! Uν (
1,c1 (L,hLν )) L On Uν , we have γν (ψ) = 0 and c1 (L, hL ν ) = c1 (L, h ) . We infer that



 Uν
(
1,c1 (L,hL ν ))

n c1 (L, hL ν) =



Xreg (
1,c1

(L,hL ))

−→

1Uν
c1 (L, hL )n

c1 (L, hL )n

as ν → ∞. (3.4.25)

Xreg (
1,c1 (L,hL ))

Thus it suffices to show that  Uν

(
1,c1 (L,hL ν ))

n c1 (L, hL ν ) −→ 0 as ν → ∞.

(3.4.26)

160

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

As γν , γν are bounded, from (3.4.17), (3.4.19) and (3.4.21), we know that c1 (L, hL ν) is dominated by ω0 on Uν . Since vol (Uν ), the volume of Uν with respect to ω0 , tends to 0 as ν → ∞, we get (3.4.26). Hence from (3.4.12), (3.4.24), (3.4.25) and (3.4.26), there exists C > 0 such that for p large enough dim H 0 (Xreg , Lp ⊗ KX )  Cpn ,

(3.4.27)

so that Xreg has n independent meromorphic functions which can be extended to X by Theorem 3.4.8. The proof of Theorem 3.4.14 is finished.


3.5 Morse inequalities for q-convex manifolds In this section, we establish the holomorphic Morse inequalities for the q-convex and weakly 1-complete manifolds. We start by defining the Kodaira Laplacian with ∂-Neumann boundary conditions. Let M be a relatively compact smooth domain in a complex manifold X of dimension n and let (E, hE ) be a holomorphic Hermitian vector bundle on X. E

Let ∂ : Ω0,• (X, E) → Ω0,• +1 (X, E) be the Dolbeault operator; we denote E,∗ E E its formal adjoint. Let ∂ : Dom(∂ ) ⊂ L20,• (M, E) → L20,• +1 (M, E) be by ∂ E,∗

its maximal extension on L20,• (M, E) (cf. Lemma 3.1.1). Let ∂ H E

space adjoint of ∂ on M . We introduced the space B Proposition 1.4.19, we have

E E,∗ ∂ s 1 , s2 = s 1 , ∂ s 2 ,

0,q

be the Hilbert

(M, E) in (1.4.70). By

for s1 ∈ Ω0,q (M , E) , s2 ∈ B 0,q+1 (M, E).

(3.5.1)

We consider the operator . / E Dom(
E ) := s ∈ B 0,q (M, E) : ∂ s ∈ B 0,q+1 (M, E) , E E,∗


E s = ∂ ∂

s+∂

E,∗ E

(3.5.2)

for s ∈ Dom(
E ) ,

∂ s,

which by (3.5.1) is positive. Let en be the inward pointing unit normal at ∂M . We decompose en as (1,0) (0,1) en = en +en ∈ T (1,0) X ⊕T (0,1)X. Then the boundary conditions of Dom(
E ) in (3.5.2) are called ∂-Neumann boundary conditions and are given by . E Dom(
E ) = s ∈ Ω0,• (M , E); ie(0,1) s = ie(0,1) ∂ s = 0 n

n

/ on ∂M .

(3.5.3)

An extension of the associated quadratic form Q is E

E

Dom(Q) := B 0,q (M, E), Q(s1 , s2 ) := ∂ s1 , ∂ s2  + ∂

E,∗

s1 , ∂

E,∗

s2  . (3.5.4)

3.5. Morse inequalities for q-convex manifolds

161

It is easy to see that Q is closable and its closure is the form Q given by (cf. (C.1.7)) . Dom(Q) = s ∈ L20,• (M, E) : there exists {sk } ⊂ B 0,• (M, E), with / E E,∗ lim sk = s and {∂ sk L2 }, {∂ sk L2 } are Cauchy sequences , (3.5.5) k→∞

E  E,∗ Q(s, s) = lim ∂ sk 2L2 + ∂ sk 2L2 , for s ∈ Dom(Q) . k→∞

By Proposition C.1.5, the self-adjoint operator associated to Q is the Friedrichs extension of
E and is called the Kodaira Laplacian with ∂-Neumann boundary conditions. We still denote this operator by
E . The ∂-Neumann problem for a domain M consists in proving Theorem 3.1.8 for the Kodaira Laplacian with ∂-Neumann boundary conditions and establishing regularity results up to the boundary ∂M for the Green operator. We have an analogue of the Andreotti–Vesentini lemma 3.3.1. E

E

Lemma 3.5.1. Ω0,• (M , E) is dense in Dom(∂ ) in the graph-norm of ∂ , and E,∗ E E,∗ B 0,q (M, E) is dense in Dom(∂ H ) and in Dom(∂ ) ∩ Dom(∂ H ) in the graphE,∗ E E,∗ norms of ∂ and ∂ + ∂ , respectively. The proof is again based on the Friedrichs lemma 3.1.3, but a more delicate convolution process in the tangential direction to ∂M is required. Proposition 3.5.2. The Kodaira Laplacian with ∂-Neumann conditions on M coincides with the Gaffney extension (3.1.4) of the Kodaira Laplacian. Proof. By Proposition C.1.4, Definition C.1.7 and Proposition 3.1.2, it suffices to show that the quadratic forms (3.5.5) and (3.1.5) are the same. But this results immediately from Lemma 3.5.1.
From now until Theorem 3.5.9, we suppose that X is a q-convex manifold of dimension n and let  : X −→ R be an exhaustion function which is q-convex outside a compact set K ⊂ X (cf. Definition B.3.2). Let (E, hE ) be a holomorphic Hermitian vector bundle on X. Let us consider a smooth sub-level set Xc = { < c} such that K ⊂ Xc . We fix u < c < v such that K ⊂ Xu . We choose now a convenient metric on X. Lemma 3.5.3. For any C1 > 0 there exists a metric g T X (with Hermitian form Θ) on X such that for any j  q, and any Hermitian vector bundle (E, hE ) on X, (∂∂)(wl , wk )w k ∧ iwl s, s  C1 |s|2 ,

s ∈ Ω0,j 0 (Xv
X u , E) .

(3.5.6)

Proof. Fix x ∈ Xv
X u . Consider local coordinates (U, z1 , . . . , zn ) centered at x, such that ∂∂ is positive definite on the subbundle of T X|U generated by ∂ ∂ TX = ∂zq , . . . , ∂zn . Let C > 0 be given. For ε > 0, consider the metric g n −1 for i < q and εi = ε for i  q. Let d1  · · ·  dn i=1 εi dzi ⊗ dz i , where εi = ε

162

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

be the eigenvalues of ∂∂ with respect to g T X . Then for ε > 0 sufficiently small, we have d1  −C −1 and dq  C. For any s ∈ Ω0,j 0 (U, E), we obtain as in (1.5.19), (∂∂)(wl , w k )w k ∧ iwl s, s  (d1 + · · · + dj )|s|2  ((j − q + 1)C − (q − 1)C −1 )|s|2 .

(3.5.7)

By choosing C > 0 large enough, we obtain (3.5.6) for s ∈ Ω0,j 0 (U, E) and j  q. Patching metrics as one constructed above with the help of a partition of unity, we conclude the proof of Lemma 3.5.3.
From now on, we use the metric g T X on X defined in Lemma 3.5.3. For a convex increasing function χ ∈ C ∞ (R), we set E −χ() hE , χ =h e

Eχ = (E, hE χ ).

(3.5.8)

Lemma 3.5.4. For any C2 > 0, there exists a constant C3 > 0 such that for χ  ()  C3 on Xv
X u and for any s ∈ B 0,j (Xc , E), with supp(s) ⊂ Xv
X u , j  q, we have

E  E∗ (3.5.9) s2L2  C2 ∂ s2L2 + ∂ s2L2 , where the L2 norm is taken with respect to g T X in Lemma 3.5.3 and hE χ on Xc . Proof. To simplify the notation, set M := Xc . From (3.5.8), REχ = RE + χ  ()∂∂ + χ  ()∂ ∧ ∂ .

(3.5.10) √ −1χ  ()∂ ∧ ∂ is positive semi-definite. By Lemma 3.5.3 and (3.5.10), there exists C4 , C5 > 0 such that for s ∈ Ω0,j 0 (Xv
X u , E), j  q,   ∗ REχ ⊗KX (wl , w k )w k ∧ iwl s, s  (χ  ()C4 − C5 ) |s|2 . (3.5.11) We denote simply by |d| := |d|gT ∗ X , 1 = ( − c)/|d| near ∂M . Then the boundary term in (1.4.84) is L1 (s, s). Note that ∂∂1 =

1 1 1 1 ∂∂ + ∂ ∧ ∂ + ∂ ∧ ∂ + ( − c)∂∂ . |d| |d| |d| |d|

(3.5.12)

Thus for s ∈ B 0,j (M, E), j  q, as  − c = ie(0,1) s = 0 on ∂M , by (3.5.6) and n (3.5.12), we get L1 (s, s) =

1 C1 (∂∂)(wl , wk )w k ∧ iwl s, s  |s|2 , |d| |d|

From (3.5.13), we deduce that  L1 (s, s) dv∂M  0 , ∂M

on ∂M.

for s ∈ B 0,j (M, E) , j  q.

(3.5.13)

(3.5.14)

3.5. Morse inequalities for q-convex manifolds

163

By the Bochner–Kodaira–Nakano formula with boundary term (1.4.84) and (3.5.11), (3.5.14) and the Hermitian torsion T is bounded, we deduce for any j  q, s ∈ B 0,j (M, E) with supp(s) ⊂ Xv
X u ,   3 E 2 E∗ (χ  ()C4 − 2C5 ) |s|2 dvX , (3.5.15) ∂ sL2 + ∂ s2L2  2 Xc X u


and Lemma 3.5.4 follows.

Theorem 3.5.5. Let X be a q-convex manifold and let c ∈ R such that K ⊂ Xc . E E Then for any j  q, ∂ : Dom(∂ ) ∩ L20,j (Xc , E) → L20,j+1 (Xc , E) has closed 0,j range and dim H(2) (Xc , E) < ∞. 0,j Proof. The L2 condition in the definition of H(2) (Xc , E) is understood with respect

to an arbitrary metric on X, since all metrics are equivalent on X c . By applying (3.5.9) to ζs where s ∈ B 0,j (Xc , E), j  q and ζ is a cutoff function with ζ = 1 near ∂Xc and supp(ζ) ⊂ Xv
X u , we deduce that the fundamental estimate (3.1.27) holds for any s ∈ B 0,j (Xc , E). Since B 0,j (Xc , E) E E∗ is dense in Dom(∂ ) ∩ Dom(∂ ), we see that (3.1.27) is satisfied. Theorem 3.5.5 follows therefore from Theorem 3.1.8.
Using weight functions η(ρ), with a rapidly growing convex function η : ] − ∞, v[→ R, in order to temper the growth of forms at the boundary ∂Xv , we can prove: Theorem 3.5.6 (H¨ ormander). Let X be a q-convex manifold, and let c ∈ R such that K ⊂ Xc . For j  q, and c < v, the restriction morphism 0,j H 0,j (Xv , E) −→ H(2) (Xc , E)

(3.5.16)

is an isomorphism, where H 0,j (Xv , E) is the Dolbeault cohomology. For any holomorphic vector bundle E on X, the sheaf cohomology of OX (E) over an open set U ⊂ X is denoted by H j (U, E). By the Dolbeault isomorphism (Theorem B.4.4), we know that H j (Xv , E)  H 0,j (Xv , E). Using similar methods as in Theorem 3.5.6, we get: Theorem 3.5.7 (Andreotti–Grauert). Let X be a q-convex manifold, and let c ∈ R such that K ⊂ Xc . The following assertions hold true for j  q: (i) (approximation theorem) The restriction morphism H j (X,E) −→ H j (Xc ,E) is an isomorphism and consequently (ii) (finiteness theorem) dim H j (X, E) < +∞. We are now ready to prove the holomorphic Morse inequalities.

164

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

Theorem 3.5.8. Let X be a q-convex manifold, and let (L, hL ), (E, hE ) be holomorphic Hermitian vector bundles with rk(L) = 1. L −χ() be the metric on For any smooth sub-level set Xc ⊃ K, let hL χ = h e  L defined in (3.5.8) with χ ()  C0 > 0 on Xv
Xu . Then as p → ∞, for any r  q, n


(−1)j−r dim H j (X, Lp ⊗ E)

j=r

pn  rk(E) n!

 Xc (r,c1 (L,hL χ ))

n n (−1)r c1 (L, hL χ ) + o(p ). (3.5.17)

√

−1 L If c1 (L, hL ) = 2π R is semi-positive outside a compact set, we can replace the  right-hand side integral by X(r,c1 (L,hL )) c1 (L, hL )n .

Proof. By Theorems 3.5.6 and 3.5.7, for any v > c, j  q, 0,j (Xc , Lp ⊗ E)  H j (Xv , Lp ⊗ E)  H j (X, Lp ⊗ E). H(2)

(3.5.18)

An easy modification of the proof of Lemma 3.5.4 shows that there exists C2 > 0 such that for any j  q, s ∈ B 0,j (Xc , Lp ⊗ E) with supp(s) ⊂ Xv
X u , s2L2 

 C2 E 2 E,∗ ∂ p sL2 + ∂ p s2L2 , p

(3.5.19)

where the L2 condition is taken with respect to the metrics g T X on Xc and hL χ, E h . We see thus that (3.2.2) is satisfied for any j  q, so by Remark 3.2.15 and (3.5.18), for any r  q, (3.5.17) follows. In order to justify the last assertion of Theorem 3.5.8, let us choose u < c such that L is semi-positive outside Xu and ∂∂ has n − q + 1 positive eigenvalues. Let us choose χ in (3.5.8) such that χ = 0 on ] − ∞, u1 [ where u < u1 < c. By (3.5.10), RLχ (i.e., R˙ Lχ ) has then n − q + 1 positive eigenvalues in Xc
X u , L L so that Xc (j, c1 (L, hL χ )) ⊂ Xu1 for j  q. But from (3.5.8), hχ = h on Xu1 , thus L c1 (L, hL χ ) = c1 (L, h ) on Xu1 and L Xc (j, c1 (L, hL χ )) = Xc (j, c1 (L, h ))

for j  q.

From (3.5.17) and (3.5.20), we get the last part of Theorem 3.5.8.

(3.5.20)


We note also a related vanishing theorem. Theorem 3.5.9. Let X be a q-convex manifold. and let (L, hL ), (E, hE ) be holomorphic Hermitian vector bundles on X with rk(L) = 1. (i) If (L, hL ) is positive in a neighborhood of the exceptional set K ⊂ X (cf. Definition B.3.2), then there exists p0 ∈ N such that H j (X, Lp ⊗ E) = 0,

for j  q, p  p0 .

(3.5.21)

3.5. Morse inequalities for q-convex manifolds

165

(ii) If X is q-complete, then H j (X, E) = 0,

for j  q.

(3.5.22)

Proof. (i). We can assume that (L, hL ) is positive on X c . Then, for j  q, and p sufficiently large, the estimate (3.5.19) holds for all s ∈ B 0,j (Xc , Lp ⊗ E), hence 0,j H(2) (Xc , Lp ⊗ E) = 0. From Theorem 3.5.7, we get (3.5.21). (ii). If K = ∅, we can take Xu = ∅ in the proof of Theorem 3.5.5, so that (3.5.9) holds for any s ∈ B 0,j (Xc , E), j  q. By Theorem 3.5.7, (3.5.22) holds.
In the same vein one can study the growth of the cohomology groups of pseudoconvex domains and weakly 1-complete manifolds. Theorem 3.5.10. Let M  X be a smooth pseudoconvex domain in a complex manifold X and let (L, hL ), (E, hE ) be holomorphic Hermitian vector bundles over X. Assume that rk(L) = 1 and L is positive in a neighborhood of ∂M . Then 0,j (M, Lp ⊗ E) < +∞ , dim H(2)

for p  1, j  1,

(3.5.23)

For r  1, as p → ∞, n


j−r

(−1)

j=r

0,j dim H(2) (M, Lp

pn ⊗ E)  rk(E) n!

 (−1)r c1 (L, hL )n + o(pn ) . M(r)

(3.5.24) Moreover dim H 0 (M, Lp ⊗ E)  rk(E)



pn n!

c1 (L, hL )n + o(pn ).

(3.5.25)

M(
1)

In particular, if L is positive over M ,  dim H 0 (M, Lp )  pn

c1 (L, hL )n /n! + o(pn ).

(3.5.26)

M

Proof. Since M is pseudoconvex, (3.5.14) holds for any j  1, so by taking an arbitrary Hermitian metric g T X on X, we deduce as in Lemma 3.5.4 that (3.5.19) holds for j  1. We conclude by Remark 3.2.15 and Theorem 3.2.16.
We have the following variant of Theorems 3.5.6 and 3.5.7 for weakly 1complete manifolds (cf. Definition B.3.14): Theorem 3.5.11. Let X be weakly 1-complete, L, E be holomorphic vector bundles over X, with rk(L) = 1, and L be positive outside a compact set K ⊂ Xc . Then for j  1 and p  1, (i) (Takegoshi) The canonical morphism H 0,j (Xc , Lp ⊗ E) −→ H 0,j (Xc , Lp ⊗ E) ,

s → [s],

(3.5.27)

166

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

(ii) (Ohsawa) The restriction morphisms H j (X, Lp ⊗ E) −→ H j (Xc , Lp ⊗ E)

(3.5.28)

are isomorphisms. Since Xc = { < c} are smooth pseudoconvex domains for a regular value c of  with  : X → R in Definition B.3.14, Theorems 3.5.10 and 3.5.11 immediately imply: Theorem 3.5.12. Let X be a weakly 1-complete manifold and let (L, hL ), (E, hE ) be holomorphic Hermitian vector bundles over X. Assume that rk(L) = 1 and (L, hL ) is positive outside a compact set K ⊂ X. Then for r  1, as p → ∞, n


(−1)j−r dim H j (X, Lp ⊗ E)  rk(E)

j=r

pn dim H (X, L ⊗ E)  rk(E) n! 0

pn n!



p

X(
1)

 (−1)r c1 (L, hL )n + o(pn ) . X(r)

√

−1 L 2π R

(3.5.29)

n

n

+ o(p ).

(3.5.30)

Corollary 3.5.13. Let X be a weakly 1-complete manifold and let (L, hL ) be a positive holomorphic Hermitian line bundle on X, then lim p−n dim H 0 (X, Lp ⊗ E) = +∞ .

p→∞

(3.5.31)

This follows immediately from (3.5.10) and (3.5.25) by taking the liberty of modifying hL to hL e−χ() , where χ : R → R is smooth, rapidly increasing and convex. This result together with the effective base point freeness methods, introduced in algebraic geometry by Angehrn–Siu, produce an answer to the conjecture of Nakano and Ohsawa about the embeddability of weakly 1-complete manifolds. Theorem 3.5.14 (Takayama). Let X be an n-dimensional weakly 1-complete manifold with a positive line bundle L. Then X is embeddable into C P2n+1 by a linear subsystem of |H 0 (X, (KX ⊗ Lp )⊗(n+2) )| for p > n(n + 1)/2. Let us finally note the Nakano vanishing theorem. Theorem 3.5.15 (Nakano). Let X be a weakly 1-complete manifold, dim X = n, and let (L, hL ), (E, hE ) be holomorphic Hermitian vector bundles over X. Assume that rk(L) = 1 and (L, hL ) is positive on X. Then for j  1 there exists p0 ∈ N such that H j (X, Lp ⊗ E) = 0, for any p  p0 . Proof. Indeed, since L is positive on X, we can obtain the fundamental estimate (3.5.19) for any s ∈ B 0,j (Xc , Lp ⊗E) and p  1, so by (3.1.29) H 0,j (Xc , Lp ⊗E) = 0. Taking into account Theorem 3.5.11, we obtain the conclusion.


3.6. Morse inequalities for coverings

167

3.6 Morse inequalities for coverings This section is organized as follows. In Section 3.6.1, we present some generalities about von Neumann dimension on covering manifolds. In Section 3.6.2, we establish the holomorphic Morse inequalities for covering manifolds by adapting the argument in Section 1.7.

3.6.1 Covering manifolds, von Neumann dimension  g T X ) be a Riemannian manifold on which a discrete group Γ acts freely Let (X,  implies that g is the unit element of Γ) and properly (i.e., g · x =x  for some x ∈X  → X,  (g, x discontinuously (i.e., the map Γ × X ) → g · x  is proper, where Γ is  TX is Γ-invariant. endowed with the discrete topology) such that g   Let X = X/Γ be the quotient and πΓ : X −→ X be the canonical projection.  is a Galois covering of X of Galois group Γ. We assume in what follows Then X  is paracompact so that Γ will be that X is compact. Moreover, we suppose that X  countable. Since g T X is Γ-invariant, there exists a Riemannian metric g T X on X   such that πΓ∗ g T X = g T X . We denote by dvX the Riemannian volume form of g T X .  a fundamental domain of the action of Γ on X  if the We call an open set U ⊂ X following conditions are satisfied: (a) (b) (c)

 = ∪γ∈Γ γ(U ), X γ1 (U ) ∩ γ2 (U ) = ∅ for γ1 , γ2 ∈ Γ, γ1 = γ2 and U
U has zero measure.

We construct a fundamental domain in the following way. Let {Uk } be a finite cover of X with open balls having the property that for each k, there exists an open set  such that πΓ : U k −→ Uk is biholomorphic with inverse φk : Uk −→ U k . k ⊂ X U Define Wk = Uk
(∪j
(3.6.1)

where U is a fundamental domain for the action of Γ. A basis for L2 Γ is formed by the functions 1 if γ = γ  , δγ (γ  ) = (3.6.2) 0 if γ = γ  .

168

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

 E),  the above identification is given by Then for f ∈ L2 (X, f  (f |γU )γ 




δγ ⊗ γ −1 · (f |γU ),

(3.6.3)

γ

 is identified with those sections of E  which vanish which means that L2 (U, E) 2  with those sections which outside U , and for any γ we identify Cδγ ⊗ L (U, E) vanish outside γU . There is a unitary action of Γ by left translations on L2 Γ by tγ δη = δγη . It is easy to check that actually Tγ = tγ ⊗ Id and that {Tγ : γ ∈ Γ} defines a unitary  E).  action of Γ on L2 (X,  E).  We give now the definition of the Γ-dimension of a Γ-module of L2 (X, 2 We consider the algebra AΓ ⊂ L (L Γ) of all operators that commute with all left translations and denote the unit element of Γ by e. We introduce a trace on AΓ by (3.6.4) TrΓ [A] = Aδe , δe , A ∈ AΓ . A left Γ-invariant subspace V ⊂ L2 Γ is called a Γ-module. If V is a Γ-module, the orthogonal projection PV on V belongs to AΓ . Imitating the usual definition of dimension as the trace of the projector of the space in question, we put dimΓ V = TrΓ [PV ] = PV δe , δe .

(3.6.5)

There is a useful formula for dimΓ V in terms of an orthonormal basis {fi } of V . 2 We complete this basis basis to an orthonormal ¯ {fi , gj } of L Γ and expanding δe = δe , fi fi + δe , gj gj = fi (e)fi + δe , gj gj , we get   ¯i (e)fi , δe = |fi (e)|2 . dimΓ V = PV δe , δe  = (3.6.6) f i i  E)  is called a Γ-module. We denote As before, a Γ-invariant space V ⊂ L2 (X, 2   by AΓ ⊂ L (L (X, E)) the algebra of all operators which commute with the action  E)),  we can associate operators aγη ∈ of Γ. Then to any operator A ∈ L (L2 (X,  such that aγη (f ) is the projection of A(δγ ⊗ f ) on Cδη ⊗ L2 (U, E).  If L (L2 (U, E)) moreover A ∈ AΓ , then the matrix (aγη ) satisfies the relation aγη = ae, γ −1 η and we can define (3.6.7) TrΓ [A] = Tr[aee ] ,  is trace class (cf. if the right-hand side is well defined. When B ∈ L (L2 (U, E)) Def. D.1.4), then (3.6.8) Tr[B] := j Bhj , hj ,  If B is a positive operator, cerwhere {hj } is an orthonormal basis of L2 (U, E). tainly Tr[B] in (3.6.8) is also well defined.

3.6. Morse inequalities for coverings

169

If A ∈ AΓ is positive, aee is positive too, so TrΓ [A] := Tr[aee ]  0 .

(3.6.9)

 E)  is a Γ-module then the projection PV ∈ AΓ . If V ⊂ L2 (X, Definition 3.6.1. The von Neumann dimension or Γ-dimension of a Γ-module V is defined by dimΓ V = TrΓ [PV ]. (3.6.10)  E)  be a Γ-module. Then Lemma 3.6.2. Let V ⊂ L2 (X,
 dimΓ V = |fi ( x)|2 dvX˜ (˜ x), i

(3.6.11)

U

where {fi } is an orthonormal basis in V . If PV ( x, y) is the Schwartz kernel of the projector PV on V ,  dimΓ V = Tr[PV ( x, x )]dvX˜ (˜ x). (3.6.12) U

Proof. We have PV ( x, y) =




x ⊗ E ∗ , fi ( x) ⊗ fi ( y )∗ ∈ E y 

i

and therefore Tr[PV ( x, x )] = also Lemma D.1.5.

i

|fi ( x)|2 . Formula (3.6.11) is proved as (3.6.6), cf.


3.6.2 Holomorphic Morse inequalities  J)  is a paracompact complex manifold and the discrete Now, we suppose that (X,  having a compact quotient group Γ acts holomorphically, freely and properly on X  T X   is Γ-equivariant and J-compatible. X = X/Γ. We assume also that the metric g  E   Let (E, h ) be a Γ-equivariant holomorphic Hermitian vector bundle on X.    Then the Kodaira Laplacian
E is Γ-invariant, i.e., for any γ ∈ Γ, Tγ
E =
E Tγ  E).  It is easy to see that g T X is complete. Thus
E is essentially selfon Ω00,• (X,  adjoint (cf. Corollary 3.3.4), and we denote still by
E its self-adjoint extension, which turns out to commute to the action of Γ (Problem 3.8). This means that for any γ ∈ Γ, 









Tγ (Dom(
E )) ⊂ Dom(
E ) and Tγ
E =
E Tγ on Dom(
E ).

(3.6.13)

   E)  (see Appendix Consider the spectral resolution Eλj (
E ) of
E acting on L20,j (X, C.2).

170

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds 

Lemma 3.6.3. For any j = 0, 1, . . . , n and λ ∈ R, we have Eλj (
E ) ∈ AΓ , its Schwartz kernel is smooth and 

dimΓ Eλj (
E ) < +∞ .

(3.6.14)



Proof. Since
E commutes to the action of Γ, we see with the help of the spectral  theorem C.2.1 that Eλj (
E ) ∈ AΓ (Problem 3.8). Using again the spectral theorem, we obtain for any m ∈ N, 



Im(Eλ (
E )) ⊂ Dom ((
E )m ).

(3.6.15)

 ⊗ E.  Note that Dom ((
E )m ) ⊂ H 2m (X,  := Λ(T ∗(0,1) X)  E,  loc), by Set E the regularity theorem A.3.4. Then due to Sobolev’s embedding theorem A.3.1,   E,  loc) ⊂ C ∞ (X,  E).  Im(Eλ (
E )) ⊂ ∩m∈N H 2m (X,

(3.6.16)

  E)  −→ C ∞ (X,  E)  is linearly continuous. By duality, Therefore Eλ (
E ) : L2 (X,  n,n   ∗ E 2    E  ∗ ) is the dual of C ∞ (X,  E),  Eλ (
) : Ω0 (X, E ) → L (X, E), where Ωn,n (X, 0 which is the space of currents with compact support (this space can be identified  ∗ and distribution coefficients). Comwith the space of (n, n)-forms with values in E  n,n   ∗ E  E)  → C ∞ (X,  E),  posing the two maps, we obtain Eλ (
) : Ω0 (X, E ) → L2 (X,  hence Eλ (
E ) is a smoothing operator. The Schwartz kernel theorem B.2.7 im  plies that the kernel Eλ (
E )( x, y) of Eλ (
E ) with respect to dvX is smooth. By (3.6.12),  



dimΓ Eλ (
E ) = U

Tr[Eλ (
E )( x, x )] dvX ( x) < +∞.


The proof of Lemma 3.6.3 is completed. 

The quantity dimΓ Eλj (
E ) is called the von Neumann spectrum distribution function.  hL ) be a Γ-equivariant holomorphic Hermitian line bundle on X.  Let (L, We use in the sequel the same notation as in Section 1.7 for the corresponding p objects on X. Following Section 1.6.1, we introduce the Γ-invariant operator D 2  −u D 0,•   p  and the Γ-invariant Kodaira Laplacian
 p . Let e p ( on Ω (X, L ⊗ E) x, x  ) 2   be the smooth kernel of the operator e−uD p with respect to dv  ( ( x, x  ∈ X) X x ). 0,j  L  p ⊗ E)  be the reduced L2 -Dolbeault cohomology group. By the Let H (2) (X, weak Hodge decomposition (3.1.21), we have:  L  p ⊗ E)   H (2) (X,  L  p ⊗ E).   p2 ) ∩ L20,j (X, Ker(D 0,j

(3.6.17)

 is the The next result shows that the local behavior of the heat kernel on X same as on X.

3.6. Morse inequalities for coverings

171

Theorem 3.6.4. For any m ∈ N, u0 > 0, ε > 0, there exists C > 0 such that for  p ∈ N∗ , u > u 0 , any x  ∈ X,  2    u 2 u ε p    − 2p D − 2p Dp2 3m+8n+8 p ( x, x ) − e (π( x), π( x))  Cp exp − . (3.6.18) e 16u Cm Proof. By the same 1.6.4, we have the estimate  proof as that of Proposition  p )(  here we use the covering of X  x, x  ) with x , x  ∈ X, (1.6.15) for G up ( u/pD induced by Uxi (cf. also the proof of Theorem 6.1.4).   u  D x,·) Now by using the finite propagation speed, Theorem D.2.1, F up p p (   u x), ·). Thus from (1.6.14), we get (3.6.18).
is the pull-back of F up p Dp (π( From (3.6.18) and Theorem 1.6.1, we deduce:  Corollary 3.6.5. For each u > 0 fixed, for any k ∈ N, under C k -norm on C ∞ (X, ∗(0,1)    X) ⊗ E)), as p → ∞, uniformly on x  ∈ X, we have End(Λ(T 2

e− p Dp ( x, x ) = (2π)−n u

det(R˙ L ) exp(2uωd) (π( x)) ⊗ IdE pn + o(pn ). det(1 − exp(−2uR˙ L))

(3.6.19)

 L  p ⊗ E);  then We denote by TrΓ,q the Γ-trace of operators acting on L20,q (X, we have the following analogue of Lemma 1.7.2. Lemma 3.6.6. For any u > 0, p ∈ N∗ , 0  q  n, we have q


0,j  L  p ⊗ E)   (−1)q−j dimΓ H (2) (X,

j=0

 u 2  (−1)q−j TrΓ,j exp(− D ) , p p j=0

q


(3.6.20)

with equality for q = n.  L  p ⊗ E).  For  p acting on L2 (X, Proof. Let Eλj,p be the spectral resolution of
0,j j,p j,p λ2 > λ1  0, we consider the projectors E j,p (]λ1 , λ2 ]) = Eλ2 − Eλ1 . Then, by the q Hodge decomposition (3.1.21), j=0 (−1)q−j E j,p (]λ1 , λ2 ]) is the projection on the  p ⊗E  L

∂ E q,p (]λ1 , λ2 ]) and thus a positive operator. Hence the Γ-invariant range of q measure j=0 (−1)q−j dEλj,p is positive on {λ > 0}. It follows that  R := λ>0

e− p λ u

q


(−1)q−j dEλj,p  0,

and R ∈ AΓ .

(3.6.21)

j=0

On the other hand,   u u 2  0,j  L  p ⊗ E)  + TrΓ TrΓ,j exp(− D ) = dim H ( X, e− p λ dEλj,p . (3.6.22) Γ (2) p 2p λ>0 From (3.6.9), (3.6.21) and (3.6.22), we obtain (3.6.20).




172

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

By (3.6.7) and (3.6.8), as in (1.7.4),    u 2  u 2  Trq e− p Dp ( x, x ) dvX ( x). TrΓ,q exp(− Dp ) = p U

(3.6.23)

Using (3.6.19) and (3.6.20) as in the proof of Theorem 1.7.1, we get: Theorem 3.6.7. As p → ∞, the following strong holomorphic Morse inequalities hold for every q = 0, 1, . . . , n: q n 
0,j  L  p ⊗ E)   rk(E)  p (−1)q−j dimΓ H (2) (X, (−1)q c1 (L, hL )n + o(pn ) , n! X(
q) j=0 (3.6.24) with equality for q = n. Corollary 3.6.8. Suppose that C =

1 n!



X(
1) c1 (L, h

) > 0. Then as p → ∞,

L n

0  L  p ⊗ E)   C rk(E)p  n + o(pn ) . dimΓ H(2) (X,

(3.6.25)

Problems Problem 3.1. Prove Lemma 3.1.1 and show that if R is any closed extension of ∗ ∗ )H . R, then R ⊂ Rmax . Especially, Rmax = (RH Problem 3.2. Prove that the quadratic form (3.1.5) is closed: If {sk } is a Cauchy sequence in the norm ·Q (cf. (C.1.5)), there exist elements s, v, w in L2 such E

E,∗

E

E

E

that sk → s, ∂ sk → v, ∂ sk → w. Show that ∂ sk → ∂ s and v = ∂ s in E E,∗ E distribution sense, hence ∂ s ∈ L2 . Since sk ∈ Dom(∂ H ), we have (∂ s, sk ) = E,∗ E E ( s, ∂ sk ) for any s ∈ Dom(∂ ). By passing to the limit, show that (∂ s, s) = E,∗ E,∗ ( s, w), thus s ∈ Dom(∂ H ) and ∂ H s = w. Problem 3.3. Let X be a complex Hermitian manifold and (E, hE ) be a holomorphic Hermitian vector bundle on X. Using Friedrichs lemma 3.1.3 show that for any s ∈ L20,• (X, E) with compact support, s is contained in the domain of the quadratic form associated to the Gaffney extension (3.1.4) if and only if s is contained in the domain of the quadratic form associated to the Friedrichs extension (C.1.7), and if this is the case, the two quadratic forms coincide on (s, s). Problem 3.4. Let A be a positive self-adjoint operator on a Hilbert space and (Eλ ) be its spectral resolution. Let λ  0 and u ∈ Dom(A), set u = u1 + u2 where u1 = Eλ u, u2 = u − u1 . (a) Show that QA (u, u) = QA (u1 , u1 ) + QA (u2 , u2 ), QA (u1 , u1 )  λu1 2 , QA (u2 , u2 )  λu2 2 . (b) Assume that QA (u, u)  λu2 and u1 = 0. Show that u = 0.

3.7. Bibliographic notes

173

Problem 3.5. Prove that for any f ∈ C ∞ (R+ ), there exists g ∈ C ∞ (R) such that f = g on R+ . (Hint: The C k version of the statement is (A.2.1). A good reference for the C ∞ version is [213].) Problem 3.6. Let A  0 and B1 , . . . , Bl > 0 and l, n ∈ N, l  n. Consider the function G : {u ∈ C : Re(u) > 0} → C, G(u) =

u

#l n−l

euA

j=1 (1

− e−uBj )

.

With the notation from (3.2.36), show that its inverse Laplace transform L−1 (G) is - 

n if l = n, j=1 αj Bj , α∈Nn δ λ + A − −1 L (G)(λ) = 1 n−l−1  t H (t) , if l < n. l l + α∈N (n−l−1)! t=λ+A− αj Bj j=1

Here δ(t) represents the Dirac measure at {0}. Thus  λ 
1  L−1 (G)(λ0 ) dλ0 = (tn−l H+ (t)) . (n − l)! t=λ+A− lj=1 αj Bj 0 l α∈N

Problem 3.7. Show that the Γ-dimension (3.6.5) has the following properties: 1. 0  dimΓ V  1. 2. dimΓ V = 0 if and only if V = 0. 3. V ⊂ V  implies dimΓ V  dimΓ V  with equality if and only if V = V  .  g T X , Γ, X) be as in Section 3.6.2. Let (E,  hE ) be a ΓProblem 3.8. Let (X,  equivariant holomorphic Hermitian vector bundle on X. 

E  E),  its (a) Let A be one of the following operators: the closure of ∂ on L20,• (X, Hilbert-space adjoint, or the unique self-adjoint extension Kodaira Laplacian 
E . Show that A commutes with the action of Γ in the following sense: for any γ ∈ Γ, Tγ (Dom(A)) ⊂ Dom(A) and Tγ A = ATγ on Dom(A). 

(b) Show that the spectral projections Eλ (
E ) commute with any Tγ , i.e.,  Eλ (
E ) ∈ AΓ .

3.7 Bibliographic notes Section 3.1. The approach to L2 Hodge theory is inspired by H¨ ormander [131], Kohn [143], Folland–Kohn [108] and Ohsawa [187]. Friedrichs lemma 3.1.3 can be found in [131, Prop. 1.2.4]. Section 3.2. Theorems 3.2.13 and 3.2.16 appeared in [169, 170] and [173, 244]. Holomorphic Morse inequalities on non-compact manifolds were proved in various contexts by Nadel–Tsuji [181], Bouche [47], Takayama [235] and recently by Berman [17].

174

Chapter 3. Holomorphic Morse Inequalities on Non-compact Manifolds

The proof of Proposition 3.2.7 is inspired by [55, §3] where the contribution from the boundary in the local index theorem is also determined. Sources for the Laplace transform are [212] and [106, XIII]. Lemma 3.2.10 is [106, XIII.1, Th. 2a]. Section 3.3. The Andreotti–Vesentini lemma 3.3.1 appeared in [7, Lemma 4, p. 92–93]. Corollary 3.3.7 is [181, Th. 1.1]. For another approach see [83]. Section 3.4. The Siu–Demailly criterion for isolated singularities appeared in [171]. The metric constructed in (3.4.17) is called a metric of Saper-type, cf. [209, 113, 114]. For the notion of Andreotti-pseudoconcavity see [2, 6]. Levi’s removable singularity theorem can be found at [107, p. 180]. Section 3.5. For the proof of Lemma 3.5.1, we refer to [131, Prop. 1.2.4]. The treatment of Andreotti–Grauert cohomology theory on q-convex manifolds is inspired by the paper of H¨ ormander [131] (where only the case of a trivial bundle E is treated; passing to the general case poses no difficulties). One can as well work with complete metrics; this is done in [7, 187]. Theorem 3.5.5 is essentially [131, Th. 3.4.1] and Theorem 3.5.6 is [131, Th. 3.4.9]. Lemma 3.5.3 appears in [7, Lemma 18]. The fundamental theorem 3.5.7 can be found in [4]. Theorem 3.5.8 was obtained by Th. Bouche [47]. His proof is based on the same principle of showing the fundamental estimate outside a compact set but he works with complete metrics. Theorem 3.5.11 was proved by Takegoshi [236]. Theorem 3.5.12 is [47, Theorem 0.2], where actually a q-positive line bundle L is considered over a K¨ahler weakly 1-complete manifold X. The K¨ ahler assumption was removed in [169] answering positively a question of Ohsawa [187, p. 218] about the polynomial growth of degree n with respect to p of dim H j (X, Lp ⊗ E) , j  1. Theorem 3.5.14 was proved by Takayama [235, Theorem 1.2]. Section 3.6. For facts about analysis on covering manifolds, the reader is referred to [9, §4], [160, 244]. The holomorphic Morse inequalities for coverings appeared in [244, 173]. The approach there was to determine the asymptotics of the von Neumann spectrum counting function NΓj,p (λ) = dimΓ Eλj,p , and was inspired by the analytic proof of Shubin [222] of the Novikov–Shubin (classical) Morse inequalities on coverings. There is a large literature about existence of holomorphic functions or sections on coverings of projective manifolds (cf. Corollary 3.6.8), motivated by the Shafarevich conjecture, see, e.g., [147] and the references therein.

Chapter 4

Asymptotic Expansion of the Bergman Kernel In this chapter, we establish the asymptotic expansion of the Bergman kernel associated to high tensor powers of a positive line bundle on a compact complex manifold. Thanks to the spectral gap property of the Kodaira Laplacian, Theorem 1.5.5, we can use the finite propagation speed of solutions of hyperbolic equations, (Theorem D.2.1), to localize our problem to a problem on R2n . Comparing with Section 1.6, the key point here is that we need to extend the connection of the line bundle L such that its curvature becomes uniformly positive on R2n . Then we still have the spectral gap property on R2n . Thus we can instead study the Bergman kernel on R2n (cf. (4.1.27)), and use various resolvent representations (4.1.59), (4.2.22) of the Bergman kernel on R2n . We conclude our results by employing functional analysis resolvent techniques. This chapter is organized as follows: In Section 4.1, we obtain the near diagonal asymptotic expansion of the Bergman kernel, and give a general way to compute the coefficients. In Section 4.2, we establish the full off-diagonal asymptotic expansion of the Bergman kernel and give a relation between the coefficients of the asymptotic expansion of the Bergman kernel and the heat kernel. We will use the notation from Section 1.6.1.

4.1 Near diagonal expansion of the Bergman kernel In this section, we establish the near diagonal asymptotic expansion of the Bergman kernel, Theorem 4.1.24. As pointed out at the beginning of this chapter, we will localize our problem to a problem on R2n by combining the spectral gap property and the finite propagation speed of solutions of hyperbolic equations. Then we will use functional analysis methods to study the resolvent in order to

176

Chapter 4. Asymptotic Expansion of the Bergman Kernel

obtain our result. Moreover, in Sections 4.1.6, 4.1.7, we develop a formal power series technique to compute the coefficients in general. This section is organized as follows. In Section 4.1.1, we state the diagonal asymptotic expansion of the Bergman kernel. In Section 4.1.2, we show that the problem is local. In Section 4.1.3, we construct in detail the extension of the Dirac operator from a small neighborhood of 0 to the whole R2n , such that the spectral gap property still holds, and we study the Taylor expansion of the operator. In Section 4.1.4, we study the Sobolev estimates of the resolvent (λ − L2t )−1 . In Section 4.1.5, we establish the existence of the near diagonal expansion, (Theorem 4.1.24), without knowing information about the coefficients Fr . In Section 4.1.6, we study the Bergman kernel of the limit operator. In Section 4.1.7, we develop a formal power series technique to compute the coefficients and we get the description of the coefficients br from Theorem 4.1.1. In Sections 4.1.8 and 4.1.9, we prove Theorems 4.1.2 and 4.1.3 by applying the results from Sections 4.1.6 and 4.1.7. In this section, we use the notation from Section 1.6.1. Especially, (X, J) is a compact complex manifold with complex structure J and dimC X = n. g T X is a Riemannian metric on T X compatible with J. (L, hL ), (E, hE ) are holomorphic Hermitian vector bundles on X and rk(L) = 1. We suppose that the positivity condition (1.5.21) holds for RL .

4.1.1 Diagonal asymptotic expansion of the Bergman kernel Let Pp be the orthogonal projection from Ω0,• (X, Lp ⊗ E) onto Ker(Dp ). Then for each p ∈ N, the Bergman kernel Pp (x, x ) (x, x ∈ X), is the smooth kernel of Pp with respect to the Riemannian volume form dvX (x ). By Theorems 1.4.1 and 1.5.5 we have for p large enough, Ker(Dp ) = Ker(Dp2 |Ω0,0 ) = H 0 (X, Lp ⊗ E).

(4.1.1)

As Dp2 preserves the Z-grading on Ω0,• (X, Lp ⊗ E), we know that for p large enough, (4.1.2) Pp (x, x ) ∈ (Lp ⊗ E)x ⊗ (Lp ⊗ E)∗x
, especially, Pp (x, x) ∈ End(Lp ⊗ E)x = End(E)x ,

(4.1.3)

where we use the canonical identification End(L ) = C for any line bundle L on X. p

d

p (dp := dim H 0 (X, Lp ⊗E)) be any orthonormal basis of H 0 (X, Lp ⊗E) Let {Sip }i=1 with respect to the inner product (1.3.14) induced by g T X , hL and hE . Then for p large enough,

Pp (x, x ) =

dp
i=1

Sip (x) ⊗ (Sip (x ))∗ ∈ (Lp ⊗ E)x ⊗ (Lp ⊗ E)∗x
.

(4.1.4)

4.1. Near diagonal expansion of the Bergman kernel

177

(1,0)

X Recall that RT X , RT are the curvatures of the Levi–Civita connection (1,0) X on T (1,0) X. ∇T X on T X and of the holomorphic Hermitian connection ∇T E E TX Let Λω (R ) be the contraction of R with respect to ω and gω := ω(·, J·). If {wω,j } is an orthonormal basis of (T (1,0) X, gωT X ), then

√ √

−1Λω (RE ) = −1 RE , ω gT X = RE (wω,j , wω,j ), ω

nRE ∧ ω n−1 = Λω (RE ) · ω n .

(4.1.5)

Theorem 4.1.1. There exist smooth coefficients br (x) ∈ End(E)x which are polynomials in RT X , RE (and dΘ, RL ) and their derivatives of order  2r − 2 (resp. 2r − 1, 2r) and reciprocals of linear combinations of eigenvalues of R˙ L at x (in the sense of Lemmas 1.2.3 and 1.2.4), such that b0 = det(R˙ L /(2π)) IdE ,

(4.1.6)

and for any k, l ∈ N, there exists Ck,l > 0 such that for any p ∈ N, k  
  br (x)pn−r  Pp (x, x) − r=0

C l (X)

 Ck,l pn−k−1 .

(4.1.7)

Moreover, the expansion is uniform in the following sense. For any fixed k, l ∈ N, assume that the derivatives of g T X , hL , hE of order  2n+2k +l+6 run over a set bounded in the C l -norm taken with respect to the parameter x ∈ X and, moreover, g T X runs over a set bounded below. Then the constant Ck, l is independent of g T X and the C l -norm in (4.1.7) includes also the derivatives with respect to the parameters. √

−1 L Theorem 4.1.2. If ω = Θ in (1.5.14), i.e., ω = 2π R is the K¨ ahler form of (T X, g T X ), then there exist smooth functions br (x) ∈ End(E)x such that (4.1.7) holds, and br are polynomials in RT X , RE and their derivatives of order  2r − 2 at x. Moreover,

b0 = IdE ,

b1 =

 1  √ 1 2 −1Λω (RE ) + rX IdE . 4π 2

(4.1.8)

Theorem 4.1.3. The term b1 in the expansion (4.1.7) is given by b1 =

  R˙ L    √ 1 det rωX − 2∆ω log(det(R˙ L )) + 4 −1Λω (RE ) , 8π 2π

(4.1.9)

here rωX , ∆ω are the scalar curvature and the Bochner Laplacian as in (1.3.19) associated to gωT X .

178

Chapter 4. Asymptotic Expansion of the Bergman Kernel

Remark 4.1.4. By the Riemann–Roch–Hirzebruch theorem 1.4.6, and relations (4.1.1) and (4.1.4), we have for p large enough,  TrE [Pp (x, x)]dvX (x) = dim H 0 (X, Lp ⊗ E) X  Td(Th X) ch(Lp ⊗ E) (4.1.10) = X     c (L)n−1 c1 (L)n n rk(E) 1 c1 (E) + p + c1 (Th X) pn−1 + O(pn−2 ). = rk(E) n! 2 (n − 1)! X X By integrating the trace of (4.1.7), and using (4.1.8) and (4.1.9), we recover the last equation of (4.1.10). Thus we can consider (4.1.8) as a local version of (4.1.10) in the spirit of local index theory.

4.1.2 Localization of the problem Recall that injX is the injectivity radius of (X, g T X ) and the function f : R → [0, 1] was defined in (1.6.12) for 0 < ε < injX /4. Set   +∞ −1  +∞ F (a) = f (v)dv eiva f (v)dv. (4.1.11) −∞

−∞

Then F (a) lies in Schwartz space S(R) and F (0) = 1. Proposition 4.1.5. For any l, m ∈ N, ε > 0, there exists Cl,m,ε > 0 such that for p  1, x, x ∈ X, |F (Dp )(x, x ) − Pp (x, x )|C m  Cl,m,ε p−l , |Pp (x, x )|C m  Cl,m,ε p−l

if d(x, x )  ε.

(4.1.12)

Here the C m norm is induced by ∇L , ∇E and hL , hE , g T X . Proof. Recall that the constant µ0 is defined in (1.5.26). For a ∈ R, set φp (a) = 1[√pµ0 ,+∞[ (|a|)F (a).

(4.1.13)

Then by Theorem 1.5.8 (cf. also Theorem 1.5.5), for p > CL /µ0 , F (Dp ) − Pp = φp (Dp ).

(4.1.14)

By (4.1.11), for any m ∈ N, there exists Cm > 0 such that sup |a|m |F (a)|  Cm .

(4.1.15)

a∈R

Let Q be a differential operator of order m ∈ N with scalar principal symbol
and with compact support in Uxi as in (1.6.17). From Dpm φp (Dp )Qs, s  =

4.1. Near diagonal expansion of the Bergman kernel

179




s, Q∗ φp (Dp )Dpm s , (1.6.6), (4.1.13) and (4.1.15), we know that for l, m ∈ N, there exists Cl,m
> 0 such that for p  1,


Dpm φp (Dp )QsL2  Cl,m
p−l+2m sL2 .

(4.1.16)

We deduce from (1.6.6) and (4.1.16) that if P, Q are differential operators of order m, m with compact support in Uxi , Uxj respectively, then for any l > 0, there exists Cl > 0 such that for p  1, P φp (Dp )QsL2  Cl p−l sL2 .

(4.1.17)

On Uxi × Uxj , by using Sobolev inequality and (4.1.14), we get the first inequality of (4.1.12). By the finite propagation speed of solutions of hyperbolic equations, Theorem D.2.1, F (Dp )(x, x ) only depends on the restriction of Dp to B X (x, ε), and is zero if d(x, x )  ε. Thus we get the second inequality of (4.1.12). The proof of Proposition 4.1.5 is complete.


4.1.3 Rescaling and Taylor expansion of the operator Dp2 Recall that ∇B,Λ is the connection on Λ(T ∗(0,1) X) defined by (1.4.27). Trivialization 4.1. We fix x0 ∈ X. From now on, we identify B Tx0 X (0, 4ε) with B X (x0 , 4ε) as in Section 1.2.1. For Z ∈ B Tx0 X (0, 4ε), we identify EZ , LZ , ∗(0,1) ∗(0,1) Λ(TZ X) to Ex0 , Lx0 , Λ(Tx0 X) by parallel transport with respect to the 0,• connections ∇E , ∇L , ∇B,Λ along the curve [0, 1]  u → uZ, cf. Section 1.6.3. 0,• Thus on B X (x0 , 4ε), (E, hE ), (L, hL ), (Λ(T ∗(0,1) X), hΛ ) are identified to the 0,• ∗(0,1) trivial Hermitian bundles (Ex0 , hEx0 ), (Lx0 , hLx0 ), (Λ(Tx0 X), hΛx0 ). Let ΓE , 0,• 0,• ΓL , ΓB,Λ be the corresponding connection forms of ∇E , ∇L and ∇B,Λ on X E L B,Λ0,• E x0 B (x0 , 4ε). Then Γ , Γ , Γ are skew-adjoint with respect to h , hLx0 , 0,• hΛx0 . 0,•

Denote by ∇U the ordinary differentiation operator on Tx0 X in the direction U . We will identify R2n to Tx0 X as in (1.6.22) by choosing an orthonormal basis {ei } of Tx0 X with dual basis {ei }. Let ϕε : R2n → R2n be the map defined by ϕε (Z) = ρ(|Z|/ε)Z with ρ defined in (1.6.19). As in (1.2.12), let R be the radial vector field defined by R=

2n


Zi ei = Z.

(4.1.18)

i=1

For the trivial vector bundle E0 := (Ex0 , hEx0 ), we defined a Hermitian connection on X0 := Tx0 X by ∇E0 = ∇ + ρ(|Z|/ε)ΓE .

(4.1.19)

180

Chapter 4. Asymptotic Expansion of the Bergman Kernel

For the trivial line bundle L0 := (Lx0 , hLx0 ), we define the Hermitian connection on X0 by 1 ∇L0 |Z = ϕ∗ε ∇L + (1 − ρ2 (|Z|/ε))RxL0 (R, ·). 2

(4.1.20)

Then we calculate easily that its curvature RL0 := (∇L0 )2 is  1  RL0 (Z) = ϕ∗ε RL + d (1 − ρ2 (|Z|/ε))RxL0 (R, ·) 2  L = 1 − ρ2 (|Z|/ε) RxL0 + ρ2 (|Z|/ε)Rϕ ε (Z) 
Zi ei  L ∧ RxL0 (R, ·) − Rϕ − (ρρ )(|Z|/ε) (R, ·) . (4.1.21) (Z) ε ε|Z| i Thus RL0 is positive in the sense of (1.5.21) for ε small enough, and the corresponding constant µ0 for RL0 is bigger than 45 µ0 . From now on, we fix ε as above. Let g T X0 (Z) := g T X (ϕε (Z)), J0 (Z) := J(ϕε (Z)) be the metric and almost complex structure on X0 . Let T ∗(0,1) X0 be the anti-holomorphic cotan∗(0,1) ∗(0,1) gent bundle of (X0 , J0 ). Then TZ,J0 X0 is naturally identified with Tϕε (Z),J X0 . ∗(0,1)

∗(0,1)

∗(0,1)

We identify Λ(TZ X0 ) with Λ(Tx0 X) by identifying first Λ(TZ X0 ) with ∗(0,1) ∗(0,1) Λ(Tϕε (Z),J X0 ), which in turn is identified with Λ(Tx0 X) by using parallel 0,•

transport with respect to ∇B,Λ along γu : [0, 1]  u → uϕε (Z). We trivialize Λ(T ∗(0,1) X0 ) in this way. We trivialize the Hermitian line bundle det(T (1,0) X0 ) by identifying at first det(T (1,0) X0 )Z to det(T (1,0) X)ϕε (Z) , and then to det(T (1,0) X)x0 by using parallel transport along γu with respect to the holomorphic Hermitian connection ∇det on det(T (1,0) X). Let Γdet be the corresponding connection form. Let ∇det0 be the Hermitian connection on det(T (1,0) X)x0 defined by ∇det0 = ∇ + ρ(|Z|/ε)Γdet .

(4.1.22)

Then g T X0 and ∇det0 define a Clifford connection ∇Cl0 on Λ(T ∗(0,1) X0 ) as in (1.3.5). The connection form ΓCl0 associated to the above trivialization of Λ(T ∗(0,1)X0 ) on R2n , satisfies ΓCl0 = 0

for |Z| > 4ε.

(4.1.23)

c be the spinc Dirac operator on R2n acting on E0,p := Λ(T ∗(0,1) X0 ) ⊗ Let D0,p p L0 ⊗ E0 , and associated to ∇L0 , ∇E0 , ∇Cl0 , as constructed in (1.3.15). Recall that the 3-form Tas was defined in (1.2.48). Set

1 A0 = − ρ(|Z|/ε)Tas,Z . 4

(4.1.24)

4.1. Near diagonal expansion of the Bergman kernel

181

Then the modified Dirac operator c Dpc,A0 = D0,p + c (A0 )

(4.1.25)

coincides with Dp on B Tx0 X (0, 2ε). Observe that all tensors (except RL0 ) in (1.3.35) are 0 outside B Tx0 X (0, 4ε), thus from (4.1.21) and the proof of Theorems 1.5.7 and 1.5.8, we see that (Dpc,A0 )2 has the following spectral gap: Spec((Dpc,A0 )2 ) ⊂ {0} ∪

8 5

 µ0 p − C, +∞ .

(4.1.26)

Let P0,p be the orthogonal projection from L2 (X0 , E0,p ) onto Ker(Dpc,A0 )2 , and let P0,p (x, x ) be the smooth kernel of P0,p with respect to the Riemannian volume form dvX0 (x ). Proposition 4.1.6. For any l, m ∈ N, there exists Cl,m > 0 such that for x, x ∈ B Tx0 X (0, ε),     (4.1.27) (P0,p − Pp )(x, x ) m  Cl,m p−l . C

Proof. Using (4.1.14) and (4.1.26), we see that P0,p − F (Dp ) verifies also (4.1.12) for x, x ∈ B Tx0 X (0, ε), thus we get (4.1.27).
As in Section 1.6.3, by means of a unit vector SL of Lx0 , we construct an isometry Ep,x0  (Λ(T ∗(0,1) X) ⊗ E)x0 =: Ex0 . Since the operator (Dpc,A0 )2 takes values in End(Ep,x0 ) = End(E)x0 under the natural identification End(Lp )  C (which does not depend on SL ), our formulas do not depend on the choice of SL . Now, under this identification, we will consider (Dpc,A0 )2 acting on C ∞ (X0 , Ex0 ). Let dvT X be the Riemannian volume form of (Tx0 X, g Tx0 X ). Let κ(Z) be the smooth positive function defined by the equation dvX0 (Z) = κ(Z)dvT X (Z),

(4.1.28)

with κ(0) = 1. Let ∇A0 be the connection induced by ∇Cl0 and A0 on Λ(T ∗(0,1) X0 ) as in (1.3.33). Let ΓA0 be the connection form of ∇A0 . Recall that ∇0 , L02 were defined in (1.6.28). For s ∈ C ∞ (R2n , Ex0 ), Z ∈ R2n , and for t = √1p , set (St s)(Z) := s(Z/t), ∇t := St−1 t κ1/2 ∇A0 κ−1/2 St , L2t := St−1 κ1/2 t2 (Dpc,A0 )2 κ−1/2 St , L20 := L02 .

(4.1.29)

182

Chapter 4. Asymptotic Expansion of the Bergman Kernel

Theorem 4.1.7. There exist polynomials Ai,j,r (resp. Bi,r , Cr ) (r ∈ N, i, j ∈ {1, . . . , 2n}) in Z with the following properties: 0,•

(i) their coefficients are polynomials in RT X (resp. RT X , RB,Λ , RE , Rdet , dΘ, RL ) and their derivatives at x0 up to order r − 2 (resp. r − 2, r − 2, r − 2, r − 2, r − 1, r) , (ii) Ai,j,r is a homogeneous polynomial in Z of degree r, the degree in Z of Bi,r is  r + 1 (resp. Cr is  r + 2), and has the same parity as r − 1 (resp. r) , (iii) if we denote Or = Ai,j,r ∇ei ∇ej + Bi,r ∇ei + Cr ,

(4.1.30)

then L2t = L20 +

m


tr Or + O(tm+1 ),

(4.1.31)

r=1

and there exists m ∈ N such that for any k ∈ N, t  1, the derivatives of order  k of the coefficients of the operator O(tm+1 ) are dominated by
Ctm+1 (1 + |Z|)m . Proof. We will add a subscript 0 to indicate the corresponding object on X0 , e.g., RE0 , Rdet0 are the curvatures of ∇E0 , ∇det0 . As in (1.6.20), let ΦE0 be the smooth self-adjoint section of End(Ex0 ) on X0 , ΦE 0 =

rX0 c E0 1 det0 + (R + R ) + c (dA0 ) − 2|A0 |2 . 4 2

(4.1.32)

Relations (1.3.35) and (1.5.17) entail (Dpc,A0 )2 = ∆A0 − p(2ω0,d + τ0 ) + ΦE0 .

(4.1.33)

Set gij (Z) = g T X0 (ei , ej )(Z) = ei , ej Z and let (g ij (Z)) be the inverse of the matrix (gij (Z)). Similar calculations as those leading to (1.6.31) and (4.1.33) imply   1 L0 E0 −1/2 0 ∇t,· =κ1/2 (tZ) ∇· + tΓA (tZ), (4.1.34) tZ + ΓtZ + tΓtZ ) κ t   L2t = − g ij (tZ) ∇t,ei ∇t,ej − tΓkij (tZ)∇t,ek − 2ω0,d,tZ − τ0,tZ + t2 ΦE0 ,tZ . Since for |tZ| < 2ε, ρ(|tZ|/ε) = 1, the operators ∇t , L2t are given by (1.6.31) for |tZ| < 2ε. Moreover κ(Z) = (det(gij (Z)))1/2 . (4.1.35) Thus from Lemmas 1.2.3, 1.2.4, (1.6.20), (1.6.31) and (4.1.35), we get Theorem 4.1.7.


4.1. Near diagonal expansion of the Bergman kernel

183

4.1.4 Sobolev estimate on the resolvent (λ − L2t )−1 Let hΛ be the Hermitian metric on Λ(T ∗(0,1) X0 ) induced by g T X0 , J0 . By the 0 0,• ) is identified to the trivial Hermitian vector trivialization 4.1, (Λ(T ∗(0,1) X0 ), hΛ 0 0,• ∗(0,1) bundle (Λ(Tx0 X), hΛx0 ). 0,•

0,•

Let hEx0 be the metric on Ex0 induced by hΛx0 , hEx0 . We also denote by ·, ·0,L2 and ·0,L2 the scalar product and the L2 -norm on C ∞ (X0 , Λ(T ∗(0,1) X0 )⊗ E0 ) induced by g T X0 , J0 , hE0 as in (1.3.14). Then ·, ·0,L2 is the same as the scalar product on C ∞ (X0 , Ex0 ) induced by hEx0 , dvX0 under our trivialization. For s ∈ C ∞ (Tx0 X, Ex0 ), set  s2t,0 := s0 = |s(Z)|2hEx0 dvT X (Z), R2n

s2t,m

=

m


2n


(4.1.36) ∇t,ei1

· · · ∇t,eil s2t,0 .

l=0 i1 ,··· ,il =1

As in Section 1.6.4, we define the corresponding Sobolev spaces H m t , and in this chapter, we will use these norms. Remark 4.1.8. Note that (Dpc,A0 )2 is self-adjoint with respect to  · 0,L2 , thus by (4.1.28), (4.1.29) and (4.1.36), L2t is a formal self-adjoint elliptic operator with respect to  · 0 , and is a smooth family of operators with respect to the parameter x0 ∈ X. Thus L20 and Or are also formal self-adjoint with respect to  · 0 . Theorem 4.1.9. There exist constants C1 , C2 , C3 > 0 such that for t ∈]0, 1] and any s, s ∈ C0∞ (R2n , Ex0 ), t L2 s, s t,0  C1 s2t,1 − C2 s2t,0 , t  (4.1.37) | L2 s, s t,0 |  C3 st,1 s t,1 . Proof. From (4.1.33), c,A0 2 (Dp ) s, s 0,L2 = ∇A0 s20,L2 + (−2pω0,d − pτ0 + ΦE0 ) s, s0,L2 .

(4.1.38)

Hence (4.1.29), (4.1.36) and (4.1.38) applied to κ−1/2 St s instead of s, yield t



L2 s, s t,0 = ∇t s2t,0 + St−1 (−2ω0,d − τ0 + t2 ΦE0 ) s, s t,0 , (4.1.39) which implies the first inequality of (4.1.37). From (4.1.34), we get the second inequality of (4.1.37).
Let δ be the counterclockwise oriented circle in C of center 0 and radius µ0 /4, and let ∆ be the oriented path in C defined by Figure 4.1. By (4.1.26) and (4.1.29), there exists t0 ∈]0, 1] such that for t ∈]0, t0 ], Spec(L2t ) ⊂ {0} ∪ [µ0 , +∞[. Thus (λ − L2t )−1 exists for λ ∈ δ ∪ ∆.

(4.1.40)

184

Chapter 4. Asymptotic Expansion of the Bergman Kernel

∆

i

δ

0

µ0 /4 µ0 /2

−i

Figure 4.1. By using Theorem 4.1.9, exactly in the same way as in proofs of Theorems 1.6.8–1.6.10, we obtain the following analogues which are uniform in x0 ∈ X. Theorem 4.1.10. There exists C > 0 such that for t ∈]0, t0 ], λ ∈ δ ∪ ∆, (λ − L2t )−1 0,0 t  C, (λ − L2t )−1 −1,1  C(1 + |λ|2 ). t

(4.1.41)

Proposition 4.1.11. Take m ∈ N∗ . There exists Cm > 0 such that for t ∈]0, 1],  ∞ 2n Q1 , . . . , Qm ∈ {∇t,ei , Zi }2n i=1 and s, s ∈ C0 (R , Ex0 ), 

  (4.1.42)  [Q1 , [Q2 , . . . , [Qm , L2t ] . . . ]]s, s t,0   Cm st,1 s t,1 . Theorem 4.1.12. For any t ∈]0, t0 ], λ ∈ δ ∪ ∆, m ∈ N, the resolvent (λ − L2t )−1 m+1 maps H m . Moreover for any α ∈ N2n , there exist N ∈ N, Cα,m > 0 t into H t such that for t ∈]0, t0 ], λ ∈ δ ∪ ∆, s ∈ C0∞ (X0 , Ex0 ),

Z α (λ − L2t )−1 st,m+1  Cα,m (1 + |λ|2 )N Z α st,m . (4.1.43) α

α

For m ∈ N, let Qm be the set of operators {∇t,ei1 · · · ∇t,eij }j
m . For k, r ∈ N , let ∗

j j 
 
 ki = k + j, ri = r, ki , ri ∈ N∗ . Ik,r = (k, r) = (ki , ri ) i=0

(4.1.44)

i=1

For (k, r) ∈ Ik,r , λ ∈ δ ∪ ∆, t ∈]0, t0 ], set Akr (λ, t) = (λ − L2t )−k0

∂ rj L2t ∂ r1 L2t t −k1 (λ − L ) · · · (λ − L2t )−kj . 2 ∂tr1 ∂trj

(4.1.45)

4.1. Near diagonal expansion of the Bergman kernel

185

Then there exist akr ∈ R such that ∂r (λ − L2t )−k = ∂tr




akr Akr (λ, t).

(4.1.46)

(k,r)∈Ik,r

Theorem 4.1.13. For any m ∈ N, k > 2(m+ r + 1), (k, r) ∈ Ik,r , there exist C > 0, N ∈ N such that for λ ∈ δ ∪ ∆, t ∈]0, t0 ], Q, Q ∈ Qm ,
QAkr (λ, t)Q st,0  C(1 + |λ|)N Z β st,0 . (4.1.47) |β|
2r

Proof. From Theorem 4.1.12, we deduce that if Q ∈ Qm , there are Cm > 0 and N ∈ N such that for any λ ∈ δ ∪ ∆, Q(λ − L2t )−m t0,0  Cm (1 + |λ|)N .

(4.1.48)

Observe that L2t is self-adjoint with respect to  · t,0 , so after taking the adjoint of (4.1.48) we have for any λ ∈ δ ∪ ∆, (λ − L2t )−m Qt0,0  Cm (1 + |λ|)N .

(4.1.49)

Thus from (4.1.48) and (4.1.49), we obtain (4.1.47) for r = 0. ∂r t Consider now r > 0. By (4.1.34) (cf. (1.6.42)), ∂t r L2 is a combination of  ∂ r3    ∂ r2  ∂ r2 ∂ r1 ∂ r1 ij ∂ r1 , (g (tZ)) ∇ ∇ (d(tZ)), (d (tZ)) ∇ t,e t,e i t,e i j i ∂tr1 ∂tr2 ∂tr3 ∂tr1 ∂tr1 ∂tr2 where d(Z), di (Z) and their derivatives in Z are uniformly bounded for Z ∈ R2n . ∂ r1 ∂ r1  β Now ∂t r1 (d(tZ)) (resp. ∂tr1 ∇t,ei ) (r1  1), are functions of the type d (tZ)Z ,  |β|  r1 (resp. r1 + 1) and d (Z) and its derivatives in Z are bounded smooth functions of Z. Let Rt be the family of operators of the type Rt = {[fj1 Qj1 , [fj2 Qj2 , . . . [fjl Qjl , L2t ] . . . ]]} with fji smooth bounded (with its derivatives) functions and Qji ∈ {∇t,el , Zl }2n l=1 . We handle now the operator Akr (λ, t)Q . We will move first all the terms Z β in d (tZ)Z β (defined above) to the right-hand side of this operator. To do so, we always use the commutator trick as in the proof of Theorem 1.6.10, i.e., each time, we perform only the commutation directly with Z β with |β| > 1). with tZi (not k  β Then Ar (λ, t)Q is as the form |β|
2r Lβ Qβ Z , and Qβ is obtained from Q and ∂ rj L t

its commutation with Z β . Now we move all the terms ∇t,ei in ∂trj 2 to the righthand side of the operator Ltβ . Then as in the proof of Theorem 4.1.12 (cf. Theorem 1.6.10), we get finally that QAkr (λ, t)Q is of the form |β|
2r Lβt Z β where Lβt is a linear combination of operators of the form








Q(λ − L2t )−k0 R1 (λ − L2t )−k1 R2 · · · Rl
(λ − L2t )−kl
Q Q ,

(4.1.50)

186

Chapter 4. Asymptotic Expansion of the Bergman Kernel

with R1 , . . . , Rl
∈ Rt , Q ∈ Q2r , Q ∈ Qm , |β|  2r, and Q is obtained from Q and its commutation with Z β . Since k > 2(m + r + 1), we can use the argument leading to (4.1.48) and (4.1.49), to split the above operator into two parts,









Q(λ − L2t )−k0 R1 (λ − L2t )−k1 R2 · · · Ri (λ − L2t )−ki ,






(4.1.51)




(λ − L2t )−(ki −ki ) · · · Rl
(λ − L2t )−kl
Q Q ,

2 N such that the  · 0,0 t -norm of each part is bounded by C(1 + |λ| ) . Thus the proof of (4.1.47) is complete.


Certainly, as t → 0, the limit of  · t,m exists, and we denote it by  · 0,m . Theorem 4.1.14. For any r  0, k > 0, there exist C > 0, N ∈ N such that for t ∈ [0, t0 ], λ ∈ δ ∪ ∆,  r t 
 ∂ L2 ∂ r L2t      s −  Ct Z α s0,1 ,   ∂tr  r ∂t t=0 t,−1 |α|
r+3  ∂ r  
  t −k k k (λ − L ) − a A (λ, 0) s  Ct(1 + |λ|2 )N  2 r r ∂tr 0,0 (k,r)∈Ik,r

(4.1.52)


Z α s0,0 .

|α|
4r+3

Proof. Note that by (4.1.34) and (4.1.36), as in (1.6.57), for t ∈ [0, 1], k  1, st,k  C




Z α s0,k .

(4.1.53)

|α|
k

An application of Taylor expansion for (4.1.34) leads to the following equation, if s, s have compact support:     ∂ r L t
∂ r L2t      2 s, s − Z α s0,1 .     Cts t,1 r r ∂t ∂t t=0 0,0

(4.1.54)

|α|
r+3

Thus we get the first inequality of (4.1.52). Note that (λ − L2t )−1 − (λ − L20 )−1 = (λ − L2t )−1 (L2t − L20 )(λ − L20 )−1 .

(4.1.55)

By passing to the limit we obtain that Theorems 4.1.10–4.1.12 still hold for t = 0. From Theorems 4.1.10, 4.1.12, (4.1.36), the first equation of (4.1.52) and (4.1.55), we get


   (λ − L2t )−1 − (λ − L20 )−1 s  Ct (1 + |λ|)N Z α s0,0 . 0,0 |α|
3

(4.1.56)

4.1. Near diagonal expansion of the Bergman kernel

187

If we denote by Lλ,t = λ − L2t , then Akr (λ, t)



 ∂ ri L2t ∂ ri L2t  −k −ki − = ··· − · · · Lλ,0 j Lλ,0 ri ri t=0 ∂t ∂t i=1 j   ∂ ri+1 L t   
−k −k0 −ki −ki 2 + Lλ,t · · · Lλ,t − Lλ,0 · · · Lλ,0 j . (4.1.57) ri+1 t=0 ∂t i=0 Akr (λ, 0)

j


−k0 Lλ,t

Now from the first inequality of (4.1.52), (4.1.46), (4.1.56) and (4.1.57), we get the second equation of (4.1.52).
Remark 4.1.15. To get the near diagonal expansion, Theorem 4.1.24, we only need Theorems 4.1.10–4.1.14 for λ ∈ δ. We need the part λ ∈ ∆ in Section 4.2.

4.1.5 Uniform estimate on the Bergman kernel The next step is to convert the estimates for the resolvent into estimates for the spectral projection P0,t : (L2 (X0 , Ex0 ),  · 0 ) → Ker(L2t ). Let P0,t (Z, Z  ) = P0,t,x0 (Z, Z  ), (with Z, Z  ∈ X0 ) be the smooth kernel of P0,t with respect to dvT X (Z  ). Let π : T X ×X T X → X be the natural projection from the fiberwise product of T X on X. Note that L2t is a family of differential operators on Tx0 X with coefficients in End(E)x0 . Thus we can view P0,t (Z, Z  ) as a smooth section of π ∗ (End(E)) over T X ×X T X by identifying a section s ∈ C ∞ (T X ×X T X, π ∗ End(E)) with the family (sx )x∈X , where sx = s|π−1 (x) . Let ∇End(E) be 0,• the connection on End(E) induced by ∇E (which is in turn induced by ∇B,Λ ∗ and ∇E ). Then ∇π End(E) induces naturally a C m -norm of s for the parameter x0 ∈ X. In the rest of this chapter, we will denote by C m (X) the C m -norm for the parameter x0 ∈ X. Theorem 4.1.16. For any m, m , r ∈ N, q > 0, there exists C > 0, such that for t ∈]0, t0 ], Z, Z  ∈ Tx0 X, |Z|, |Z  |  q,   ∂ |α|+|α
| ∂ r    C.  α  α
r P0,t (Z, Z  )  m

∂t C (X) ∂Z ∂Z |α|+|α |
m sup

Proof. By (4.1.40), for any k ∈ N∗ ,  1 P0,t = λk−1 (λ − L2t )−k dλ. 2πi δ

(4.1.58)

(4.1.59)

From (4.1.48), (4.1.49) and (4.1.59), we obtain QP0,t Q 0,0  Cm , t

for Q, Q ∈ Qm .

(4.1.60)

188

Chapter 4. Asymptotic Expansion of the Bergman Kernel

Let | · |(q),m be the usual Sobolev norm on C ∞ (B Tx0 X (0, q + 1), Ex0 ) induced by hEx0 and the volume form dvT X (Z) as in (4.1.36). Observe that by (4.1.34) and (4.1.36), for m > 0, there exists Cq > 0 such that for s ∈ C ∞ (X0 , Ex0 ), supp(s) ⊂ B Tx0 X (0, q + 1), 1 st,m  |s|(q),m  Cq st,m . Cq

(4.1.61)

Now (4.1.60) and (4.1.61) together with Sobolev’s inequalities imply sup |Z|,|Z
|
q

|QZ QZ
P0,t (Z, Z  )|  C ,

for Q, Q ∈ Qm .

(4.1.62)

Thanks to (4.1.34) and (4.1.62), estimate (4.1.58) holds for r = m = 0. To obtain (4.1.58) for r  1 and m = 0, note that from (4.1.59),  r 1 ∂r k−1 ∂ P = λ (λ − L2t )−k dλ , for k  1 . (4.1.63) 0,t ∂tr 2πi δ ∂tr By (4.1.46), (4.1.47), (4.1.63) and the above argument, we get the estimate (4.1.58) with m = 0. Finally, for any vector U on X,  1 π ∗ End(E) π ∗ End(E) P0,t = λk−1 ∇U (λ − L2t )−k dλ. (4.1.64) ∇U 2πi δ π ∗ End(E)

Now we use a similar formula as (4.1.46) for ∇U

∂ r1 L2t ∂tr1 by on Tx0 X

π ∗ End(E) ∇U L2t ,

π ∗ End(E) ∇U L2t

(λ − L2t )−k by replacing

and remark that is a differential operator with the same structure as L2t , i.e., it has the same type as (1.6.42). Then using the above argument, we conclude that (4.1.58) holds for m  1. The proof of Theorem 4.1.16 is complete.
For k large enough, set Fr = Fr,t

1 2πi r!

 λk−1 δ


(k,r)∈Ik,r

akr Akr (λ, 0)dλ, (4.1.65)

1 ∂r = P0,t − Fr . r! ∂tr

Let Fr (Z,Z  ) (Z,Z  ∈ Tx0 X) be the smooth kernel of Fr with respect to dvT X (Z  ). Then Fr ∈ C ∞ (T X ×X T X, π ∗ End(E)). Theorem 4.1.17. For q > 0, there exists C > 0 such that for t ∈]0, 1], Z, Z  ∈ Tx0 X, |Z|, |Z  |  q,     (4.1.66) Fr,t (Z, Z  ) Ct1/(2n+1) .

4.1. Near diagonal expansion of the Bergman kernel

189

Proof. By (4.1.52), (4.1.63) and (4.1.65), there exists C > 0 such that for t ∈]0, 1], Fr,t (q),0  Ct.

(4.1.67)

By Theorem 4.1.16 and (4.1.67), the same as in (1.6.63) and (1.6.64), we get (4.1.66).
Finally, we obtain the following near diagonal estimate for the kernel of P0,t . Theorem 4.1.18. For any k, m, m ∈ N, q > 0, there exists C > 0 such that if t ∈]0, t0 ], Z, Z  ∈ Tx0 X, |Z|, |Z  |  q, k   ∂ |α|+|α
|  
  sup Fr tr (Z, Z  )
 Ctk+1 .  α  α
P0,t −
C m (X) |α|+|α |
m ∂Z ∂Z r=0

(4.1.68)

Proof. By (4.1.65) and (4.1.66), 1 ∂r P0,t |t=0 = Fr . r! ∂tr Now by Theorem 4.1.16 and (4.1.65), Fr has the same estimate as (4.1.58). Again from (4.1.58), (4.1.65), and the Taylor expansion G(t) −

 k
1 ∂rG ∂ k+1 G 1 t r (0)t = (t − t0 )k k+1 (t0 )dt0 , r r! ∂t k! 0 ∂t r=0

(4.1.69) 1 ∂r r! ∂tr P0,t

in

(4.1.70)


we have (4.1.68).

4.1.6 Bergman kernel of L Now we discuss the eigenvalues and eigenfunctions of L20 in detail. (1,0) holds. We choose {wi }ni=1 an orthonormal basis of Tx0 X, such that (1.5.18) √ Let {wj }nj=1 be its dual basis. Then e2j−1 = √12 (wj + wj ) and e2j = √−1 (wj − 2 wj ) , j = 1, . . . , n forms an orthonormal basis of Tx0 X. We use the coordinates on Tx0 X  R2n induced by {ei } as in (1.6.22) and in what follows we also introduce n 2n Z = z+ the complex √ ∂ coordinates √ ∂ z = (z1 , . . . , zn ) on C  R . Thus z, and ∂ ∂ wi = 2 ∂zi , w i = 2 ∂zi . We will also identify z to i zi ∂z and z to i z i ∂z i i when we consider z and z as vector fields. Remark that  2  2 1 1  ∂    (4.1.71)  ∂zi  =  ∂z∂ i  = , so that |z|2 = |z|2 = |Z|2 . 2 2 From (1.6.28), set 1 ∇0,· = ∇· + RxL0 (R, ·), 2

L =−


(∇0,ei )2 − τx0 . i

(4.1.72)

190

Chapter 4. Asymptotic Expansion of the Bergman Kernel

It is very useful to rewrite L by using the creation and annihilation operators. Set bi = −2∇

∂ 0, ∂z

,

b+ i = 2∇

∂ 0, ∂z

i

,

b = (b1 , . . . , bn ) .

(4.1.73)

i

Then by (1.5.18) and (4.1.72), we have 1 ∂ bi = −2 ∂z + ai z i , i 2

1 ∂ b+ i = 2 ∂z i + ai zi , 2

(4.1.74)

and for any polynomial g(z, z) on z and z, + + [bi , b+ j ] = bi bj − bj bi = −2ai δi j , + [bi , bj ] = [b+ i , bj ] = 0 ,

(4.1.75)

[g(z, z), bj ] = 2 ∂z∂ j g(z, z), ∂ [g(z, z), b+ j ] = −2 ∂z j g(z, z) .

By (1.5.22), R˙ L ∈ End(T (1,0) X) is positive, thus ai in (4.1.74) are strictly positive. From (1.5.19), (4.1.29), (4.1.72)–(4.1.75),
L = bi b+ L20 = L − 2ωd,x0 = L + 2aj wj ∧ iwj . (4.1.76) i , i

As bj is the formal adjoint of b+ j , from (4.1.76), we get Ker(L ) = ∩nj=1 Ker(b+ j ).

(4.1.77)

Remark 4.1.19. Let L = C be the trivial holomorphic line bundle on Cn with the canonical frame 1, define by Cn → L, z → (z, 1). Let hL be the metric on L defined by n 2 1 |1|hL (z) := e− 4 j=1 aj |zj | = h(Z) for z ∈ Cn . Let g T C be the canonical metric on Cn . Then L is twice the corL∗ L responding Kodaira Laplacian ∂ ∂ under the trivialization of L by using the n 2 1 ∗ unit section e 4 j=1 aj |zj | 1. Let ∂ be the adjoint of the Dolbeault operator ∂ n associated to L with the trivial metric on (Cn , g T C ). In fact, under the canonical trivialization by 1, L L∗ ∗ ∂ = ∂, ∂ = h−2 ∂ h2 . (4.1.78) n

Set

∗

∂ h = h∂h−1 ,

∗

∂ h = h−1 ∂ h.

(4.1.79)

Then b+ j = 2[i ∂h =

∂ ∂z j

, ∂ h ],

∗

bj = [∂ h , dz j ∧],

1
dz j ∧ b+ j , 2 j

∗

∂h =


j

(4.1.80) i

∂ bj . ∂z j

4.1. Near diagonal expansion of the Bergman kernel

191

Under the trivialization by h−1 ·1, we know the Kodaira Laplacian ∂ ∗ ∗ is ∂ h ∂ h + ∂ h ∂ h , and its restriction on functions is 12 L .

L∗ L

L L∗

∂ +∂ ∂

Theorem 4.1.20. The spectrum of the restriction of L on L2 (R2n ), the space of square integrable functions on R2n , is given by n  
Spec(L |L2 (R2n ) ) = 2 αi ai : α = (α1 , . . . , αn ) ∈ Nn

(4.1.81)

i=1

n and an orthogonal basis of the eigenspace of 2 i=1 αi ai is given by



1
bα z β exp − ai |zi |2 , with β ∈ Nn . (4.1.82) 4 i 

Proof. At first z β exp − 14 i ai |zi |2 , β ∈ Nn are annihilated by b+ i (1  i  n), thus they are in the kernel of L |L2 (R2n ) . Now, by (4.1.75), (4.1.82) are eigenfunc n tions of L |L2 (R2n ) with eigenvalue 2 i=1 αi ai . But the span of functions (4.1.82) 

includes all the rescaled Hermite polynomials multiplied by exp − 14 i ai |zi |2 , which is an orthogonal basis of L2 (R2n ) by Lemma E.1.3. Thus the eigenfunctions in (4.1.82) are all the eigenfunctions of L |L2 (R2n ) . The proof of Theorem 4.1.20 is complete.
Especially an orthonormal basis of Ker(L |L2 (R2n ) ) is 

n n 1/2   1
 aβ β 2 , a z exp − a |z | i j j 4 j=1 (2π)n 2|β| β! i=1

β ∈ Nn .

(4.1.83)

Let P(Z, Z  ) be the smooth kernel of P, the orthogonal projection from (L2 (R2n ),  · 0 ) onto Ker(L ), with respect to dvT X (Z  ). From (4.1.83), we get P(Z, Z  ) =

n  1


  ai exp − ai |zi |2 + |zi |2 − 2zi z i . 2π 4 i i=1

(4.1.84)

We denote by IC⊗E the projection from Λ(T ∗(0,1) X) ⊗ E onto C ⊗ E under the decomposition Λ(T ∗(0,1) X) = C ⊕ Λ>0 (T ∗(0,1) X). Let P N be the orthogonal projection from (L2 (R2n , Ex0 ),  · 0 =  · t,0 ) onto N = Ker(L20 ), and P N (Z, Z  ) its smooth kernel with respect to dvT X (Z  ). Then from (4.1.76), we get P N (Z, Z  ) = P(Z, Z  )IC⊗E .

(4.1.85)

4.1.7 Proof of Theorem 4.1.1 Let f (λ, t) be a formal power series on t with values in End(L2 (R2n , Ex0 )), f (λ, t) =

∞
r=0

tr fr (λ),

fr (λ) ∈ End(L2 (R2n , Ex0 )).

(4.1.86)

192

Chapter 4. Asymptotic Expansion of the Bergman Kernel

By (4.1.31), consider the equation of formal power series for λ ∈ δ, (−L20 + λ −

∞


tr Or )f (λ, t) = IdL2 (R2n ,Ex0 ) .

(4.1.87)

r=1 ⊥

Let N ⊥ be the orthogonal space of N = Ker(L20 ) in L2 (R2n , Ex0 ), and P N be the orthogonal projection from L2 (R2n , Ex0 ) onto N ⊥ . We decompose f (λ, t) according to the splitting L2 (R2n , Ex0 ) = N ⊕ N ⊥ , gr (λ) = P N fr (λ),

⊥

fr⊥ (λ) = P N fr (λ).

(4.1.88)

Using (4.1.88) and identifying the powers of t in (4.1.87), we find that g0 (λ) =

1 N P , λ

⊥

f0⊥ (λ) = (λ − L20 )−1 P N ,

fr⊥ (λ) = (λ − L20 )−1

r


⊥

P N Oj fr−j (λ),

j=1

gr (λ) =

1 λ

r


(4.1.89)

P N Oj fr−j (λ).

j=1

Theorem 4.1.21. There exist Jr (Z, Z  ) ∈ End(Λ(T ∗(0,1) X) ⊗ E)x0 polynomials in Z, Z  with the same parity as r and deg Jr (Z, Z  )  3r, whose coefficients are 0,• polynomials in RT X , RB,Λ , Rdet , RE (dΘ and RL ) and their derivatives of order  r − 2 ( resp. r − 1, r), and reciprocals of linear combinations of eigenvalues of R˙ L at x0 , such that Fr (Z, Z  ) = Jr (Z, Z  )P(Z, Z  ), J0 (Z, Z  ) = IdC⊗E . (4.1.90)  1 (λ−L2t )−1 dλ . Thus by (4.1.63), (4.1.65), (4.1.69) Proof. By (4.1.59), P0,t = 2πi δ and (4.1.88), we know that the operator Fr does not depend on the choice k  1 in (4.1.66), thus   1 1 Fr = gr (λ)dλ + f ⊥ (λ)dλ. (4.1.91) 2πi δ 2πi δ r From Theorem 4.1.20, (1.5.22) and (4.1.76), the only eigenvalue of L20 inside δ is 0. From (4.1.85), (4.1.89) and (4.1.91), we get F0 = P N = P IC⊗E .

(4.1.92)

Generally, from Theorems 4.1.7, 4.1.20, Remark 4.1.8, (4.1.85), (4.1.89), (4.1.91) and the residue formula, we conclude that Fr has the form (4.1.90).
From Theorem 4.1.20, (4.1.76), (4.1.89), (4.1.91) and the residue formula, ⊥ we can get Fr by using the operators (L20 )−1 , P N , P N , Ok (k  r). This gives us a general method to compute Fr in view of Theorem 4.1.20. Especially, from (4.1.89) and (4.1.91), we get the following formula for F1 , F2 .

4.1. Near diagonal expansion of the Bergman kernel

193

Theorem 4.1.22. The following identities hold, ⊥

⊥

F1 = − P N (L20 )−1 O1 P N − P N O1 (L20 )−1 P N , ⊥

⊥

⊥

F2 =(L20 )−1 P N O1 (L20 )−1 P N O1 P N − (L20 )−1 P N O2 P N ⊥

⊥

⊥

⊥

⊥

+ P N O1 (L20 )−1 P N O1 (L20 )−1 P N − P N O2 (L20 )−1 P N ⊥

(4.1.93)

+ (L20 )−1 P N O1 P N O1 (L20 )−1 P N − P N O1 (L20 )−2 P N O1 P N ⊥

⊥

− P N O1 P N O1 (L20 )−2 P N − P N (L20 )−2 O1 P N O1 P N . Remark 4.1.23. In Theorem 8.3.9 and Problem 8.5, we will show that even in the symplectic case, we have (4.1.94) P N O1 P N = 0. Thus the last two terms in (4.1.93) are zero. The following near diagonal expansion of the Bergman kernels is the main result of this section. Theorem 4.1.24. For any k, m, m ∈ N, q > 0, there exists C > 0 such that if √ p  1, Z, Z  ∈ Tx0 X, |Z|, |Z  |  q/ p,  ∂ |α|+|α
|  1   α  α
n Pp (Z, Z  )
p |α|+|α |
m ∂Z ∂Z sup −

k


 1 1 r √ √  Fr ( pZ, pZ  )κ− 2 (Z)κ− 2 (Z  )p− 2 

r=0

C m
(X)

 Cp−

k−m+1 2

. (4.1.95)

Proof. By (4.1.28) and (4.1.29) as in (1.6.66), we have for Z, Z  ∈ R2n , P0,p (Z, Z  ) = t−2n κ− 2 (Z)P0,t (Z/t, Z  /t)κ− 2 (Z  ). 1

1

By Proposition 4.1.6, Theorem 4.1.18 and (4.1.96), we get (4.1.95).

(4.1.96)


Proof of Theorem 4.1.1. Set now Z = Z  = 0 in (4.1.95). Since Theorem 4.1.21 implies F2r+1 (0, 0) = 0, we obtain (4.1.7) when we restrict to degree 0 and br (x0 ) = IC⊗E F2r (0, 0)IC⊗E .

(4.1.97)

Hence we obtain b0 from (4.1.84), (4.1.90) and (4.1.97). 0,• By (1.2.38), (1.2.42), (1.2.48), (1.2.51) and (1.4.27), the curvatures RB,Λ and Rdet are linear combinations of RT X and functions on dΘ and its first derivatives. Now the statement about the structure of br follows from Theorem 4.1.21. In fact, we claim that for any r ∈ N, Fr preserves the Z-grading of Ex0 , and Fr (Z, Z  )|Λ>0 (T ∗(0,1) X)⊗E = 0.

(4.1.98)

194

Chapter 4. Asymptotic Expansion of the Bergman Kernel 0,•

Recall that in Section 4.1.3 we use the connection ∇B,Λ to trivialize the bundle Λ(T ∗0,1) X) on B X (x0 , 4ε), thus the trivialization preserves the Z-grading of Λ(T ∗0,1) X). As Dp2 preserves the Z-grading of Λ(T ∗0,1) X), we know that on B Tx0 X (0, 2ε/t), L2t (thus Or ) preserves the Z-grading of Ex0 = (Λ(T ∗0,1) X) ⊗ E)x0 . As a consequence, we know that Fr (Z, Z  ) also preserves the Z-grading of Ex0 . But from (4.1.76), (λ − L20 )−1 |Λ>0 (T ∗(0,1) X)⊗E is holomorphic for |λ|  µ0 , thus (4.1.89) and (4.1.91) yield (4.1.98). To prove the uniformity part of Theorem 4.1.1, we notice that in the proof of Theorem 4.1.16, we only use the derivatives of the coefficients of L2t with order  2n + m + m + r + 2. Thus by (4.1.70), the constants in Theorems 4.1.16, 4.1.17, (resp. Theorem 4.1.18) are uniformly bounded, if with respect to a fixed metric

g0T X , the C 2n+m+m +r+4 (resp. C 2n+m+m +k+5 )-norms on X of the data (g T X , hL , hE ) are bounded, and g T X is bounded below (as the coefficients of L2t are functions of g T X , hL , hE and their derivatives with order  2). Moreover, taking derivatives with respect to the parameters we obtain a similar equation as (4.1.64),
where x0 ∈ X plays now a role of a parameter. Thus the C m -norm in (4.1.95)
can also include the parameters if the C m -norms (with respect to the parameter x0 ∈ X) of the derivatives of the above data with order  2n + m + k + 5 are bounded. Thus we can take Ck, l in (4.1.7) independent of g T X under our condition (we apply (4.1.95) with k replaced by 2k +1 to get (4.1.7)). This achieves the proof of Theorem 4.1.1.


4.1.8 The coefficient b1 : a proof of Theorem 4.1.2 From the end of Section 4.1.7, to compute the coefficient br in (4.1.7), we only need to do our computation on C ∞ (X, Lp ⊗ E). Thus in this subsection, our operators Lt , ∇t,· , L0 , Or are the restriction of the operators L2t , ∇t,· , L20 , Or in (4.1.31) on C ∞ (R2n , Ex0 ). Recall that {ei } is an orthonormal basis of Tx0 X as in Section 4.1.6. We assume here ω = Θ. Thus in (1.5.19) and (4.1.74), for 1  j  n, aj = 2π.

(4.1.99)

Theorem 4.1.25. The following identities hold:
L0 = bj b+ O1 = 0, j = L, j

rX 1 O2 = RxT0X (R, ei )R, ej ∇0,ei ∇0,ej − RxE0 (wj , wj ) − x0 3 6   1 TX π TX E R (R, ek )ek + Rx0 (z, z)R, ej − Rx0 (R, ej ) ∇0,ej . + 3 x0 3

(4.1.100)

Proof. By (1.2.19) and (4.1.28), κ(Z) = | det(gij (Z))|1/2 = 1 +

1 TX Rx0 (R, ej )R, ej x0 + O(|Z|3 ). 6

(4.1.101)

4.1. Near diagonal expansion of the Bergman kernel

195

If Γlij are the Christoffel symbols of ∇T X with respect to the frame {ei }, then ∂ (∇TeiX ej )(Z) = Γlij (Z)el . By (1.2.1) and (1.2.19), with ∂j := ∂Z , j 1
lk g (∂i gjk + ∂j gik − ∂k gij )(Z) (4.1.102) 2 k

 1 TX Rx0 (R, ej )ei , el x + RxT0X (R, ei )ej , el x + O(|Z|2 ). = 0 0 3

Γlij (Z) =

Observe that J is parallel with respect to ∇T X , thus J ei , ej Z = Jei , ej x0 . By (1.2.12), (1.2.21) and (1.2.27), √ −1 L R (R, el ) = θki (Z)θlj (Z) J ei ,  ej Z Zk (4.1.103) 2π Z

1 TX Rx0 (R, JR)R, el x0 + O(|Z|4 ). = JR, el x0 + 6 By (1.2.30), (4.1.29) and (4.1.103), for t =

= ∇0,ei

we get

 t ∇ei κ (tZ) (4.1.104) t 2κ

t2 t2 T X πRx0 (z, z)R − RxT0X (R, ek )ek , ei + RxE0 (R, ei ) + O(t3 ). − 6 2

∇t,ei = ∇ei +

1

√1 , p

ΓL (ei ) + tΓE (ei ) −

Let {w i (Z)}i be the parallel transport of {wi }i along the curve [0, 1]  u → uZ. By (1.4.31), as in (4.1.34), when we restrict on C ∞ (R2n , Ex0 ), we get,   i , w i )(tZ) − 2πn. Lt = −g ij (tZ) ∇t,ei ∇t,ej − tΓlij (t·)∇t,el (Z) − t2 RE (w (4.1.105) From the fact that RT X is a (1,1)-form with values in End(T X), we get

∇ej RxT0X (z, z)R, ej    = 2 ∂z∂ j RxT0X (z, z)z, ∂z∂ j +

∂ ∂zj



RxT0X (z, z)z, ∂z∂ j

 = 0. (4.1.106)


From (1.2.5), (4.1.102), (4.1.104)–(4.1.106), we derive (4.1.100). ∞

Proof of Theorem 4.1.2. Recall that we do all our computations on C (R , Ex0 ), thus here we still use P to denote the orthogonal projection from L2 (R2n , Ex0 ) onto Ker(L ) and set P ⊥ = 1 − P. By Theorem 4.1.22 and (4.1.100), we know that F2 = − L −1 P ⊥ O2 P − PO2 L −1 P ⊥ .

2n

(4.1.107)

The second term is the adjoint of the first term by Remark 4.1.8. Thus we only need to compute the first term of (4.1.107).

196

Chapter 4. Asymptotic Expansion of the Bergman Kernel

Note that by (4.1.74), (4.1.77), (4.1.84) and (4.1.99),  (b+ i P)(Z, Z ) = 0 ,

(bi P)(Z, Z  ) = 2π(z i − z i )P(Z, Z  ).

(4.1.108)

Thus from Theorem 4.1.20, (4.1.73), (4.1.75), (4.1.100) and (4.1.108), and RT X is a (1, 1)-form, we get 1    RT X (R, ∂z∂ i )R, ∂z∂ j bi bj (P ⊥ O2 P)(Z, 0) = P ⊥ 3  2  4π  T X ∂ R (R, ∂zk )R, ∂z∂ k − RT X (R, ∂z∂ k ) ∂z∂k , ∂z∂ j bj − 3 3    π  TX E ∂ R (z, z)z, ∂zj bj + R (R, ∂z∂ j )bj P (Z, 0) − 3 (4.1.109)       ⊥ 1 TX E ∂ ∂ ∂ = P R (z, ∂z i )z, ∂zj bi bj + R (z, ∂z j )bj P (Z, 0) 6  4b    b b  i j i RT X (z, ∂z∂ i )z, ∂z∂ j + RT X (z, ∂z∂ i ) ∂z∂ j , ∂z∂ j = 6 3 E ∂ + bj R (z, ∂zj ) P (Z, 0). Thus by Theorem 4.1.20, (1.2.5), (4.1.74), (4.1.84), (4.1.99) and (4.1.109), we obtain   b b  i j RT X (z, ∂z∂ i )z, ∂z∂ j − (L −1 P ⊥ O2 P)(0, 0) = − 48π  b   bi  T X j ∂ ∂ R (z, ∂zi ) ∂zj , ∂z∂ j + RE (z, ∂z∂ j ) P (0, 0) + 3π 4π (4.1.110)  1 E ∂ 1  TX ∂ ∂ ∂ ∂ R ( ∂zi , ∂z i ) ∂zj , ∂zj + R ( ∂zj , ∂z∂ j ) = 2π 2π 1 X 1 E ∂ ∂ r + R ( ∂zj , ∂zj ). = 16π x0 2π From (4.1.97), (4.1.107) and (4.1.110), we get (4.1.8). (1,0) X . Hence the 2r-order Since ω = Θ, we have dΘ = 0 and RT X = RT L derivatives of R are the (2r − 2)-order derivatives of RT X . This follows by (4.1.103), as J is parallel with respect to ∇T X . Applying Theorem 4.1.1 we conclude the proof of Theorem 4.1.2.
Remark 4.1.26. From Theorem 4.1.22 and (4.1.100), we see that under the hypothesis of Theorem 4.1.2, we have F1 = 0.

(4.1.111)

4.1.9 Proof of Theorem 4.1.3 ˙L

R −1 E We define a metric on E by hE h . Let RωE be the curvature assoω := det( 2π ) ciated to the holomorphic Hermitian connection of (E, hE ω ); then

RωE = RE − ∂∂ log det(R˙ L ).

(4.1.112)

4.2. Off-diagonal expansion of the Bergman kernel

197

Let  ·, ·ω be the Hermitian product on C ∞ (X, Lp ⊗ E) induced by gωT X , hL , hE ω. Therefore, (C ∞ (X, Lp ⊗ E), ·, ·ω ) = (C ∞ (X, Lp ⊗ E), ·, ·) , dvX,ω = (2π)−n det(R˙ L )dvX .

(4.1.113)

Observe that H 0 (X, Lp ⊗ E) does not depend on g T X , hL or hE . If Pω,p (x, x ), (x, x ∈ X) denotes the smooth kernel of the orthogonal projection from (C ∞ (X, Lp ⊗ E),  ·, ·ω ) onto H 0 (X, Lp ⊗ E) with respect to dvX,ω (x), we have Pp (x, x ) = (2π)−n det(R˙ L )(x )Pω,p (x, x ).

(4.1.114)

Now we can apply Theorem 4.1.2 for the kernel Pω,p (x, x ), since gωT X (·, ·) = ω(·, J·) is a K¨ahler metric on T X. Hence (4.1.9) follows from (4.1.8), (4.1.112) and (4.1.114). Remark 4.1.27. Since X is compact, (4.1.114) allowed us to reduce the general situation considered in Theorem 4.1.1 to the case ω = Θ and apply Theorem 4.1.2. However, if X is not compact, the trick of using (4.1.114) does not work anymore, because the operator associated to gωT X , hL , hE ω might not have a spectral gap (cf. Section 6.1). More generally, the approach here still holds if the curvature RL is just non-degenerate and our argument extends to the symplectic case without any modification, cf. Chapter 8.

4.2 Off-diagonal expansion of the Bergman kernel In this section, we study in detail the relation on the asymptotic expansion of the Bergman kernel and of the heat kernel. Especially, we establish the off-diagonal asymptotic expansion of the Bergman kernel by using the heat kernel. Again the spectral gap property plays an essential role in our approach. This section is organized as follows. In Section 4.2.1, we explain the relation between the asymptotic expansions of the Bergman kernel and of the heat kernel, and we state the off-diagonal asymptotic expansion of the Bergman kernel. In Section 4.2.2, we study the uniform estimate on the heat kernel and the Bergman kernel. In Section 4.2.3, we prove the results stated in Section 4.2.1. We use the notation from Sections 1.6.1, 4.1, and we suppose that the positivity condition (1.5.21) holds for RL .

4.2.1 From heat kernel to Bergman kernel In Section 4.1.3, on B X (x0 , 4ε)  B Tx0 X (0, 4ε), we trivialize Ep (cf. (1.6.1)) by using the parallel transport with respect to the connection ∇B,Ep defined by (1.4.27) along the curve [0, 1]  u → uZ. Recall that E = Λ(T ∗(0,1) X) ⊗ E. Recall that we denote by C m (X) the C m -norm for the parameter x0 ∈ X.

198

Chapter 4. Asymptotic Expansion of the Bergman Kernel

Under this trivialization, for Z, Z  ∈ Tx0 X, Pp (Z, Z  ) ∈ End(Ep,x0 ) = End(Ex0 ). Thus as in Section 4.1.5, we view Pp,x0 (Z, Z  ) := Pp (Z, Z  ), (Z, Z  ∈ Tx0 X, |Z|, |Z  |  2ε), as a smooth section of π ∗ (End(E)) over T X ×X T X. Recall that µ0 , κ, Fr were defined in (1.5.26), (4.1.28), (4.1.65) and (4.1.91). The following off-diagonal expansion of the Bergman kernel which extends Theorem 4.1.24, is the main result of this section. Theorem 4.2.1. There exists C  > 0 such that for any k, m, m ∈ N, there exists C > 0 such that for p  1, Z, Z  ∈ Tx0 X, |Z|, |Z  |  2ε, α, α ∈ N2n , |α|+|α |  m, we have  $ % k  ∂ |α|+|α
| 
1 r 1 √ √  −1    −  − 2 (Z)κ 2 (Z )p 2 P (Z, Z ) − F ( pZ, pZ )κ  α  α
 p r  ∂Z ∂Z  m
pn r=0 C (X)  √ √ √ −(k+1−m)/2  Mk+1,m,m
  Cp (1 + p|Z| + p|Z |) exp(− C µ0 p|Z − Z  |) + O(p−∞ ),

(4.2.1)

with Mk,m,m
= 2(n + k + m + 1) + m.

(4.2.2)

The term O(p−∞ ) means that for any l, l1 ∈ N, there exists Cl,l1 > 0 such that its C l1 -norm is dominated by Cl,l1 p−l . Remark 4.2.2. In (4.2.2), we make precise the exponent Mk+1,m,m
in (4.2.1). For the most applications, we only need to know the existence of the constant Mk+1,m,m
. The following result (and more generally (4.2.35)) relates the coefficients of the expansion of the Bergman kernel and the heat kernel. Theorem 4.2.3. There exist smooth sections br,u of End(Λ(T ∗(0,1) X) ⊗ E) on X which are polynomials in RT X , RE (dΘ and RL ) and their derivatives with order  2r − 2 (resp. 2r − 1, 2r) and functions on the eigenvalues of R˙ L at x, and b0,u = (2π)−n

det(R˙ L ) exp(2uωd) , det(1 − exp(−2uR˙ L ))

(4.2.3)

such that for each u > 0 fixed, we have the asymptotic expansion in the sense of (4.1.7) as p → ∞, k
u exp(− Dp2 )(x, x) = br,u (x)pn−r + O(pn−k−1 ). p r=0

(4.2.4)

Moreover, as u → +∞, with br in (4.1.7), we have br,u (x) = br (x) IC⊗E + O(e− 8 µ0 u ). 1

(4.2.5)

4.2. Off-diagonal expansion of the Bergman kernel

199

Remark 4.2.4. The formula (4.2.4) holds without the assumption (1.5.21) and b0,u is computed in Theorem 1.6.1. In fact, we only need to replace our contour δ ∪ ∆ by Γ in Section 1.6.4, and Theorems 4.1.10–4.1.14 hold for λ ∈ Γ without the assumption (1.5.21). Thus we still get the second equation of (4.2.30).

4.2.2 Uniform estimate on the heat kernel and the Bergman kernel Let e−uL2 (Z, Z  ), (L2t e−uL2 )(Z, Z  ) (Z, Z  ∈ Tx0 X) be the smooth kernels of t t t the operators e−uL2 , L2t e−uL2 with respect to dvT X (Z  ). We view e−uL2 (Z, Z  ), t (L2t e−uL2 )(Z, Z  ) as smooth sections of π ∗ (End(E)) on T X ×X T X. t

t

Theorem 4.2.5. There exists C  > 0 such that for any m, m , r ∈ N, u0 > 0, there exists C > 0 such that for t ∈]0, t0 ], u  u0 , Z, Z  ∈ Tx0 X,  ∂ |α|+|α
| ∂ r  t   sup  α  α
r e−uL2 (Z, Z  )  m

∂t C (X) |α|+|α |
m ∂Z ∂Z  1 2C  |Z − Z  |2 ,  C(1 + |Z| + |Z  |)Mr,m,m
exp µ0 u − 2 u  ∂ |α|+|α
| ∂ r  t   sup  α  α
r (L2t e−uL2 ) (Z, Z  )  m
∂t C (X) |α|+|α
|
m ∂Z ∂Z   1 2C  |Z − Z  |2 .  C(1 + |Z| + |Z  |)Mr,m,m
exp − µ0 u − 4 u

(4.2.6)

Proof. By (4.1.40), for any k ∈ N∗ , t ∈]0, t0 ],  (−1)k−1 (k − 1)! −uL2t e = e−uλ (λ − L2t )−k dλ, (4.2.7) 2πiuk−1 δ∪∆    t (−1)k−1 (k − 1)! −uλ t −k t −k+1 λ(λ − L dλ. L2t e−uL2 = e ) − (λ − L ) 2 2 2πiuk−1 ∆ From (4.1.48), (4.1.49) and (4.2.7), there exists Cm > 0 such that for u  u0 , Q, Q ∈ Qm , we have Qe−uL2 Q t0,0  Cm e 4 µ0 u , t

1

− 2 µ0 u . Q(L2t e−uL2 )Q 0,0 t  Cm e t

1

(4.2.8)

For m  0, let  m be the usual Sobolev norm on C ∞ (R2n , Ex0 ) induced ∗(0,1) = hΛ(Tx0 X)⊗Ex0 and the volume form dvT X (Z) as in (A.1.10). Observe by h that by (4.1.34) and (4.1.36), there exists C > 0 such that for s ∈ C ∞ (X0 , Ex0 ), supp s ⊂ B Tx0 X (0, q), m  0, Ex0

1 (1 + q)−m st,m  sm  C(1 + q)m st,m. C

(4.2.9)

200

Chapter 4. Asymptotic Expansion of the Bergman Kernel

Now (4.2.8), (4.2.9) together with Sobolev’s inequalities implies that if Q, Q ∈ Qm , |QZ QZ
e−uL2 (Z, Z  )|  C(1 + q)2n+2 e 4 µ0 u , t

sup |Z|,|Z
|
q

1

|QZ QZ
(L2t e−uL2 )(Z, Z  )|  C(1 + q)2n+2 e− 2 µ0 u . t

sup |Z|,|Z
|
q

1

(4.2.10)

1

Thus by (4.1.34) and (4.2.10), we derive (4.2.6) with the exponentials e 4 µ0 u , 1 e− 2 µ0 u for the case when r = m = 0 and C  = 0. To obtain (4.2.6) in general, we proceed as follows. Note that the function f is defined in (1.6.12). For h > 1, put  Ku,h (a) =

+∞

−∞

 v 2   dv √ 1√ 1 − f( exp(iv 2ua) exp − 2uv) √ . 2 h 2π

(4.2.11)

Then there exist C  , C1 > 0 such that for any c > 0, m, m ∈ N, there is C > 0 such that for u  u0 , h > 1, a ∈ C, |Im(a)|  c, we have  C1 2  (m
) |a|m |Ku,h (a)|  C exp C  c2 u − h . u

(4.2.12)

For any c > 0, let Vc be the images of {λ ∈ C, |Im(λ)|  c} by the map λ → λ2 . Then Vc = {λ ∈ C, Re(λ) 

1 Im(λ)2 − c2 }, 4c2

(4.2.13)

 u,h be the holomorphic function such that and δ ∪ ∆ ⊂ Vc for c big enough. Let K 2  Ku,h (a ) = Ku,h (a). Then by (4.2.12), for λ ∈ Vc ,  
 (m ) (λ)|  C exp C  c2 u − C1 h2 . |λ|m |K u,h u

(4.2.14)

 u,h (L t )(Z, Z  ) be the kernel of K  u,h (L t ) with respect to dvT X (Z  ). UsLet K 2 2 ing finite propagation speed of solutions of hyperbolic equations, Theorem D.2.1, (D.2.17) and (4.2.11), we find that there exists a fixed constant c > 0 (which depends on ε), such that  u,h (L t )(Z, Z  ) = e−uL2t (Z, Z  ) if |Z − Z  |  c h. K 2

(4.2.15)

By (4.2.14), we see that given k ∈ N, there is a unique holomorphic function  u,h,k (λ) defined on a neighborhood of Vc which verifies the same estimates as K   u,h,k (λ) → 0 as λ → +∞; moreover Ku,h in (4.2.14) and K  (k−1) (λ)/(k − 1)! = K  u,h (λ). K u,h,k

(4.2.16)

4.2. Off-diagonal expansion of the Bergman kernel

201

As in (4.2.7),  u,h (L2t ) = 1 K 2πi  u,h (L2t ) L2t K

 

 u,h,k (λ)(λ − L2t )−k dλ, K

δ∪∆

1 = 2πi

   u,h,k (λ) λ(λ − L2t )−k − (λ − L2t )−k+1 dλ. K

(4.2.17)

∆

Using (4.1.48), (4.1.49) and proceeding as in (4.2.8)–(4.2.10), we find that  u,h (a) or aK  u,h (a), for α, α with |α| + |α |  m, and |Z|, |Z  |  q, for K(a) = K     ∂ |α|+|α
| C1 2  t   2n+m+2  2  C(1 + q) h K(L )(Z, Z ) exp C c u − . (4.2.18)   α  α
2 u ∂Z ∂Z Setting h =

1 c
|Z

− Z  | in (4.2.18), we get for any α, α verifying |α| + |α |  m,

 ∂ |α|+|α
|     α  α
K(L2t )(Z, Z  ) ∂Z ∂Z

  C1  C(1 + |Z| + |Z  |)2n+m+2 exp C  c2 u −  2 |Z − Z  |2 . (4.2.19) 2c u

By (4.2.6) with the exponentials e 4 µ0 u , e− 2 µ0 u for r = m = C  = 0, (4.2.15) and (4.2.19), we infer (4.2.6) for r = m = 0. To prove (4.2.6) for r  1, note that (4.2.7) imply that for k  1,  ∂ r −uL2t (−1)k−1 (k − 1)! ∂r e = e−uλ r (λ − L2t )−k dλ. (4.2.20) r k−1 ∂t 2πiu ∂t δ∪∆ 1

r

1

∂ t −uL2 ). We have a similar equation for ∂t r (L2 e By (4.1.46), (4.1.47) and (4.2.20), we get the similar estimates (4.2.6) with t ∂ r −uL2t ∂ r , ∂tr (L2t e−uL2 ) with the weight 2n+2r+2+m m = C  = 0, (4.2.19) for ∂t re instead of 2n + m + 2. Thus we get (4.2.6) for m = 0. Finally, by using the argument after (4.1.64) and the above argument, we
obtain (4.2.6) for m  1. The proof of Theorem 4.2.5 is complete. t

Recall that P0,t is the orthogonal projection from C ∞ (X0 , Ex0 ) to Ker(L2t ) with respect to  , t,0 . Set  1 t e−uλ (λ − L2t )−1 dλ. (4.2.21) Fu (L2 ) = 2πi ∆ Let Fu (L2t )(Z, Z  ) be the smooth kernel of Fu (L2t ) with respect to dvT X (Z  ). Then by (4.1.40),  +∞ t t L2t e−u1 L2 du1 . (4.2.22) Fu (L2t ) = e−uL2 − P0,t = u

202

Chapter 4. Asymptotic Expansion of the Bergman Kernel

Corollary 4.2.6. With the notation in Theorem 4.2.5,  ∂ |α|+|α
| ∂ r     α  α
r Fu (L2t ) (Z, Z  )  m
∂t C (X) |α|+|α
|
m ∂Z ∂Z  1  C(1 + |Z| + |Z  |)Mr,m,m
exp(− µ0 u − C  µ0 |Z − Z  |). (4.2.23) 8 √

Proof. Note that 18 µ0 u + 2Cu |Z − Z  |2  C  µ0 |Z − Z  |, thus sup



+∞





e

− 14 µ0 u1 − 2C |Z−Z
|2 u 1

du1  e

√ − C

µ0 |Z−Z
|

u



+∞

e− 8 µ0 u1 du1 1

u

8 − 1 µ0 u−√C

µ0 |Z−Z
| = e 8 . (4.2.24) µ0


By (4.2.6), (4.2.22) and (4.2.24), we get (4.2.23). In view of (4.1.46),we set for k large enough Fr,u =

(−1)k−1 (k − 1)! 2πi r! uk−1 (−1) (k − 1)! 2πi r! uk−1 k−1

Jr,u =






e−uλ

∆

 δ∪∆

akr Akr (λ, 0)dλ,

(k,r)∈Ik,r

e−uλ




akr Akr (λ, 0)dλ,

(k,r)∈Ik,r

(4.2.25)

1 ∂r Fu (L2t ) − Fr,u , r! ∂tr 1 ∂ r −uL2t = e − Jr,u . r! ∂tr

Fr,u,t = Jr,u,t

Theorem 4.2.7. There exist C > 0, N ∈ N such that for t ∈]0, t0 ], u  u0 , q ∈ N∗ , Z, Z  ∈ Tx0 X, |Z|, |Z  |  q,   1   Fr,u,t (Z, Z  ) Ct1/(2n+1) (1 + q)N e− 8 µ0 u ,   (4.2.26) 1   Jr,u,t (Z, Z  ) Ct1/(2n+1) (1 + q)N e 2 µ0 u . Proof. Let Jx00 ,q be the vector space of square integrable sections of Ex0 over {Z ∈ Tx0 X, |Z|  q + 1}. Let A(q) be the operator norm of A ∈ L (Jx00 ,q ) with respect to  (q),0 in (4.1.61). By (4.1.52), (4.2.20) and (4.2.25), we get: There exist C > 0, N ∈ N such that for t ∈]0, t0 ], u  u0 , q > 1, Fr,u,t (q)  Ct(1 + q)N e− 2 µ0 u , 1

1

Jr,u,t (q)  Ct(1 + q)N e 4 µ0 u .

(4.2.27)

4.2. Off-diagonal expansion of the Bergman kernel

203

Let φ : R2n → [0, 1] be a smooth function with compact support, equal 1 near 0, such that Tx X φ(Z)dvT X (Z) = 1. Take ν ∈]0, 1]. By the proof of Theorem 0 4.2.5, Fr,u verifies a similar inequality as (4.2.23). Thus by (4.2.23), there exists C > 0 such that if |Z|, |Z  |  q, U, U  ∈ Ex0 ,     Fr,u,t (Z, Z  )U, U   −

Fr,u,t (Z − W, Z  − W  )U, U  

Tx0 X×Tx0 X

1 φ(W/ν) ν 4n

 1  × φ(W /ν)dvT X (W )dvT X (W  )  Cν(1 + q)N e− 8 µ0 u |U ||U  |. (4.2.28) 

On the other hand, by (4.2.27), we have for |Z|, |Z  |  q,   

Fr,u,t (Z − W, Z  − W  )U, U  

Tx0 X×Tx0 X

1 φ(W/ν) ν 4n

 1 1  × φ(W  /ν)dvT X (W )dvT X (W  )  Ct 2n (1 + q)N e− 2 µ0 u |U ||U  |. (4.2.29) ν

By taking ν = t1/(2n+1) , we deduce the first estimate (4.2.26). In the same way,
we obtain (4.2.26) for Jr,u,t . Theorem 4.2.8. There exists C  > 0 such that for any k, m, m ∈ N, there exists C > 0 such that if t ∈]0, t0 ], u  u0 , Z, Z  ∈ Tx0 X,   k  ∂ |α|+|α
|   
 t r   Fr,u t (Z, Z ) sup  α  α
Fu (L2 ) −  |α|+|α
|
m  ∂Z ∂Z r=0

C m
(X)

 1  Ctk+1 (1 + |Z| + |Z  |)Mk+1,m,m
exp(− µ0 u − C  µ0 |Z − Z  |), 8   k  ∂ |α|+|α
|   
t   sup Jr,u tr (Z, Z  )  α  α
e−uL2 −   m

|α|+|α |
m ∂Z ∂Z r=0

 Ctk+1 (1 + |Z| + |Z  |)Mk+1,m,m


C

(4.2.30)

(X)

1 2C  |Z − Z  |2 ). exp( µ0 u − 2 u

Proof. By (4.2.25) and (4.2.26), 1 ∂r Fu (L2t )|t=0 = Fr,u , r! ∂tr 1 ∂ r −uL2t e |t=0 = Jr,u . r! ∂tr

(4.2.31)

Now by Theorem 4.2.5 and (4.2.25), Jr,u , Fr,u have the same estimates ∂ r −uL2t ∂r t , ∂t as ∂t re r Fu (L2 ) in (4.2.6), (4.2.23). Again from the Taylor expansion (4.1.70), (4.2.6), (4.2.23) and (4.2.25), we get (4.2.30).


204

Chapter 4. Asymptotic Expansion of the Bergman Kernel

4.2.3 Proof of Theorem 4.2.1 By (4.2.22) and (4.2.30), for any u > 0 fixed, there exists Cu > 0 such that for t = √1p , Z, Z  ∈ Tx0 X, x0 ∈ X, we have   k  ∂ |α|+|α
|   
  sup tr (Jr,u − Fr,u ) (Z, Z  )  α  α
P0,t −   m

|α|+|α |
m ∂Z ∂Z r=0 C (X)  k+1  Mk+1,m,m
  Cu t (1 + |Z| + |Z |) exp(− C µ0 |Z − Z  |). (4.2.32) Comparing with Theorem 4.1.18, we get Fr = Jr,u − Fr,u .

(4.2.33)

By taken the limit of (4.2.23) as t → 0,   √


1   Fr,u (Z, Z  )  C(1 + |Z| + |Z  |)2(n+r+1) e− 8 µ0 u− C µ0 |Z−Z | .

(4.2.34)

Thus from (4.2.33) and (4.2.34), when u → ∞, Jr,u (Z, Z  ) = Fr (Z, Z  ) + O(e− 8 µ0 u ), 1

(4.2.35)

uniformly on any compact set of Tx0 X × Tx0 X. We now observe that, as a consequence of (4.1.96), (4.2.32) and (4.2.33), we obtain the following important estimate. Theorem 4.2.9. There exists C  > 0 such that for any k, m, m ∈ N, there exists C > 0 such that for α, α ∈ Nn , |α| + |α |  m, Z, Z  ∈ Tx0 X, |Z|, |Z  |  ε, x0 ∈ X, p  1,  $ % k   ∂ |α|+|α
|
1 √ √  − 12   − 12  − r2  P (Z, Z ) − F ( pZ, pZ )κ (Z)κ (Z )p   α  α
0,p r  m
 ∂Z ∂Z pn r=0 C (X)  √ √ √  Cp−(k+1−m)/2 (1 + | pZ| + | pZ  |)Mk+1,m,m
exp(− C  µ0 p|Z − Z  |). (4.2.36) By Theorem 4.2.9 and (4.1.27), we conclude Theorem 4.2.1.

4.2.4 Proof of Theorem 4.2.3 Using (4.1.29) and proceeding as in (4.1.96), we obtain for Z, Z  ∈ Tx0 X,   u t exp − (Dpc,A0 )2 (Z, Z  ) = pn e−uL2 (Z/t, Z  /t)κ−1/2 (Z)κ−1/2 (Z  ). (4.2.37) p

4.2. Off-diagonal expansion of the Bergman kernel

205

From Proposition 1.6.4, we deduce as in Lemma 1.6.5, that for any u > 0 fixed and for any l ∈ N, there exists C > 0 such that for x0 ∈ X, Z, Z  ∈ Tx0 X, |Z|, |Z  |  ε,    u u    Cp−l . (4.2.38)  exp(− Dp2 ) − exp(− (Dpc,A0 )2 ) (Z, Z  ) m
p p C (X) Thus from (4.1.28), (4.2.30), (4.2.37) and (4.2.38) we get k 1 
k+1 u   Jr,u (0, 0)p−r/2  m
 Cp− 2 .  n exp(− Dp2 )(x0 , x0 ) − p p C (X) r=0

(4.2.39)

By Theorem 4.1.7, (4.1.46), (4.2.25) and the discussion after (4.1.97), we know that Jr,u (Z, Z  ) are polynomials in RT X , RE (resp. dΘ, RL ) and their derivatives with order  r − 2 (resp r − 1, r) and functions on the eigenvalues of R˙ L at x0 . If we can prove that for any r ∈ N, J2r+1,u (0, 0) = 0,

(4.2.40)

br,u = J2r,u (0, 0).

(4.2.41)

then we have (4.2.4) and at x0 ,

Relations (4.1.97), (4.1.98), (4.2.35) and (4.2.41) imply (4.2.5). In the rest, we give a proof of (4.2.40). Let {σk } be an orthonormal basis of Ex0 . We denote by

2 2 1 1 (4.2.42) ψα,β,k = cαβ bα z β e− 4 i ai |zi | σk = ϕαβ (Z)e− 4 i ai |zi | σk , such that ψα,β,k 0 = 1. Then ϕαβ (Z) is a polynomial in Z with the same parity as |α| + |β|. Set F even = span{ψα,β,k ; |α| + |β| is even}, F odd = span{ψα,β,k ; |α| + |β| is odd}.

(4.2.43)

Then from Theorem 4.1.20, L2 (R2n , Ex0 ) = F even ⊕ F odd . From Theorems 4.1.7, 4.1.20, (4.2.25), we know that for r odd, Jr,u exchanges F even and F odd . Thus for any α, β ∈ Nn , we have (Jr,u ψα,β,k )(0)∗ ψα,β,k (0) = 0. We claim that Jr,u (Z, Z  ) =




(Jr,u ψα,β,k )(Z  )∗ ψα,β,k (Z),

α,β,k

converges uniformly on any compact set of R2n × R2n .

(4.2.44)

(4.2.45)

206

Chapter 4. Asymptotic Expansion of the Bergman Kernel

To prove the uniform convergence of (4.2.45), set Il := {(α, β); α, β ∈ Nn , |α| + |β|  l} for l ∈ N. For s ∈ L2 (R2n , Ex0 ), we define


JIl s =

s, Jr,u ψα,β,k  ψα,β,k .

(4.2.46)

(Jr,u ψα,β,k )(Z  )∗ ψα,β,k (Z).

(4.2.47)

(α,β)∈Il ,k

Then the smooth kernel of JIl is


JIl (Z, Z  ) =

(α,β)∈Il ,k

From Theorem 4.1.20, we know that for k1 , k2 ∈ N, s ∈ L2 (R2n , Ex0 ), (L20 )k1 JIl (L20 )k2 s =






s, (L20 )k2 Jr,u (L20 )k1 ψα,β,k ψα,β,k

(α,β)∈Il ,k

=






(L20 )k1 JIl (L20 )k2 s, ψα,β,k ψα,β,k . (4.2.48)

(α,β)∈Il ,k

Thus for Theorem 4.1.13, for any q > 1, k1 , k2 ∈ N, there exists Cq > 0 such that for s ∈ L2 (R2n , Ex0 ), supp(s) ⊂ B Tx0 X (0, q + 1), l ∈ N, (L20 )k1 JIl (L20 )k2 s0  (L20 )k1 Jr,u (L20 )k2 s0  Cq s0 .

(4.2.49)

Thus by Theorems A.1.6 and A.1.7, JIl (Z, Z  ) and its derivatives are uniformly bounded for |Z|, |Z  |  q. When l → ∞, JIl → Jr,u in the | · |(q),0 -norm from (4.1.61). Applying the argument from the proof of Theorem 4.2.7, we get (4.2.45). Now, (4.2.44) and (4.2.45) entail (4.2.40). The proof of Theorem 4.2.3 is complete.

Problems Problem 4.1. Let (X, ω) be a K¨ahler manifold and let g T X be the K¨ ahler metric associated to the K¨ahler form ω. Verify that   (1,0) X ), ω . rX = 4π c1 (T (1,0) X, ∇T Problem 4.2. Verify Remark 4.1.19 and Theorem 4.1.22. Problem 4.3. Verify (4.2.12) and (4.2.15).

4.2. Off-diagonal expansion of the Bergman kernel

207

Problem 4.4 (cohomology of O(k)). We continue here Problem 1.8. Let n  1. ˇ Using Cech cohomology show that: H q (C Pn , O(k)) = 0, for (a) q = n, k < 0 and (b) q = 0, k > −n − 1. Moreover for any k ∈ N,   H 0 (CPn , O(k)) = C sα , |α| = k, and α ∈ Nn+1 . Here the holomorphic section sα of O(|α|) is defined by the map Cn+1  z → # αj n 0 n j=0 zj . Calculate the dimension of H (CP , O(k)). n n Describe H (C P , O(k)), k  −n − 1. (One may consult [124, III.5]; for the general cohomology groups H q (C Pn , OCr Pn (O(k))) see [79, VII.10.7], [217, (4.3)]) Problem 4.5. We continue Problems 1.8 and 4.4. Verify that the Riemannian volume form on (CPn , ωF S ) is 1 ωF S (z)n = n!

 √ n −1 dz1 dz 1 · · · dzn dz n . 2π (1 + nj=1 |zj |2 )n+1

Show that for α ∈ Nn+1 , sα 2L2 =

α! . (n + |α|)!

Prove that the Bergman kernel Pp (z, w) associated to O(p) is Pp (z, w) =


α∈Nn+1 ,|α|=p

(n + p)! sα (z) ⊗ sα (w)∗ . α!

Especially for any z ∈ CPn , we have Pp (z, z) = (n + p)!/p!. Problem 4.6 (moment map). In this problem, we use the notation from Problems 1 1.8 and 4.4, and we use the metric g T CP associated to the K¨ahler form ω := 2ωF S 1 on CP . Let K be the canonical basis of Lie S 1 = R, i.e., for t ∈ R, exp(tK) = √ 2π −1t e ∈ S1. We define an S 1 -action on CP1 by g · [z0 , z1 ] = [z0 , gz1 ] for g ∈ S 1 . 1 On our local coordinate U0 , g · z = gz, and the vector field K CP on CP1 induced by K is  √  ∂ 1 ∂ ∂ K CP (z) := ∂t exp(−tK) · z|t=0 = −2π −1 z ∂z − z ∂z .

208

Chapter 4. Asymptotic Expansion of the Bergman Kernel

We define the S 1 -action on (L, hL ) := (O(2), hO(2) ) by exp(tK) · s(2−j,j) = e s(2−j,j) . Verify that it defines an S 1 -action on L which preserves the L metric h . (Hint: (g · s)(x) = g · (s(g −1 · x)) for g ∈ S 1 .) ∂ Let LK s = ∂t exp(tK)·s be the associated Lie derivative for s ∈ C ∞ (CP1 , L). L Let ∇ be the holomorphic Hermitian connection of (L, hL ). On C, set √ 2π −1(1−j)t

µ(K) = 2|z|2 (1 + |z|2 )−1 − 1. Prove that √ − LK , 2π −1µ(K) = ∇L K CP1 and dµ(K) = iK CP1 ω. The map µ is called the moment map associated to the S 1 -action. 1 Let PpS be the smooth kernel of the orthogonal projection from C ∞ (CP1 , Lp ) 1 onto H 0 (CP1 , Lp )S , the S 1 -invariant sub-space of H 0 (CP1 , Lp ). Verify that 1

PpS (z, z) =

(2p + 1)! |z|2p . 2 2 (p!) (1 + |z|2 )2p

Use Stirling’s formula, 1/2 p −p

p! = (2πp)

p e

   1 1+O p

as p → +∞,

to show that for z ∈ C fixed, when p → +∞,   p/π(1 + O( p1 )) for |z| = 1, S1 Pp (z, z) = O(p−∞ ) for |z| = 1. Problem 4.7. We use the notation and assumption from Theorem 4.1.2. Let dν be any volume form on X. Let η be the positive function on X defined by dvX = η dν. The L2 -scalar product ·, ·ν on C ∞ (X, Lp ) is given by  σ1 , σ2 ν := σ1 (x), σ2 (x)Lp dν(x) . X

Let Pν,p (x, x ) (x, x ∈ X) be the smooth kernel of the orthogonal projection from (C ∞ (X, Lp ), ·, ·ν ) onto H 0 (X, Lp ) with respect to dν(x ). Note that Pν,p (x, x ) ∈ Lpx ⊗ Lp∗ x
. Set Kp (x, x ) := |Pν,p (x, x )|2hLp ⊗hLp∗ , x

0

p

x


Rν,p := (dim H (X, L ))/ vol(X, ν),  where vol(X, ν) := X dν. Set vol(X) := X dvX . 

4.3. Bibliographic notes

209

Let QKp be the integral operator associated to Kp which is defined for f ∈ C ∞ (X) by  1 Kp (x, y)f (y)dν(y). QKp (f )(x) := Rν,p X Recall that ∆ is the (positive) Laplace operator on (X, g T X ) acting on the functions on X. We denote by  · L2 the L2 -norm on the function on X with respect to dvX . Prove that there exists a constant C > 0 such that for any f ∈ C ∞ (X), p ∈ N∗ ,       C  QKp − vol(X, ν) η exp − ∆ f   2  p f L2 , vol(X) 4πp L       ∆ ∆ ∆ vol(X, ν) C    p QKp − vol(X) p η exp − 4πp f  2  p f L2 . L (Hint: Use the heat kernel expansion in [15, Theorems 2.23, 2.26], Theorem 4.2.1 and (4.1.114)). Problem 4.8 (Open problem). Let (L, hL ) be a holomorphic Hermitian line bundle on a compact connected complex √ Hermitian manifold (X, Θ). Donnelly [96] showed through an example that if −1RL is just semipositive, the spectral gap property expressed in Theorem 1.5.5 cannot hold. This begs the question whether the asymptotic expansion of the Bergman kernel from Theorems 4.1.1, 4.1.24 and 4.2.1 still hold for semipositive line bundles?

4.3 Bibliographic notes Section 4.1. The existence of the asymptotic expansion (4.1.7), started in the paper of Tian [241] (cf. also Bouche [48], Ruan [208]) following a suggestion of Yau [258], [259], was first established by Catlin [62] and Zelditch [261]. They computed also b0 . √ −1 L R is the K¨ ahler form of (X, g T X ). Lu [156] obtained Assume now ω = 2π Theorem 4.1.2 when E = C, i.e., more information on the coefficients br via RT X , and he computed b1 , b2 , b3 , cf. also [157]; Wang [250] got also b1 in (4.1.8) for general E. When E = C, the existence of the asymptotic expansion similar to (4.1.95) was also obtained in [219, Th. 1] (cf. also [38]). For other versions of the asymptotic expansion (cf. also [135], [63], [18]). In [62], [261], [63], [135] and [219], the Boutet de Monvel–Sj¨ ostrand parametrix for the Szeg¨o kernel [53], [105] was applied, and in [241], [156], [250] the coefficients were computed by the peak section trick in complex geometry [132]. The coefficient b1 (when E = C) can be also obtained from Bismut–Vasserot’s result on the asymptotic of the analytic torsion [35] (cf. Section 5.5) and Donaldson’s moment map picture [89].

210

Chapter 4. Asymptotic Expansion of the Bergman Kernel

Sections 4.1 and 4.2. The techniques here are basically from [69] and [161] which are inspired by local index theory, especially the analytic localization techniques of Bismut and Lebeau [33, §11], [29, §11]. In [69], by using the heat kernel method in Section 4.2, Dai–Liu–Ma established the full off-diagonal expansion of the Bergman kernel, Theorem 4.2.1, and the asymptotic relation between the Bergman kernel and the heat kernel, Theorem 4.2.3, for the spinc Dirac operator on compact symplectic manifolds. By using the asymptotic expansion of the heat kernel, in [69, §5], they also compute b1 in (4.1.8). In [161], by using the resolvent method presented in Section 4.1, we establish the near-diagonal asymptotic expansion of the generalized Bergman kernel, Theorem 4.1.18, associated to the renormalized Bochner Laplacian. Moreover, in [161], we found the formal power series trick in Section 4.1.7 which gives us a general and algorithmic way to compute the coefficients in various cases. In [165], Ma–Zhang have established the corresponding family version in the spirit of Bismut’s family local index theory. They obtain further in [164] the asymptotic expansion of the G-invariant Bergman kernel with a Hamiltonian action of a Lie group. In both papers the spectral gap plays again an essential role. The approach here is slightly different to what was explained in [161, §3.4] ∗ (To extend the operator ∂ + ∂ from a local coordinate to Cn , we need to use the holomorphic coordinate therein.) Here we work in normal coordinates, and we give a unified approach for the spinc Dirac operators and the Kodaira Laplacian by using modified Dirac operators, thus we also work on the full algebra of differential forms; but the final result for the Kodaira Laplacian is in degree zero. Problem 4.6 is taken from [164, §3.3]. Problem 4.7 is from [155] and QKp was defined in Donaldson’s paper [95].

Chapter 5

Kodaira Map In this chapter we present some applications of the asymptotic expansion of the Bergman kernel. We start with an analytic proof of the Kodaira embedding theorem in Section 5.1. In Section 5.2, we explain very briefly Donaldson’s approach to the existence of K¨ ahler metrics of constant scalar curvature and it’s relation to the Bergman kernel. In Section 5.3, we give an introduction to the distribution of zeros of random sections. In Section 5.4, by applying the results in Chapter 4, we establish the asymptotic expansion of the Bergman kernel in the orbifold case, and then we study the analytic aspect of Baily’s orbifold embedding theorem. Finally, in Section 5.5, we explain the asymptotics of the Ray-Singer analytic torsion by Bismut and Vasserot. This Section is quite independent from other topics touched in this book. It can be read after Chapter 1; we only use the spectral gap property (Theorem 1.5.5) and the technique from Section 1.6. We will use the notation from Chapter 4.

5.1 The Kodaira embedding theorem In this section, as an application of the asymptotic expansion of the Bergman kernel, we present an analytic proof of the Kodaira embedding theorem and study the convergence of the induced Fubini–Study metrics. In Section 5.1.1, we recall some facts on the projective spaces and Grassmannian manifolds. In Section 5.1.2, we present an analytic proof of the Kodaira embedding theorem, cf. also Section 8.3.5. In Section 5.1.3, we explain the classical proof of the Kodaira embedding theorem. In Section 5.1.4, we study the Grassmannian embedding.

212

Chapter 5. Kodaira Map

5.1.1 Universal bundles Let V be an m-dimensional complex vector space and let V ∗ be its dual. The projective space P(V ) is the set of complex lines through the origin in V . A line l ⊂ V is determined by the hyperplane {f (l) = 0; f ∈ V ∗ } ⊂ V ∗ . A line l ⊂ V is also determined by any 0 = u ∈ l, so we can write P(V ) = V
{0}/ ∼; here u ∼ v if and only if there is λ ∈ C∗ such that u = λ v. For v ∈ V
{0}, we denote by [v] the complex line through v, and [v] ∈ P(V ) the corresponding point. In the following, we will use these different points of view about P(V ). Let O(−1) be the universal (tautological) line bundle on P(V ∗ ), then O(−1) = {(h, f ) ∈ P(V ∗ ) × V ∗ , f ∈ h ⊂ V ∗ }.

(5.1.1)

For k ∈ Z, set O(k) := O(−1)−k , especially, O(1) is the dual bundle of O(−1). Let hV be a Hermitian metric on V ; it induces naturally a Hermitian metric V∗ h on V ∗ , thus it induces a Hermitian metric hO(−1) on O(−1), as a sub-bundle of the trivial bundle V ∗ on P(V ∗ ). Let hO(1) be the Hermitian metric on O(1) induced by hO(−1) . For any v ∈ V , the linear map V ∗  f → (f, v) ∈ C defines naturally a holomorphic section σv of O(1) on P(V ∗ ). By the definition, for f ∈ V ∗
{0}, at [f ] ∈ P(V ∗ ), we have |σv ([f ])|2hO(1) = |(f, v)|2 /|f |2hV ∗ .

(5.1.2)

Let RO(1) be the curvature of the holomorphic Hermitian (Chern) connection on O(1) on P(V ∗ ). The Fubini–Study form ωF S is the K¨ ahler form associated ∇ to the Fubini–Study metric on P(V ∗ ), and is defined by : for any 0 = v ∈ V , √ √ −1 O(1) −1 R ωF S = = ∂∂ log |σv |2hO(1) on {x ∈ P(V ∗ ), σv (x) = 0}. (5.1.3) 2π 2π O(1)

The Grassmannian G(k, V ) is defined to be the set of k-dimensional complex linear subspaces of V ; we write G(k, m) for G(k, Cm ). G(k, V ) is also identified to G(m− k, V ∗ ). The universal bundle U on G(k, V ∗ ) is a k-dimensional holomorphic vector bundle defined by U = {(h, f ) ∈ G(k, V ∗ ) × V ∗ , f ∈ h ⊂ V ∗ }.

(5.1.4)

Let hU be the Hermitian metric on U induced by hV , by considering U as a ∗ ∗ sub-bundle of the trivial bundle V ∗ on G(k, V ∗ ). Let hU , hdet U be the Hermitian metrics on U ∗ , det U ∗ induced by hU . For any v ∈ V , the linear map V ∗  f → (f, v) ∈ C defines naturally a holomorphic section σv of U ∗ on G(k, V ∗ ). At [V1∗ ] ∈ G(k, V ∗ ), assume that {f 1 , . . . , f k } is a basis of V1∗ , {ej } is an or∗ thonormal basis of (V ∗ , hV ), and {ej } its dual basis. Then there exist Aij ∈ C such

5.1. The Kodaira embedding theorem



213

. Then fi = ((AA∗ )−1/2 A)ij ej is an orthonormal basis of V1∗ . Thus σv ([V1∗ ]) = ki=1 fi (fi , v), where {fi } is the dual basis of {fi }. Thus that f i = Aij ej , v =

|σv ([V1∗ ])|hU ∗ = 2

j

vj ej . Set A = (Aij )

1ik

1
j
m

k 
m 2
  ((AA∗ )−1/2 A)ij vj  = (A∗ (AA∗ )−1 A)ij vj v i .  i=1

(5.1.5)

j=1

∗

Let Rdet U be the curvature of the holomorphic Hermitian connection of det U ∗ → G(k, V ∗ ). Set √ −1 det U ∗ R ωF S = . (5.1.6) 2π We verify that ωF S is actually a positive (1, 1)-form. At [V1∗ ] ∈ G(k, V ∗ ), we choose ∗ j k {ej }m j=1 an orthonormal basis of V such that {e }j=1 is an orthonormal basis of V1∗ , then B = (Bjl )1lm−k → span{ej + Bjl ek+l }j is a local chart of G(k, V ∗ ) 1
j
k

near [V1∗ ], and A = (I, B), where I is the k × k identity matrix. Especially   (I + BB ∗ )−1 (I + BB ∗ )−1 B ∗ ∗ −1 A (AA ) A = . (5.1.7) B ∗ (I + BB ∗ )−1 B ∗ (I + BB ∗ )−1 B From (5.1.5) and (5.1.7), we have |σe1 ∧ · · · ∧ σek |2hdet U ∗ = det(σei , σej hU ∗ )1
i,j
k = det((A∗ (AA∗ )−1 A)ij )1
i,j
k = det(I + BB ∗ )−1 .

(5.1.8)

As σe1 ∧ · · · ∧ σek is a holomorphic frame of det U ∗ , we get from (1.5.8) and (5.1.8), √ −1 ∗ dBjl ∧ dB jl . ωF S ([V1 ]) = (5.1.9) 2π In analogy to the projective space we call ωF S and its associated Hermitian metric the Fubini–Study form and Fubini–Study metric on G(k, V ∗ ), respectively.

5.1.2 Convergence of the induced Fubini–Study metrics Let (X, J) be a compact complex manifold with complex structure J and dim X = n. We introduce a Riemannian metric g T X on T X compatible with J and denote by dvX the Riemannian volume form on (X, g T X ). We consider a holomorphic positive line bundle L on X, endowed with a Hermitian metric hL such that the curvature RL associated to hL verifies (1.5.21). p The metric hL on L induces metrics hL on Lp . We denote by ω and Θ the (1, 1)-forms on X associated to RL and g T X as in (1.5.14). Then dvX = Θn /n!.

214

Chapter 5. Kodaira Map

On C ∞ (X, Lp ) there is an L2 -scalar product ·, · induced by hL and g T X as in (1.3.14); for s, s ∈ C ∞ (X, Lp ) we set  s(x), s (x)Lp dvX (x) . (5.1.10) s, s  = X 0

p

The scalar product ·, · induces an L2 -metric hH (X,L ) on the space H 0 (X, Lp ) of holomorphic sections of Lp on X. Let Pp (x, x ), (x, x ∈ X), be the smooth kernel of the orthogonal projection Pp from (C ∞ (X, Lp ), ·, ·) onto H 0 (X, Lp ), with respect to dvX (x ). d

p , (dp = dim H 0 (X, Lp )), is an orthonormal basis of (H 0 (X, Lp ), If {Sip }i=1 H 0 (X,Lp ) h ), then by (4.1.4),

Pp (x, x) =

dp


|Sip (x)|2hLp .

(5.1.11)

i=1

The Kodaira map for Lp was defined in (2.2.10) and (2.2.11): Φp : X
Blp −→ P(H 0 (X, Lp )∗ ),

(5.1.12)

Φp (x) = {s ∈ H 0 (X, Lp ) : s(x) = 0},

where Blp = {x ∈ X : s(x) = 0 for all s ∈ H 0 (X, Lp )} is the base locus of H 0 (X, Lp ). Definition 5.1.1. Let F be a holomorphic line bundle over X. F is called semi-ample if there exists p0 such that Blp = ∅ for all p  p0 (so that Φp are holomorphic maps on X for p  p0 ). F is called ample if F is semi-ample and Φp is an embedding for p large enough. F is called very ample if Bl1 = ∅ and Φ1 is an embedding. Due to the Cartan–Serre–Grothendieck lemma 5.1.11, we can require in the definition of ampleness that Φp be an embedding for just one p. Thus, F is ample if and only if F p is very ample for p large enough. Lemma 5.1.2. The positive line bundle L is semi-ample on X. Proof. By Theorem 4.1.1, we know that uniformly for x ∈ X, Pp (x, x) = b0 (x)pn + O(pn−1 ),

with b0 (x) = det

 R˙ L  2π

(x) > 0.

(5.1.13)

Since X is compact, there exists p0 such that Pp (x, x) = 0 for all p  p0 and x ∈ X. By (5.1.11), Blp = ∅ for all p  p0 .
Theorem 5.1.3. For p  p0 , Φp : X −→ P(H 0 (X, Lp )∗ ) is holomorphic and the map Ψp : Φ∗p O(1) → Lp , Ψp ((Φ∗p σs )(x)) = s(x),

for any s ∈ H 0 (X, Lp )

(5.1.14)

5.1. The Kodaira embedding theorem

215

defines a canonical isomorphism from Φ∗p O(1) to Lp on X, and under this isomorphism, we have ∗

hΦp O(1) (x) = Pp (x, x)−1 hL (x). p

(5.1.15)

∗

Here hΦp O(1) is the metric on Φ∗p O(1) induced by hO(1) on O(1) on P(H 0 (X, Lp )∗ ). Proof. Let {sj } be any basis of H 0 (X, Lp ), and {sj } be its dual basis. Let eL be any 0 p local frame of L near x0 ∈ X, and write sj = fj e⊗p L . Then for any s ∈ H (X, L ), dp 


dp 
fj (x)sj , s e⊗p (x) = sj (x)(sj , s) = s(x). L

j=1

(5.1.16)

j=1

By (5.1.12) and (5.1.16), we have Φp (x) =

dp 


 fj (x)sj ∈ P(H 0 (X, Lp )∗ ).

(5.1.17)

j=1

Taking a holomorphic local frame eL near x0 , then fj (x) are holomorphic; from (5.1.17), we know Φp is holomorphic. dp By definition of σs and (5.1.17), for j=1 fj (x)sj ∈ O(−1)Φp (x) , we have dp dp  

(Φ∗p σs )(x), fj (x)sj e⊗p (x) = fj (x)(s, sj )e⊗p L L (x) = s(x). j=1

(5.1.18)

j=1

Thus (Φ∗p σs )(x) = 0 is equivalent to s(x) = 0. For any ζ ∈ (Φ∗p O(1))x , from Lemma 5.1.2, take s ∈ H 0 (X, Lp ) such that ζ = (Φ∗p σs )(x). Thus the map Ψp in (5.1.14) is well defined from (5.1.18). As (Φ∗p σs ), s are holomorphic sections of Φ∗p O(1), Lp , from (5.1.14), we know Ψp is holomorphic. 0 p To get (5.1.15), we take {sj } an orthonormal basis of (H 0 (X, Lp ), hH (X,L ) ). Then by (5.1.2), (5.1.11), (5.1.16) and (5.1.17), for ζ ∈ (Φ∗p O(1))x , there is s ∈ H 0 (X, Lp ) such that ζ = (Φ∗p σs )(x), then under the isomorphism Ψp , |ζ|2 Φ∗p O(1) = |σs (Φp (x))|2hO(1) h dp fj (x)sj , s)|2 |( j=1 |s(x)|2hLp = dp = dp = Pp (x, x)−1 |ζ|2hLp . (5.1.19) 2 2 j | j=1 fj (x)s |hH 0 (X,Lp )∗ j=1 |sj (x)|hLp The proof of Theorem 5.1.3 is completed. The next result shows that the Kodaira map tends to be isometric.




216

Chapter 5. Kodaira Map

Theorem 5.1.4 (Tian–Ruan). Assume that the line bundle (L, hL ) is positive. Then the induced Fubini–Study form p1 Φ∗p (ωF S ) converges in the C ∞ topology to ω ; for any l  0 there exists Cl > 0 such that 1  Cl   .   Φ∗p (ωF S ) − ω  l p p C (X) Proof. From (1.5.8), (5.1.3) and (5.1.15), we get √ √ 1 ∗ −1 L −1 ∂∂ log Pp (x, x) . Φ (ωF S ) − R =− p p 2π 2πp

(5.1.20)

(5.1.21)

By (5.1.13), ∂∂ log Pp (x, x) = ∂∂ log(b0 (x)) + O(1/p).

(5.1.22)


From (5.1.21) and (5.1.22), we get (5.1.20). √ −1 L 2π R

(i.e., Remark 5.1.5. Assume that g T X is the metric associated to ω = ω = Θ). Then b0 (x) = 1. Thus from (5.1.21) and (5.1.22), for any l ∈ N, there exists Cl > 0 such that 1  Cl    2 . (5.1.23)  Φ∗p (ωF S ) − ω  l p p C (X) From Theorem 5.1.4, we deduce immediately: Lemma 5.1.6. Φp is an immersion for p large enough. (1,0)

Proof. Let (Φp )∗,x be the linear tangential map of Φp at x ∈ X. For v ∈ Tx X, v = 0, we have ω(v, v) > 0. By evaluating p1 Φ∗p (ωF S ) − ω on (v, v) and using the estimate (5.1.20), we see that there exists p1 (not depending on x and v) such that (Φp