Holomorphic Morse Inequalities And Bergman Kernels

Progress in Mathematics Volume 254 Series Editors H. Bass J. Oesterlé A. Weinstein Xiaonan Ma George Marinescu Holo...

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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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,6%1%LUNKlXVHU9HUODJ$*%DVHO%RVWRQ%HUOLQ 7KLVZRUNLVVXEMHFWWRFRS\ULJKW$OOULJKWVDUHUHVHUYHGZKHWKHUWKHZKROHRUSDUW RIWKHPDWHULDOLVFRQFHUQHGVSHFL¿FDOO\WKHULJKWVRIWUDQVODWLRQUHSULQWLQJUHXVHRI LOOXVWUDWLRQVEURDGFDVWLQJUHSURGXFWLRQRQPLFUR¿OPVRULQRWKHUZD\VDQGVWRUDJHLQ GDWDEDQNV)RUDQ\NLQGRIXVHZKDWVRHYHUSHUPLVVLRQIURPWKHFRS\ULJKWRZQHUPXVW be obtained. © 2007 Birkhäuser Verlag AG Basel · Boston · Berlin 32%R[&+%DVHO6ZLW]HUODQG Part of Springer Science+Business Media Printed on acid-free paper produced of chlorine-free pulp. TCF Printed in Germany ,6%1

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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 Université de Paris VI Joseph Oesterlé Université de Paris VI Oriol Serra Universitat Politècnica de Catalunya, Barcelona Alan Weinstein University of California at Berkeley

Ferran Sunyer i Balaguer Prize winners since 1997: 1997

Albrecht Böttcher 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

André 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 José M. Mazón 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 José 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

Contents

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 Kähler 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 Kähler 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 (Poincaré 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 Poincaré 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 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 = 2p 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


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.


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


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)


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


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 .


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


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 }jm . 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).


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


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


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).


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 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


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 Kähler 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.)


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


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-Kähler 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–Hörmander 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


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:


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|

74


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


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


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 .


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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.

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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.


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


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 1jm

Jxkj : H 0 (X, L) −→

Lxj ⊗ (OX,xj /mk+1 xj ),

(2.2.8)

1jm

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


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)


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


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)

1jm

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


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


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 ) 1im

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.


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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


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.


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


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)


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 Poincaré duality), which associates to each homology class [Y ] ∈ H2k (X, Z) of a

92


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


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 Poincaré 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


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.


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 Kähler 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


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 à 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


∗ 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 (Poincaré–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 Poincaré–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)


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


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.


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


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.


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


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


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


" 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 .


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


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)


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 Kähler 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)


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


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.


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


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:


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


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


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 Kähler, 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


In this section, we present an example of Kollár, 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)


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 (Kollár–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


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


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 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 = GE = 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 CE 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


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


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)


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 = 2p,X the operator Dp2 = 2p with Dirichlet boundary condition (cf. (D.1.25)) on X.

138


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)


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


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)


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


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)


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


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 (aI −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. 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)


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


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) ∈ ∪jq A j , we obtain (3.2.57) invoking (3.2.39).


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


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 Kähler 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 Kähler, 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


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 Kähler 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

3L s, s (2pC0 − C)s2L2 . We can thus proceed just as in the proof of (i).

(3.3.15)

152


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 Kähler. 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 Kähler 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


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.


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 XK1 (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


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.


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


, 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 Poincaré–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


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


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


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)


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


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)


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


(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 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


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.


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


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 Kähler 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


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 =


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


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)


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


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.


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


∆

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 }jm . 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)


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


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)


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


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) .


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


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


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


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 .


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)


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


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 Kähler 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


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)


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


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)


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


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)


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


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


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


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 Kähler manifold and let g T X be the K¨ ahler metric associated to the Kähler 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).


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 Kähler 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


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 .


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 Szegö 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


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

1jm

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 ∗ ) 1jk

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 ∗ )1i,jk = det((A∗ (AA∗ )−1 A)ij )1i,jk = 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


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)


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


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 )∗,x (v) = 0 for all p p1 . Our next goal is to show that Φp is injective for p large enough. For this we need the following important notion. Definition 5.1.7. For p p0 , a peak section of Lp at x ∈ X is a unit norm generator Sxp of the orthonormal space to Φp (x) such that |Sxp (x)|2hLp = Pp (x, x). The concrete construction goes as follows. Let x0 ∈ X be fixed. Since Φp is base point free for p p0 , we can consider the hyperplane Φp (x0 ) ⊂ H 0 (X, Lp ). dp of H 0 (X, Lp ) such that the first dp − 1 We construct an orthonormal basis {Sip }i=1 p elements belong to Φp (x0 ). Then Sdp is a unit norm generator of the orthogonal complement of Φp (x0 ). This is a peak section at x0 . Indeed, it follows from (5.1.11) that |Sxp0 (x0 )|2 = Pp (x0 , x0 ) and Pp (x, x0 ) = Sxp0 (x) ⊗ Sxp0 (x0 )∗ and therefore Sxp0 (x) =

1 Pp (x, x0 ) · Sxp0 (x0 ). Pp (x0 , x0 )

(5.1.24)


217

From (4.1.68), (4.1.90) and (5.1.24), we deduce that for a sequence {rp } with √ rp → 0 and rp p → ∞, |Sxp0 (x)|2 dvX (x) = 1 − o(1) , for p → ∞. (5.1.25) B X (x0 ,rp )

Relation (5.1.25) explains the term ‘peak section’: when p grows, the mass of Sxp0 concentrates near x0 . Lemma 5.1.8. For any p p0 there exists kp such that Φpk is injective for all k kp . Proof. Let Diag(X × X) := {(x, x) : x ∈ X}. Consider (x, y) ∈ Diag(X × X). If Φp (x) = Φp (y), we can construct as before the peak section Sxp = Syp . For large p, we learn from relation (5.1.25) that the mass of Sxp = Syp (which is by definition 1) concentrates both near x and y and approaches therefore 2 if p −→ ∞. This is a contradiction which proves the existence of some p(x, y) such that Φp (x) = Φp (y) for all p p(x, y) . For p p0 , we define the analytic set Ap := {(x, y) ∈ X × X : Φp (x) = Φp (y)} . Consider (x, y) ∈ Ap . Hence, there exist sections S1 , S2 ∈ H 0 (X, Lp ) such that S1 (x) = 0, S1 (y) = 0 and S2 (x) = 0, S2 (y) = 0. Then the tensor powers S1⊗k , S2⊗k ∈ H 0 (X, Lpk ) have the same properties: S1⊗k (x) = 0, S1⊗k (y) = 0 and S2⊗k (x) = 0, S2⊗k (y) = 0. Therefore, (x, y) ∈ Apk and hence Apk ⊂ Ap for all p p0 and k 1. Let us fix some p p0 and consider the descending sequence {Apk }k1 of analytic sets. Lemma 5.1.9 (Noetherian property of analytic sets). Every descending chain of analytic sets in a complex space is stationary over any relatively compact subset. Since X is compact, by Lemma 5.1.9, we know that there exists kp such that Apk = Apkp for all k kp . We show now that ∩k1 Apk = Diag(X × X). Indeed, let (x, y) ∈ Diag(X × X). If (x, y) ∈ Ap , of course (x, y) ∈ ∩k1 Apk . If (x, y) ∈ Ap , there exists S ∈ H 0 (X, Lp ), S ∈ Φp (x) = Φp (y); thus S(x) = 0, S(y) = 0. On the other hand, by the first part of the proof, there exists k = k(x, y) and S1 ∈ H 0 (X, Lk ) with S1 (x) = 0, S1 (y) = 0. But then S ⊗ S1⊗p ∈ H 0 (X, Lp(k+1) ) and S ⊗ S1⊗p (x) = 0, S ⊗ S1 (y) = 0 and therefore (x, y) ∈ Apk ⊃ ∩k1 Apk . We deduce that Apk = Diag(X × X) for all k kp and therefore Φpk is injective for this range of k. Theorem 5.1.10 (Chow). Every closed analytic subset of C PN is a projective algebraic variety. Thus, a projective manifold is necessarily algebraic. We leave the proof as an exercise (Problems 5.4 and 6.3).

218


Lemma 5.1.11 (Cartan–Serre–Grothendieck). Let F be a holomorphic line bundle over a compact complex manifold X. The following assertions are equivalent: (i) There exists p such that the Kodaira map Φp for F p is an embedding. (ii) (Serre vanishing theorem) For any coherent analytic sheaf F on X, there exists p0 (F ) > 0 such that H q (X, F p ⊗ F ) = 0 for q 1, p p0 (F ). (iii) The Kodaira map Φp for F p is an embedding for p large enough. Proof. (i) =⇒ (ii). Since Φp is an embedding, we identify, X with a complex submanifold of C Pdp −1 . By Chow’s theorem 5.1.10, the complex structure of X is induced by a structure of algebraic subvariety of C Pdp −1 . The GAGA principle of Serre implies that F is a coherent algebraic sheaf. Since F p ∼ = Φ∗p O(1), we identify p F with OX (1) := O(1)|X . We next divide the proof in two steps. Step 1: F is very ample, i.e., F = OX (1). Let i : X → C Pdp −1 be the inclusion and i∗ F be the extension by zero outside X which is a coherent algebraic sheaf. Then for any r ∈ N, H q (X, OX (r) ⊗ F ) = H q (C Pdp −1 , O(r) ⊗ i∗ F ),

(5.1.26)

so we can and will consider only the case X = C Pdp −1 in the sequel of this step. The cohomology of the sheaves O(m) is well known: by Problem 4.4, we know that (ii) holds for F = OX (m) where m ∈ Z. Hence: (ii) For any sheaf F = ⊕m OX (m), where m runs over a finite subset of Z, the Serre vanishing theorem holds. For a general coherent algebraic sheaf, we prove the statement (ii) by descendˇ ing induction on q. By a basic result about Cech cohomology, H q (X, OX (m) ⊗ F ) ˇ is canonically isomorphic to the Cech cohomology of an affine cover of X. Since X has an affine cover U with dp elements, we have H dp (X, OX (m) ⊗ F ) = H dp (U , OX (m) ⊗ F ) = 0.

(5.1.27)

Assume that (ii) holds for q dp . We show it holds for q − 1 too. By a theorem of Serre, there exists an exact sequence E := ⊕N i=1 OX (mi ) −→ F −→ 0,

(mi ∈ Z).

(5.1.28)

The kernel G of this morphism is an algebraic coherent sheaf. We tensor the exact sequence 0 −→ G −→ E −→ F −→ 0 with OX (r) and take the following piece of the exact cohomology sequence: H p (X, OX (r) ⊗ E ) −→ H q (X, OX (r) ⊗ F ) −→ H q+1 (X, OX (r) ⊗ G ). (5.1.29) For r 0, the left-hand side vanishes by (ii) and the right-hand side vanishes by the induction hypothesis. Therefore H q (X, OX (r) ⊗ F ) = 0 for q 1 and r 0.


219

Step 2: the general case. By Step 1, we know that H q (X, F pr ⊗ F ) = 0 for q 1, r 0. We apply this for F replaced by F ⊗ F , . . . , F p−1 ⊗ F and get (ii) in full generality. (ii) =⇒ (iii). For (x, y) ∈ X × X, we define the sheaf I (x, y) in the following way: for x = y, I (x, y) is the sheaf of holomorphic functions on X which vanish at x and y; for x = y set I (x, y) := I (x)2 , the sheaf of holomorphic functions on X which vanish at x to second order. I (x, y) are coherent analytic sheaves. By hypothesis there exists p0 (x, y) such that H 1 (X, F p ⊗ OX /I (x, y)) = 0 for p p0 (x, y). Upon taking the exact cohomology sequence of the exact sequence of sheaves 0 → F p ⊗ I (x, y) −→ F p −→ F p ⊗ OX /I (x, y) → 0, we obtain that the sequence H 0 (X, F p ) −→ H 0 (X, F p ⊗ OX /I (x, y)) −→ 0

(5.1.30)

is exact for p p0 (x, y). But H 0 (X, F p ⊗ OX /I (x, y)) equals Fx ⊗ (OX,x /mx ) ⊕ Fy ⊗ (OX,y /my ) for x = y and Fx ⊗ (OX,x /m2x) for x = y, where mx ∈ OX,x is the maximal ideal of the ring of germs of holomorphic functions at x. It follows that there exist neighborhoods Ux and Uy of x and y, respectively, such that H 0 (X, F p ) −→ H 0 (X, F p ⊗ OX /I (x , y )) −→ 0

(5.1.31)

for all x ∈ Ux , y ∈ Uy and p p0 (x, y). Since X × X is compact, we find a single positive integer p0 such that (5.1.30) is exact for all (x, y) ∈ X × X and p p0 . By applying this result for x = y (resp. x = y) we see that Φp is injective (resp. immersion) for p p0 . (iii) =⇒ (i). Obvious. From Lemmas 5.1.2, 5.1.6, 5.1.8 and 5.1.11 follows immediately: Theorem 5.1.12 (Kodaira embedding theorem). (i) A compact complex manifold X is a projective manifold if and only if X possesses a positive line bundle. (ii) A holomorphic line bundle L on a compact complex manifold X is ample if and only if it is positive.

5.1.3 Classical proof of the Kodaira embedding theorem In the proof of the Kodaira embedding theorem for orbifolds, we will make use of the classical proof of the Kodaira embedding theorem, which we include here. In Section 8.3.5, we find another analytic proof of the Kodaira embedding theorem. For more information on Theorem 5.1.12, we refer to Problems 5.1–5.6. Let L be a holomorphic positive line bundle over a compact complex manifold X. =X x,y → X of X with For (x, y) ∈ X × X, we define the blow-up of π : X center {x, y} as follows. If x = y, it is the usual blow-up at the points x and y.

220


x,y is the We set Ex = π −1 (x), Ey = π −1 (y) and E = Ex,y = Ex + Ey . If x = y, X blow-up at x as a manifold, and we set Ex,x := 2Ex = 2E. Lemma 5.1.13. Fix m ∈ N. Then there exists p0 such that the line bundle ∗ OX (−mE) ⊗ π ∗ (Lp ) ⊗ KX

is positive and hence O (−mE) ⊗ π ∗ (Lp )) = 0, for all p p0 and (x, y) ∈ X × X. (5.1.32) H 1 (X, X Proof. By Proposition 2.1.7, we have ∗ ∗ p OX (−mE) ⊗ π ∗ (Lp ) ⊗ KX ((−m − n + 1)E) . = π (L ⊗ KX ) ⊗ OX

(5.1.33)

Remark first that Lp ⊗ KX is positive on X for p 0. Thanks to Proposition 2.1.8 and the compactness of X, (5.1.33) is positive for m > −n + 1, all (x, y) ∈ X × X and p 0. We conclude by applying Theorem 1.5.4. Let us consider the ideals I (x, y) (resp. IEx,y ) of holomorphic functions vanishing on {x, y} (resp. on Ex,y ). For x = y, we make the convention I (x, y) = I (x)2 , IEx,y = IE2x . Then π∗ IEmx,y = I (x, y)m . This boils down to showing that: Lemma 5.1.14. Let U be a neighborhood of x. f ∈ OX (U ) vanishes to order m at x, if and only if, π ∗ f = f ◦ π ∈ OX (π −1 (U )) vanishes to order m along E. The lemma is obvious when looking at the local coordinates of the blow-up (2.1.8). Thus we have the commutative diagram: 0 → OX (π ∗ Lp ) ⊗ IEmx,y O π1∗

/ O (π ∗ Lp ) X O

/ OX (π ∗ Lp ) ⊗ (OX /IEm ) → 0 x,y O

π∗

π2∗

/ OX (Lp ) ⊗ (OX /I (x, y)m ) → 0 (5.1.34) where the vertical arrows are induced by π ∗ . We obtain a commutative diagram 0 → OX (Lp ) ⊗ I (x, y)m

π ∗ Lp ) H 0 (X, O

α

/ H 0 (X, π ∗ Lp ⊗ O /I m ) Ex,y X O

/ H 1 (X, π ∗ Lp ⊗ I m ) Ex,y

π2∗

π∗

H 0 (X, Lp )

/ OX (Lp )

β

/ H 0 (X, Lp ⊗ OX /I (x, y)m ) .

(5.1.35)

We observe now that (i) π ∗ is bijective, (ii) π2∗ is injective. Indeed, (i) is trivial = X). Otherwise (i) follows from Hartog’s Kugelsatz: any for n = 1 (then X holomorphic function on a punctured ball in Cn , n > 1, extends to a holomorphic


221

function in that ball. Moreover, (ii) follows from Lemma 5.1.14. Assertions (i) and (ii) show that in order to get the surjectivity of β, it suffices to prove the surjectivity of α. But, by Proposition 2.1.4, IEmx,y is the locally free sheaf of sections in the bundle OX (−mEx,y ), so Lemma 5.1.13 entails that α is surjective for all p p0 and all (x, y) ∈ X × X. For x = y, H 0 (X, Lp ⊗ OX /I (x, y)) = Lpx ⊕ Lpy , thus the surjectivity of β implies that Φp is injective. For x0 = y0 , we choose a local holomorphic frame of L ∗(1,0) around x0 which gives an identification Lp ⊗ OX /I (x0 )2 ∼ = Lpx0 ⊗ (C ⊕ Tx0 X). d ∗(1,0) p Thus, H 0 (X, Lp ⊗ OX /I (x0 , y0 )) = Lpx0 ⊗ (C ⊕ Tx0 X). Let {si }i=1 be a basis 0 p of H (X, L ) such that s1 (x0 ) = 0, ds1 (x0 ) = 0 and sj (x0 ) = 0 i.e., β(sj ) ∈ ∗(1,0) Lpx0 ⊗ Tx0 X for j 2. We write sj (x) = fj (x)s1 (x) in the neighborhood of x0 , where fj are holomorphic functions with fj (x0 ) = 0 for j 2. By (5.1.17), we have (1,0) near x0 , Φp (x) = [1, f2 (x), . . . , fdp (x)]. Hence, for a tangent vector U ∈ Tx0 X, we have (Φp )∗,x0 (U ) = (0, df2 (x0 ) · U, . . . , dfdp (x0 ) · U ) ∈ TΦp (x0 ) P(H 0 (X, Lp )∗ ). (1,0)

d

∗(1,0)

p Since {dfj (x0 )}j=2 span Tx0 X, we deduce that Ker(Φp )∗,x0 = {0}, so Φp is an immersion. We conclude that the Kodaira map Φp is an embedding for p p0 .

5.1.4 Grassmannian embedding We use the assumption in Section 5.1.2 now. Let E be a holomorphic vector bundle on X, and rk(E) = k. Let hE be a p Hermitian metric on E. Let hL ⊗E be the Hermitian metric on Lp ⊗ E induced by hL , hE . Let Pp (x, x ) be the Bergman kernel on Lp ⊗ E associated to g T X , hL , hE . Theorem 5.1.15. There exists p0 ∈ N such that for p p0 , the Kodaira map Φp is well defined: Φp : X −→ G(k, H 0 (X, Lp ⊗ E)∗ ), Φp (x) = {s ∈ H 0 (X, Lp ⊗ E) : s(x) = 0}.

(5.1.36)

Proof. By Theorem 4.1.1, we know that uniformly for x ∈ X, Pp (x, x) = b0 (x) IdE pn + O(pn−1 ),

with b0 (x) = det

R˙ L 2π

(x) > 0. (5.1.37)

Let eL be a unit frame of L, and let {ξl } be a frame of E; from (5.1.37), we have Pp (x, x)e⊗p (x)ξl (x) − b0 (x)e⊗p (x)ξl (x)pn Cpn−1 . (5.1.38) L L Thus there is p0 ∈ N such that the evaluation map H 0 (X, Lp ⊗ E) → (Lp ⊗ E)x is surjective for x ∈ X, p p0 . Thus the Kodaira map Φp in (5.1.36) is defined on X.

222


Theorem 5.1.16. For p p0 , Φp : X −→ G(k, H 0 (X, Lp ⊗ E)∗ ) is holomorphic and the map Ψp : Φ∗p U ∗ → Lp ⊗ E, Ψp ((Φ∗p σs )(x)) = s(x)

for any s ∈ H 0 (X, Lp ⊗ E),

(5.1.39)

defines a canonical isomorphism from Φ∗p U ∗ to Lp ⊗ E on X, and under this isomorphism, we have ∗

∗

hΦp U (x) = hL ∗

p

⊗E

(x) ◦ Pp (x, x)−1 .

∗

(5.1.40)

∗

Here hΦp U is the metric on Φ∗p U ∗ induced by hU on U ∗ on G(k, H 0 (X, Lp ⊗E)∗ ). Proof. Let {sj } be any basis of H 0 (X, Lp ⊗ E), and {sj } be its dual basis. Let {hl } be a frame of Lp ⊗ E near x0 ∈ X, and write sj (x) = Alj (x)hl (x).

(5.1.41)

Then for any s ∈ H 0 (X, Lp ⊗ E), dp

dp Alj (x)s , s hl = sj (x)(sj , s) = s(x). j

j=1

(5.1.42)

j=1

By (5.1.36) and (5.1.42), we have dp ∈ G(k, H 0 (X, Lp ⊗ E)∗ ). Φp (x) = span Alj (x)sj j=1

(5.1.43)

l

By the same argument as in the proof of Theorem 5.1.3, we get Theorem 5.1.16, except (5.1.40). Let ψx : H 0 (X, Lp ⊗ E) → (Lp ⊗ E)x be the evaluation map ψx s = s(x). Let ∗ ψx be its adjoint. 0 p Now we take {sj } an orthonormal basis of (H 0 (X, Lp ⊗ E), hH (X,L ⊗E) ), and hl an orthonormal frame of Lp ⊗ E near x0 . Then by (5.1.41), s(x) = ψx s = Alj (x)(sj , s)hl (x),

ψx∗ hl (x) = Alj (x)sj .

(5.1.44)

Thus ψx ψx∗ hl (x) = (A(x)A(x)∗ )ml hm (x). By (4.1.4), (5.1.44) and (5.1.45), we get Pp (x, x) = j sj (x)·, sj (x) = ψx ψx∗ ∈ End(E)x .

(5.1.45)

(5.1.46)


From (5.1.44) and (5.1.45), we get ((A(x)A∗ (x))−1/2 A)lj sj , s hl = (ψx ψx∗ )−1/2 s(x).

223

(5.1.47)

Now from (5.1.5), (5.1.43), (5.1.46) and (5.1.47), we get for ζ ∈ (Φ∗p U ∗ )x , there is s ∈ H 0 (X, Lp ⊗ E) such that ζ = (Φ∗p σs )(x); then under the isomorphism Ψp , |ζ|2 Φ∗p U ∗ = |(ψx ψx∗ )−1/2 s(x)|2hLp ⊗E = |(Pp (x, x))−1/2 s(x)|2hLp ⊗E . h

The proof of Theorem 5.1.16 is complete.

(5.1.48)

Theorem 5.1.17. The induced Fubini–Study form 1p Φ∗p (ωF S ) converges in the C ∞ topology to k ω ; for any l 0 there exists Cl > 0 such that 1 Cl . (5.1.49) Φ∗p (ωF S ) − k ω l p p C Proof. From (1.5.8), (5.1.6) and (5.1.40), we get √ √ 1 ∗ −1 L 1 −1 E Φp (ωF S ) − k R = Tr[R ] − ∂∂ log(det Pp (x, x)) . p 2π p 2πp From (5.1.37) and (5.1.50), we get (5.1.49).

(5.1.50)

Theorem 5.1.18. Φp is an embedding for p large enough. Proof. From Theorem 5.1.17, as in Lemma 5.1.6, we deduce immediately that Φp is an immersion for p large enough. We need to prove Φp is injective for p large enough. Let ex0 be a unit vector of (L, hL ) at x0 , and let {ζl,x0 }kl=1 be an orthonormal basis of (E, hE ) at x0 , let [Φp (x0 )]⊥ be the orthogonal space of Φp (x0 ) in H 0 (X, Lp ⊗ E); then from Theorem 4.1.24, for p large enough, ⊥ span{Pp (x, x0 )e⊗p x0 ⊗ ζl,x0 }l = [Φp (x0 )] .

(5.1.51)

But from Theorem 4.1.24, the mass of all unit vectors in span{Pp (x, x0 )e⊗p x0 ⊗ ζl,x0 }l , will concentrate on B(x0 , rp ) in the sense of (5.1.25). Assume (x, y) ∈ Diag(X × X). From the above discussion, there exists p(x, y) ∈ N such that Φp (x) = Φp (x) for all p p(x, y) . Now take p0 ∈ N such that Theorems 5.1.12, 5.1.15 hold for p p0 . For p p0 , we define the analytic set Ap := {(x, y) ∈ X × X : Φp (x) = Φp (y)} . We claim that Ap+l ⊂ Ap for any l, p p0 . In fact, if (x, y) ∈ Ap , then there exists s ∈ H 0 (X, Lp ⊗ E) such that s(x) = 0, s(y) = 0. From the Kodaira embedding theorem, for l p0 , there exists σ1 ∈

224


H 0 (X, Ll ) such that σ1 (x) = 0, σ1 (y) = 0. Thus σ1 ⊗ s ∈ H 0 (X, Ll+p ⊗ E) such that (σ1 ⊗ s)(x) = 0, (σ1 ⊗ s)(y) = 0. Thus (x, y) ∈ Al+p . By Lemma 5.1.9, there exists p1 such that Ap = Ap1 for p p1 . Summarizing, for any p2 p1 , we have Diag(X × X) = ∩p Ap = Ap2 . Therefore Φp is injective for p p1 .

(5.1.52)

5.2 Stability and Bergman kernel In this section, we give an introduction to Donaldson’s approach to the existence of K¨ ahler metrics of constant scalar curvature (briefly, csc K¨ ahler metrics) and especially, the relation with the Bergman kernel, which originates from his attempt to understand a conjecture of Yau. In Section 5.2.1, we explain very briefly the history of the csc K¨ ahler metrics. Section 5.2.2 is an introduction to Donaldson’s approach on the existence of csc K¨ ahler metrics. In Section 5.2.3, we explain the vector bundle version. We will use the notation from Section 5.1.

5.2.1 Extremal Kähler metrics The existence of special metrics is one of the central problems in differential geometry. Usually, once we know the existence of special metrics, it will have many interesting applications. In K¨ ahler geometry, many of these questions have been instigated by the seminal work of Calabi. Let (X, J, ω) be a compact Kähler manifold and dim X = n. Let g T X be the Riemannian metric on T X associated to ω. We call the cohomology class associated to ω the K¨ ahler class of ω. ∗ ∗ , Let RKX be the curvature of the holomorphic Hermitian connection on KX ∗(1,0) X) of X, with metric induced the dual of the canonical bundle KX := det(T by ω. Then the Ricci form Ricω of (X, ω) is a real (1, 1)-form on X given by (cf. Problem 1.7), √ ∗ Ricω = −1RKX . (5.2.1) We denote by [Ricω ] the cohomology class of Ricω , then ∗ [Ricω ] = 2πc1 (KX ) =: 2πc1 (X) ∈ H 2 (X, R).

(5.2.2)

Definition 5.2.1. ω is a K¨ ahler–Einstein metric on X if there exists λ ∈ R such that Ricω = λω.

(5.2.3)

5.2. Stability and Bergman kernel

225

By (5.2.2), if ω is a K¨ ahler–Einstein metric, it verifies the topological condition λ[ω] = 2πc1 (X) ∈ H 2 (X, R).

(5.2.4)

In the 1950s, Calabi initiated the study of Kähler–Einstein metrics. In the 1970s, the existence of Kähler–Einstein metrics was proved by Aubin and Yau ∗ is negative. If c1 (X) = 0, Yau established independently, if c1 (X) < 0, i.e., KX the existence of the Ricci flat metric in each K¨ ahler class on X: the renowned ahler–Einstein metric. If Calabi–Yau metrics. If c1 (X) > 0, X need not have a K¨ dim X = 2 and c1 (X) > 0, Tian solved completely this problem in the 1990s: X admits K¨ ahler–Einstein metrics if and only if the algebra of holomorphic vector fields is reductive. In the late 1980s, Yau conjectured the relation between the notions of stability of manifolds and existence of special metrics such as K¨ ahler–Einstein metrics and csc Kähler metrics. Tian has made enormous progress towards understanding precisely when X has a K¨ ahler–Einstein metrics if c1 (X) > 0. Tian introduced K-stability (later modified by Donaldson) which is thought to be the right notion of stability equivalent to the existence of csc K¨ ahler metrics. Now, we relax the topological condition (5.2.4) on the K¨ ahler form ω on X; then the natural metrics we are looking for are the csc K¨ ahler metrics. Certainly, a K¨ ahler–Einstein metric has constant scalar curvature (cf. Problem 1.7). Proposition 5.2.2. If there exists λ ∈ R such that (5.2.4) holds and ω is a csc K¨ ahler metric on X, then ω is a K¨ ahler–Einstein metric. Proof. By the ∂∂-Lemma 1.5.1, there exists a real function f on X such that √ Ricω −λω = −1 ∂∂f. (5.2.5) Taking the trace of these (1, 1)-forms, we get from Problem 4.1 and (5.2.2) that 1 X 1 r − λn = − ∆f. 2 2 Since 12 rX − λn is a constant function, (5.2.6) implies 2 X |∆f | dvX = −(rω − 2λn) ∆f dvX = 0. X

(5.2.6)

(5.2.7)

X

This means ∆f = 0. Thus f is constant on X (cf. Problem 1.9). We conclude that ω is a K¨ ahler–Einstein metric. More generally, in the early 1980s, Calabi introduced the extremal K¨ ahler metrics which are critical points of the functional Ca(ω) = (rωX )2 ω n , (5.2.8) X

226


in a given Kähler class, where rωX is the scalar curvature of (X, J, ω). Especially, a csc Kähler metric is an extremal K¨ ahler metric. The uniqueness problem for extremal metrics is well understood now. The uniqueness of K¨ ahler–Einstein metrics of non-positive scalar curvature was already known to Calabi in the 1950s. In 1986, Bando and Mabuchi proved the uniqueness of K¨ ahler–Einstein metrics of positive scalar curvature. In 2000, following a suggestion of Donaldson, Chen proved the uniqueness of csc K¨ ahler metrics in any K¨ ahler class on a compact Kähler manifold with non-positive first Chern class. Donaldson proved the uniqueness of csc K¨ ahler metrics with rational K¨ ahler class on any projective manifolds without non-trivial holomorphic vector fields and Mabuchi was able to extend Donaldson’s result to some cases that holomorphic vector fields are not zero. Finally, Chen and Tian established the uniqueness of extremal metrics in the most general case.

5.2.2 Scalar curvature and projective embeddings Let X be a compact complex manifold. Let L be a positive line bundle on X. Following Donaldson, we consider the existence and uniqueness problem of csc Kähler metrics on X in the K¨ ahler class c1 (L). By Proposition 1.5.3, the K¨ ahler forms on X in the K¨ ahler class c1 (L) are given by the Hermitian metrics on L (unique up to multiplication by positive √ constants) such that the associated curvature is −2π −1 times the Kähler form. By Theorem 5.1.12, there exists p0 ∈ N such that the Kodaira map Φp : X −→ P(H 0 (X, Lp )∗ ) is an embedding for any p p0 . By Theorem 5.1.3, Φ∗p O(1) is canonically isomorphic to Lp . Thus the K¨ ahler metrics on X in the K¨ ahler class c1 (L) is closed related to the Hermitian metrics on H 0 (X, Lp ). We explore this point now. Set dp , c1 (L)n /n!, dp = dim H 0 (X, Lp ), Rp = vol(X) = vol(X) X (5.2.9) 4π c1 (L)n−1 X . r = c1 (X) vol(X) X (n − 1)! Then r X is the average of the scalar curvature of any K¨ ahler metric in the K¨ ahler class c1 (L) (cf. Problem 4.1). Set p

p

Met(Lp ) = {hL ; hL is a metric on Lp with associated curvature √ − 2π −1p ωhLp where ωhLp is a positive (1,1)-form}. (5.2.10) p

In this section, given hL ∈ Met(Lp ), we will always use the Riemannian metric on T X associated to ωhLp . We assume from now on that p p0 .


227

p

p

Definition 5.2.3. For hL ∈ Met(Lp ), set Hilb(hL ) as the Hermitian metric on H 0 (X, Lp ) defined by 2 |s(x)|2hLp ωhnLp /n! . (5.2.11) sHilb(hLp ) = Rp X

Definition 5.2.4. For a Hermitian metric hH on H 0 (X, Lp ), the metric FS(hH ) on dp Lp is defined as follows: For {sj }j=1 an orthonormal basis of (H 0 (X, Lp ), hH ), we have |sj (x)|2FS(hH ) = 1 for any x ∈ X. (5.2.12) j

In other x0 ∈ X, and write sj = fj e⊗p L , words,2let eL be a local frame of L near 0 then j |fj (x)| > 0; as the evaluation map H (X, Lp ) → Lpx is surjective for all x ∈ X and p p0 , we define 1 . 2 j |fj (x)|

2 |e⊗p L (x)|FS(hH ) =

(5.2.13)

O(1)

Let hH be the metric on O(1) → P(H 0 (X, Lp )∗ ) induced by hH . Then, using the same argument as in (5.1.19), we see that FS(hH ) is the metric induced O(1) by hH under the isomorphism (5.1.14). ahler form which is the restriction on X of the We denote by ωFS(hH ) the K¨ standard Fubini–Study form on P(H 0 (X, Lp )∗ ) defined by hH as in (5.1.3). p

Definition 5.2.5. A pair (G, hL ) is balanced if p

G = Hilb(hL ),

p

and hL = FS(G).

(5.2.14)

p

In this situation, we will say the metrics hL on Lp and G on H 0 (X, Lp ) are balanced, or they are balanced metrics. p

By (5.1.2), (5.2.11) and (5.2.12), if hL is a balanced metric on Lp , then L C h is also a balanced metric for any constant C > 0. p

p

Proposition 5.2.6. Let hL ∈ Met(Lp ) be a metric on Lp and as in (5.1.11), we p denote by Pp (x, x) the diagonal of the Bergman kernel on Lp associated to hL ; Lp then h is balanced if and only if Pp (x, x) = constant on X. Proof. If (5.2.15) holds, then 1 dp Pp (x, x) = = Rp . Pp (x, x)dvX (x) = vol(X) X vol(X)

(5.2.15)

(5.2.16)

228

Chapter 5. Kodaira Map 0

p

If {sj } is an orthonormal basis of (H 0 (X, Lp ), hH (X,L ) ), then by (5.2.11), p −1/2 {Rp sj } is an orthonormal basis of (H 0 (X, Lp ), Hilb(hL )). p p Thus from (5.2.12), FS(Hilb(hL )) = hL if and only if Rp−1 Pp (x, x) = Rp−1 |sj (x)|2hLp = 1. Let {sj } be a basis of H 0 (X, Lp ), and {sj } its dual basis. Then we can define ψ : H 0 (X, Lp )∗ Cdp by ψ(zj sj ) = (z1 , . . . , zdp ). This gives us an embedding Φp,{sj } : X → C Pdp −1 . If we choose another basis { sj }, then there exists σ ∈ SL(dp , C), c ∈ C such that si = c σij sj , thus Φp,{sj } = σ ◦ Φp,{sj } .

(5.2.17)

This inspires us to introduce the following notation. Definition 5.2.7. Let X ⊂ C PN be a smooth compact complex manifold. We define M (X) to be the (N + 1) × (N + 1) matrix with entries zz ωn i j 2 FS . (5.2.18) M (X)ij = n! X l |zl | X is said to be balanced if there exists σ ∈ SL(N + 1, C) such that M (σ(X)) is a multiple of the identity matrix. For a pth embedding X → C Pdp −1 , we have two classical notions of stability in geometric invariant theory (G.I.T.): Chow poly-stable, corresponding to dp −1 p the Chow with de point nassociated to (X, L ) in the Chow group of C P n gree p X c1 (L) ; and Hilbert poly-stable, corresponding to the Hilbert point associated to (X, Lp ) in the Hilbert scheme of C Pdp −1 with Hilbert polynomial χ(m) := dmp . It would be quite long to give an elementary introduction to stability, thus we will only give some references. The following result explains the relation between the balance condition and the Chow poly-stable condition in algebraic geometry. Theorem 5.2.8 (Zhang). (X, Lp ) is Chow poly-stable if and only if the projective embedding X → C Pdp −1 induced by Lp can be balanced. Definition 5.2.9. The automorphism group Aut(X, L) of the pair (X, L) is Aut(X, L) = {ϕ : L → L; ϕ is a biholomorphic, fiberwise linear map, and it induces an automorphism on X}. (5.2.19) Theorem 5.2.10 (Donaldson). Suppose that Aut(X, L) is discrete. If X has a csc K¨ ahler form ω in the K¨ ahler class c1 (L), then for large enough p there is a unique ahler form pωp on X, and ωp → ω as p → ∞. balanced metric on Lp inducing a K¨ Conversely, if there are balanced metrics on Lp for all large p and the sequence ωp converges to ω, then ω has constant scalar curvature.


229

Proof. The first part is the hard and central part. Donaldson needs to combine Theorem 4.1.2 and his moment map picture to complete the proof. We refer the readers to Donaldson’s original paper for the details. The second part is a consequence of Theorems 4.1.1 and 4.1.2. In fact, if there are balanced metrics on Lp for all large p and the sequence ωp converges to ω, we denote by Pp,ωp (x, x) the diagonal of the Bergman kernel on Lp associated to the balanced metric on Lp and ωp . Then from Theorem 4.1.1, we know there exists C > 0 such that for p ∈ N, x ∈ X, Pp,ωp (x, x) − pn − 1 rωX (x)pn−1 Cpn−2 . (5.2.20) 8π p But by (4.1.10), (5.2.9) and (5.2.16), we get Pp,ωp (x, x) = Rp = pn +

1 X n−1 r p + O(pn−2 ). 8π

(5.2.21)

Thus |rωXp (x) − rX | C/p.

(5.2.22)

But rωXp (x) converges to rωX when p → ∞, since ωp → ω. Thus rωX is constant.

5.2.3 Gieseker stability and Grassmannian embeddings We explain the vector bundle version of Section 5.2.2 which was established by Wang. Let X be an n-dimensional compact complex manifold, and L a positive line bundle. √ Suppose L is very ample now. We fix a metric hL on L such that −1RL is a positive (1, 1)-form, where RL is the curvature associated to hL . We also fix the √ −1 L K¨ ahler form ω = 2π R on X. Let E be a holomorphic vector bundle on X and rk(E) = k. Set deg(E) = c1 (E)c1 (L)n−1 . (5.2.23) X

As L is very ample, then for any OX -coherent sheaf F , we have a projective resolution of vector bundles on X, 0 → OX (E n ) → · · · → OX (E 0 ) → OX (F ) → 0,

(5.2.24)

here E i are holomorphic vector bundles on X and (5.2.24) is an exact sequence of OX -sheaves. Especially c1 (F ) and deg(F ) are well defined and c1 (F ) = (−1)i c1 (E i ), ch(F ) = (−1)i ch(E i ). (5.2.25) i

i

230


Definition 5.2.11. We call E Mumford stable (resp. semi-stable) if for any torsion free proper sub-sheaf F ⊂ OX (E), we have deg(E) deg(F ) > (resp. ) . rk(E) rk(F )

(5.2.26)

Recall that the Euler characteristic number χ(X, E) is defined in (1.4.25). Definition 5.2.12. We call E Gieseker stable (resp. semi-stable) if for any torsion free proper sub-sheaf F ⊂ OX (E), there exists p0 ∈ N such that for any p p0 , we have χ(X, Lp ⊗ F) χ(X, Lp ⊗ E) > (resp. ) . rk(E) rk(F )

(5.2.27)

By the Riemann–Roch–Hirzebruch theorem 1.4.6, we know that Mumford stable =⇒ Gieseker stable =⇒ Mumford semi-stable.

(5.2.28)

We assume from now on that p p0 as in Section 5.1.4. Set dE,p = dim H 0 (X, Lp ⊗ E) = χ(X, Lp ⊗ E),

RE,p =

dE,p . vol(X)k

(5.2.29)

Definition 5.2.13. Let hE be a Hermitian metric on E and define the Hermitian metric Hilb(hE ) on H 0 (X, Lp ⊗ E) by ωn 2 . (5.2.30) sHilb(hE ) = RE,p |s(x)|2hLp ⊗E n! X Definition 5.2.14. For a Hermitian metric hH on H 0 (X, Lp ⊗E), the metric FS(hH ) ∗ on E is defined as follows: Let hU be the Hermitian metric on U ∗ (the dual of the universal bundle on G(k, H 0 (X, Lp ⊗ E)∗ )), induced by hH . Then FS(hH ) is the metric satisfying, ∗

hL ⊗ FS(hH ) = Φ∗p hU , p

∗

(5.2.31)

∗

where Φ∗p hU is the pull-back of hU by the canonical map (5.1.39). Definition 5.2.15. A pair (G, hE ) is balanced for the Kodaira map Φp if G = Hilb(hE ),

and hE = FS(G).

(5.2.32)

In this situation, we will say the metrics hE on E and G on H 0 (X, Lp ⊗ E) are balanced. By (5.1.5), (5.2.30) and (5.2.31), if hE is a balanced metric on E, then ChE is also a balanced metric for any constant C > 0.

5.3. Distribution of zeros of random sections

231

Theorem 5.2.16. Let hE be a metric on E. hE is balanced for Φp if and only if the corresponding Bergman kernel Pp (x, x ) on Lp ⊗ E verifies Pp (x, x) = RE,p IdE .

(5.2.33)

Proof. By using (5.1.40) as in the proof of Proposition 5.2.6, we get Theorem 5.2.16. Theorem 5.2.17 (Wang). E is Gieseker stable if and only if there exists p0 ∈ N E such that for all p p0 , there exists hE p on E such that hp is balanced for Φp . Theorem 5.2.18 (Wang). Suppose E is Gieseker stable. If for p → +∞, hE p in E , then the metric h solves the following weakly Theorem 5.2.17 converges to hE ∞ ∞ Hermitian–Einstein equation, √ deg(E) 1 X −1 1 X E Λω (R∞ ) + r IdE = + r (5.2.34) IdE , 2π 8π vol(X)k (n − 1)! 8π E where R∞ is the curvature associated to hE ∞ on E. Conversely, suppose there is a Hermitian metric hE ∞ on E solving (5.2.34), E l for any p p such that hE then there exist balanced metrics hE 0 p p → h∞ in C for any l ∈ N.

Proof of Theorem 5.2.18. From (4.1.10) and (5.2.29), we have for p p0 : deg(E) 1 X + r (5.2.35) RE,p = pn + pn−1 + O(pn−2 ). vol(X)k (n − 1)! 8π Now from Theorems 4.1.1 and 5.2.17, we get (5.2.34). The converse is the hard part of the proof. One needs to adapt Donaldson’s argument for vector bundles. We refer to Wang’s original paper for the details.

5.3 Distribution of zeros of random sections As an application of Theorem 5.1.4, we study the distribution of zeros of random sections. We use the same notation and assumption in Section 5.1.2. Especially, (X, J) is a compact complex manifold with complex structure J and dim X = n. g T X is a Riemannian metric on T X compatible with J. (L, hL ) is a positive holomorphic √ −1 L Hermitian line bundle on X. ω = 2π R , Θ = g T X (J·, ·) are positive (1, 1)-forms on X. Let dS be the usual volume form on the 2m − 1-dimensional unit sphere S 2m−1 := {λ ∈ Cm ; |λ| = 1}; then 2 πm vol(S 2m−1 ) = . (5.3.1) dS = (m − 1)! S 2m−1

232

Chapter 5. Kodaira Map 0

p

Let hH (X,L ) be the L2 -metric on H 0 (X, Lp ) induced by g T X , hL . Let dµp be the equidistribution probability measure on the unit sphere SH 0 (X, Lp ) := {s ∈ H 0 (X, Lp ); |s|hH 0 (X,Lp ) = 1}.

(5.3.2) d

p Recall that dp = dim H 0 (X, Lp ). We fix an orthonormal basis {Sip }i=1 of p H 0 (X,Lp ) 2dp −1 0 p 2dp −1 (H (X, L ), h ), then we can identify S to SH (X, L ) by S (z1 , . . . , zdp ) → i zi Sip , and we have

0

dµp =

dS . vol(S 2dp −1 )

(5.3.3)

As in (B.2.17), we denote by [Div(s)] the current of integration on the divisor Div(s) of the section s ∈ H 0 (X, Lp ). We view [Div(s)] as a Ω 1,1 (X)valued Random variable as s varies over the probability space (SH 0 (X, Lp ), dµp ). The expected value of [Div(s)] is the current E([Div(s)]) ∈ Ω 1,1 (X) defined for ϕ ∈ Ωn−1,n−1 (X), E([Div(s)])(ϕ) =

S 2dp −1

dp Div λi Sip , ϕ dµp (λ).

(5.3.4)

i=1

# 0 p Now we consider the probability space Ω = ∞ p=1 SH (X, L ) with the prob#∞ ability measure µ = p=1 µp . We denote s = {sp } ∈ Ω. Theorem 5.3.1. If the Kodaira map Φp is defined on X for p ∈ N, then E([Div(sp )]) = Φ∗p (ωFS ).

(5.3.5)

Proof. By the Poincaré–Lelong formula, Theorem 2.3.3, for s ∈ H 0 (X, Lp ), √ −1 ∂∂ log |s|2hLp + p ω. (5.3.6) [Div(s)] = 2π Thus the current 1p [Div(s)] is a representative of c1 (L) for any s ∈ H 0 (X, Lp ). Let ϕ ∈ Ωn−1,n−1 (X). By (5.1.21), (5.3.6), for s ∈ H 0 (X, Lp ), √ −1 2 p ∂∂ log |s|hL + p ω ∧ ϕ ([Div(s)], ϕ) = 2π X √ −1 ∗ = Φp (ωFS ) ∧ ϕ + ∂∂ log |Pp (x, x)−1/2 s(x)|2hLp ∧ ϕ 2π X X √ −1 ∗ Φp (ωFS ) ∧ ϕ + log |Pp (x, x)−1/2 s(x)|2hLp ∂∂ϕ. = 2π X X Let eL (x) be a unit vector of L at x ∈ X and define from (5.1.11), 2dp −1 ψ(x) = Pp (x, x)−1/2 (S1p (x), . . . , Sdpp (x))/e⊗p ; L (x) ∈ S

(5.3.7)

5.3. Distribution of zeros of random sections

233

then ψ(x) is well defined up to multiplication by unit numbers. For u = (u1 , . . . , udp ), v = (v1 , . . . , vdp ) ∈ Cdp , we denote u·v =

dp

ui vi ,

u · Sp =

i=1

dp

ui Sip .

(5.3.8)

i=1

Observe that for any u ∈ S 2dp −1 , 2 −1/2 p log Pp (x, x) λ · S Lp dµp (λ) = log(|λ · ψ(x)|2 )dµp (λ) h S 2dp −1 S 2dp −1 log(|λ · u|2 )dµp (λ) = cp (5.3.9) =

S 2dp −1

is a constant function on X. Thus from (5.3.4), (5.3.7) and (5.3.9), we get ([Div(λ · S p )] , ϕ) dµp (λ) E([Div(sp )])(ϕ) = S 2dp −1 √ −1 ∗ Φp (ωFS ) ∧ ϕ + cp ∂∂ϕ = Φ∗p (ωFS ) ∧ ϕ. (5.3.10) = 2π X X X The proof of Theorem 5.3.1 is complete. Lemma 5.3.2. For u, v ∈ S

2dp −1

Ap (u, v) =

, set

S 2dp −1

log(|λ · u|) (log |λ · v|)dµp (λ).

(5.3.11)

Then there exists a constant Cp (depending only on p) such that Ap (u, v) − Cp is uniformly bounded for (u, v) ∈ S 2dp −1 × S 2dp −1 , p ∈ N. 2 m Proof. Set |Z|2 = m i=1 |zi | , and dZ the standard volume form on C for m ∈ N. We consider the Gaussian integral 2 Ap (u, v) := e−|Z| log |z · u| log |z · v| dZ. (5.3.12) Cd p

We evaluate (5.3.12) in two ways. First we choose the coordinates in Cdp such that u = (1, 0, . . . , 0), v = (v1 , v2 , 0, . . . , 0). Set z = (z1 , z2 ), z = (z3 , . . . , zdp ), v = (v1 , v2 ), and with cp from (5.3.9) 2 e−|Z | log |z1 | log |z1 v1 + z2 v2 | dZ , ρ(v ) = 2 ∞C (5.3.13) 2 e−r r2dp −1 vol(S 2dp −1 )(log r)2 + cp log r dr. Cp = 0

234


Then from (5.3.12), we have p (u, v) = A

Cdp −2

2 e−|Z| dZ ρ(v ) = π dp −2 ρ(v ).

(5.3.14)

By the Cauchy-Schwarz inequality, for v ∈ S 3 , we have |ρ(v )|

C2

2

e−|Z | (log |z1 |)2 dZ

1/2

e−|Z

2

C2

= C2

|

1/2 2 log |z1 v1 + z2 v2 | dZ

2

e−|Z | (log |z1 |)2 dZ < +∞. (5.3.15)

Now we use spherical coordinates Z = rλ with λ ∈ S 2dp −1 . We have ∞ 2 Ap (u, v) = dr e−r r2dp −1 (log r + log |λ · u|)(log r + log |λ · v|)dS(λ). 0

S 2dp −1

By using (5.3.1), (5.3.3), (5.3.9), (5.3.13) and the above equation, we have ∞ 2 p (u, v) = Cp + A e−r r2dp −1 dr log |λ · u| log |λ · v| dS(λ) 0 S 2dp −1 (5.3.16) dp = Cp + π log |λ · u| log |λ · v| dµp (λ). S 2dp −1

From (5.3.14), (5.3.16), we get p (u, v) − C ) = π −2 ρ(v ) − C π −dp . Ap (u, v) = π −dp (A p p

(5.3.17)

The proof of Lemma 5.3.2 is completed.

The main result of this section is the following theorem which states that for a random variable s = {sp } ∈ Ω, the sequence of zeros of the sections sp is asymptotically uniformly distributed. Theorem 5.3.3. For µ-almost all s = {sp } ∈ Ω, p1 [Div(sp )] → ω weakly in the sense of distribution, i.e., for all continuous (n − 1, n − 1)-forms ϕ, 1 lim [Div(sp )], ϕ = ω ∧ ϕ. (5.3.18) p→∞ p X Proof. At first, from (5.3.6), 1 1

|([Div(sp )], ϕ)| [Div(sp )], ω n−1 |ϕ|C 0 = |ϕ|C 0 p p

c1 (L)n .

(5.3.19)

X

By considering a countable C 0 -dense family of ϕ, we need only to consider one ϕ. Consider the random variables 1 1 [Div(sp )] − Φ∗p (ωFS ), ϕ . Yp (s) = (5.3.20) p p

5.4. Orbifold projective embedding theorem

235

By Theorem 5.1.4, we need to prove that µ-almost surely for s ∈ Ω, as p → ∞, Yp (s) → 0.

(5.3.21)

By Theorem 5.3.1, we get 2

1 1 2 E |Yp (s)|2 = 2 E |([Div(sp )], ϕ)| − 2 (Φ∗p (ωFS ), ϕ) . p p

(5.3.22)

By (5.3.4), (5.3.7) and (5.3.9), 2 2 E |([Div(sp )], ϕ)| = (Φ∗p (ωFS ), ϕ) 1 + 2 (∂∂ϕ(x))(∂∂ϕ(y)) log |Pp (x, x)−1/2 λ · S p (x)|2hLp 4π X X S 2dp −1 × log |Pp (y, y)−1/2 λ · S p (y)|2hLp dµp (λ).

(5.3.23)

From Lemma 5.3.2, (5.3.22) and (5.3.23), we get

1 E |Yp (s)|2 = (∂∂ϕ(x))(∂∂ϕ(y))Ap (ψ(x), ψ(y))dµp (λ) 4π 2 p2 X X (5.3.24) = O(p−2 ). Thus ∞ ∞ ∞

2 |Yp (s)| dµ(s) = |Yp (s)|2 dµ(s) = E |Yp (s)|2 < +∞. Ω p=1

p=1

Ω

(5.3.25)

p=1

Hence µ-almost surely, Yp → 0 as p → ∞.

5.4 Orbifold projective embedding theorem In Section 5.4.1, we recall the basic definitions about orbifolds. In Section 5.4.2, we recall the orbifold version of Dolbeault cohomology and Hodge theory of the Kodaira Laplacian. In Section 5.4.3, we establish the asymptotic expansion of the Bergman kernel on orbifolds. In Section 5.4.4, we explain the metric aspect of Baily’s version of Kodaira embedding for orbifolds.

5.4.1 Basic definitions on orbifolds We define at first a category Ms as follows: The objects of Ms are the class of pairs (G, M ) where M is a connected smooth manifold and G is a finite group acting effectively on M (i.e., if g ∈ G satisfies gx = x for any x ∈ M , then g is the unit element of G). If (G, M ) and (G , M ) are two objects, then a morphism Φ : (G, M ) → (G , M ) is a family of open embeddings ϕ : M → M satisfying:

236


i) For each ϕ ∈ Φ, there is an injective group homomorphism λϕ : G → G that makes ϕ be λϕ -equivariant. ii) For g ∈ G , ϕ ∈ Φ, we define gϕ : M → M by (gϕ)(x) = gϕ(x) for x ∈ M . If (gϕ)(M ) ∩ ϕ(M ) = ∅, then g ∈ λϕ (G). iii) For ϕ ∈ Φ, we have Φ = {gϕ, g ∈ G }. Definition 5.4.1. Let X be a paracompact Hausdorff space and let U be a covering of X consisting of connected open subsets. We assume U satisfies the condition : For any x ∈ U ∩ U , (U, U ∈ U), there is U ∈ U such that x ∈ U ⊂ U ∩ U .

(5.4.1)

Then an orbifold structure V on X consists in the following data: τ ) → i) a ramified covering V(U ) = ((GU , U U ), for any U ∈ U, giving an /GU . identification U U ) → (GV , V ), for any U, V ∈ U, U ⊂ V , which ii) a morphism ϕV U : (GU , U covers the inclusion U ⊂ V and satisfies ϕW U = ϕW V ◦ϕV U for any U, V, W ∈ U, with U ⊂ V ⊂ W . If U is a refinement of U satisfying (5.4.1), then there is an orbifold structure V such that V ∪V is an orbifold structure. We consider V and V to be equivalent. Such an equivalence class is called an orbifold structure over X. So we may choose U arbitrarily fine. In the above definition, we can replace Ms by a category of manifolds with an additional structure such as orientation, Riemannian metric, almost-complex structure or complex structure. We understand that the morphisms (and the groups) preserve the specified structure. So we can define oriented, Riemannian, almost-complex or complex orbifolds. ) ∈ Ms , we can always construct a G-invariant RieCertainly, for any (G, U . By a partition of unity argument, there exists a Riemannian mannian metric on U metric on the orbifold (X, V). Remark 5.4.2. Let P be a smooth manifold, and let H be a compact Lie group acting on P . We assume that the action of H is locally free. Then the quotient space P/H is an orbifold. Reciprocally, any orbifold X can be presented in this way, with H = O(m), (m = dimR X), the orthogonal group of degree m over R.

Let (X, V) be an orbifold. For each x ∈ X, we can choose a small neighborx ) → Ux such that x ∈ U x is a fixed point of Gx (such Gx is unique hood (Gx , U up to isomorphisms for each x ∈ X from the definition). We denote by |Gx | the cardinal of Gx . If |Gx | = 1, then x is a smooth point of X. If |Gx | > 1, then x is a singular point of X. We denote by Xsing = {x ∈ X; |Gx | > 1} the singular set of X. x ⊂ Rm such that the finite Lemma 5.4.3. We can choose the local coordinates U m −1 group Gx acts linearly on R , and {0} = τx (x).


237

x y → (ϕ1 (y), . . . , ϕm (y)) ∈ Rm be local coordinates near Proof. Let ϕ : U x with an open subset ϕ(U x ) in Rm , and x which vanish at x . We identify U x ∈U x ) in the sequel. we work on ϕ(U x ). Let dg ∈ End(Rm ) be the differential of For g ∈ Gx , g acts now on ϕ(U the action g at 0. Set σ=

1 (dg)−1 ◦ g. |Gx |

(5.4.2)

g∈Gx

Then for any g ∈ Gx , we have σ ◦ g = dg ◦ σ. In the new coordinates σ ◦ ϕ, g ∈ Gx acts as the linear map dg. Definition 5.4.4. An orbifold vector bundle E over an orbifold (X, V) is defined as U → U ) is a GE -equivariant U : E follows: E is an orbifold and for U ∈ U, (GE U,p U E E E vector bundle and (GU , EU ) (resp. (GU = GU /KU , U ), KUE = Ker(GE U → E Diffeo(U ))) is the orbifold structure of E (resp. X). If GU acts effectively on for U ∈ U, i.e., K E = {1}, we call E a proper orbifold vector bundle. U U Note that any structure on X or E is locally Gx or GE Ux -equivariant. pr * Remark 5.4.5. Let E be an orbifold vector bundle on (X, V). For U ∈ U, let E U pr * ) defines a U on U . Then (GU , E be the maximal KUE -invariant sub-bundle of E U pr proper orbifold vector bundle on (X, V), denoted by E . The (proper) orbifold tangent bundle T X on an orbifold X is defined by → U), for U ∈ U. (GU , T U

Let E → X be an orbifold bundle. A section s : X → E is called C ∞ (or k C ) if for each U ∈ U, s|U is covered by a GE U -invariant smooth (or C ) section k k → E U . We denote by C (X, E) (k ∈ N) the space of C sections of E sU : U on X. If X is oriented, we define the integral X ω for a form ω over X (i.e., a section of Λ(T ∗ X) over X) as follows: if supp(ω) ⊂ U ∈ U, then 1 ω := ω U . (5.4.3) |GU | U X k

Assume now that (X, V) is compact. For x, y ∈ X, put ti ∂ d(x, y) = Infγ i (t)|dtγ : [0, 1] → X, γ(0) = x, γ(1) = y, i ti−1 | ∂t γ such that there exist t0 = 0 < t1 < · · · < tk = 1, γ([ti−1 , ti ]) ⊂ Ui , i that covers γ|[t ,t ] . Ui ∈ U, and a C ∞ map γ i : [ti−1 , ti ] → U i−1

i

Then (X, d) is a metric space. For x ∈ X, set d(x, Xsing ) := inf y∈Xsing d(x, y).

238


5.4.2 Complex orbifolds Let X be a compact complex orbifold of complex dimension n with complex structure J. Let E be a holomorphic orbifold vector bundle on X. Let OX be the sheaf over X of local GU -invariant holomorphic functions over for U ∈ U. The local GE -invariant holomorphic sections of E →U define a U, U • sheaf OX (E) over X. Let H (X, OX (E) be the cohomology of the sheaf OX (E) over X. Notice that by definition, we have OX (E) = OX (E pr ).

(5.4.4)

Thus without lost generality, we may and will assume that E is a proper orbifold vector bundle on X. ∞ sections of Λq (T ∗(0,1) X) ⊗ E over X. By the Let Ω0,q X (E) be the sheaf of C x ⊂ Cn proof of Lemma 5.4.3, we can choose the holomorphic local coordinate U E such that Gx acts C-linearly on Cn . As the usual Dolbeault operator ∂ is Gx x ), it induces the Dolbeault operator ∂ E : invariant on each local chart (Gx , U 0,q+1 x ), we (E). By the usual ∂-Lemma on each local chart (Gx , U Ω0,q X (E) → ΩX have the exact sequence of OX -sheaves ∂

E

∂

E

0,n 0 −→ OX (E) −→ Ω0,1 X (E) −→ · · · −→ ΩX (E) −→ 0.

(5.4.5)

0,q Put Ω0,q (X, E) = Γ(X, Ω0,q X (E)), the space of sections of ΩX (E) on X. Set E

Ω0,• (X,E) = ⊕q Ω0,q (X, E). Then we have the Dolbeault complex (Ω0,• (X,E),∂ ) of C ∞ sections of Λ(T ∗(0,1) X) ⊗ E over X : ∂

E

∂

E

0 −→ Ω0,0 (X, E) −→ · · · −→ Ω0,n (X, E) −→ 0.

(5.4.6)

Note that the partition of unity property holds for the sheaves Ω0,q X (E), thus (E) is fine, so their higher cohomology groups vanish. From the abstract de Ω0,q X Rham theorem, we get E

H • (Ω0,• (X, E), ∂ ) H • (X, OX (E)).

(5.4.7)

In the sequel, we also denote H • (X, OX (E)) by H • (X, E). In the same way, we define the Dolbeault cohomology group H r,q (X, E) and (B.4.10) holds. Let g T X be a Riemannian metric on T X, and let hE be a Hermitian metric on E. Let dvX be the Riemannian volume form on X associated to g T X . Then they induce a scalar product on Ω• (X, E) defined in (1.3.14) and (5.4.3), and E,∗ we have the operator ∂ , D, E as in Section 1.4.1. One of the most important properties is that Hodge theory, Theorem 1.4.1, still holds for compact complex orbifolds. This is seen by adapting the arguments from Section 3.1.


239

Theorem 5.4.6 (Hodge theory). For any q ∈ N, we have the direct sum decomposition E

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

E,∗

|Ω0,q+1 ).

(5.4.8)

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

(5.4.9)

Especially, H q (X, E) is finite-dimensional. In the same way, the de Rham cohomology group H • (X, R) is well defined by using C ∞ differential forms on X. Let F be a holomorphic proper orbifold line bundle on X, let hF be a Hermitian metric on F , and let ∇F be the holomorphic Hermitian connection on x ), the corresponding form c1 (F, hF ) is a (F, hF ). Then on each local chart (Gx , U Gx -invariant, real, closed (1, 1)-form and defines a cohomology class c1 (F ) in the de Rham cohomology group H • (X, R). Moreover, by the argument from Appendix B.5, c1 (F ) does not depend on the choice of the metric hF . We call c1 (F ) the first Chern class of F . As we indicated in Section 5.4.1, when we say an orbifold has a specified x ), this structure is induced from structure, this means on each local chart (Gx , U x , and the morphisms preserve this a corresponding Gx -invariant structure on U structure. Especially, analogous to Definition 1.2.7, we can define K¨ ahler orbifolds. Definition 5.4.7. If there is a real, d-closed, (1, 1)-form ω on a complex orbifold x ), ω(·, J·) defines a Gx -invariant metric (X, J) such that on each local chart (Gx , U x , then we say ω is a K¨ on U ahler form on X, ω(·, J·) is a Kähler metric on X, and X is a K¨ ahler orbifold. By the same argument as in Proposition 1.5.3, we have Proposition 5.4.8. Let F be a holomorphic proper orbifold line bundle on a compact K¨ ahler orbifold M . F is positive if and only if its first Chern class may be represented by a positive form in H 2 (M, R).

5.4.3 Asymptotic expansion of the Bergman kernel Let X be a compact complex orbifold with singular set Xsing , and dim X = n. Let (L, hL ) be a holomorphic Hermitian proper orbifold line bundle on X, and let (E, hE ) be a holomorphic Hermitian proper orbifold vector bundle on X. In the rest of this section, we assume that the associated curvature RL of L (L, h ) verifies (1.5.21), i.e., (L, hL ) is a positive proper orbifold line bundle on X. √ −1 L R . Let g T X = ω(·, J·) be the Riemannian metric on X induced by ω := 2π 2 As in (1.5.20), we define the operators Dp , p . Then Dp preserves the Zgrading on Ω0,• (X, Lp ⊗ E).

240


By the same proof as in Sections 1.5.1, 1.5.2, we get vanishing results and the spectral gap property. Theorem 5.4.9. (a) H r,q (X, L) = 0 if r + q > n; especially, (b) H q (X, L ⊗ KX ) = 0 if q > 1. (c) There exists C > 0 such that for any p ∈ N, Spec(Dp2 ) ⊂ {0} ∪ ]4πp − C, +∞[,

(5.4.10)

and Dp2 |Ω0,>0 is invertible for p large enough. Especially, for p large enough, H q (X, Lp ⊗ E) = 0

for any q > 0.

(5.4.11)

By Theorems 5.4.6 and 5.4.9, we can define the Bergman kernel Pp (x, x ) C , as the smooth kernel of the orthogonal projection from (x, x ∈ X), for p > 2π ∞ p C (X, L ⊗ E) onto H 0 (X, Lp ⊗ E), with respect to the Riemannian volume form dvX (x ). dp C From now on, we assume p > 2π . Let {Sip }i=1 , (dp = dim H 0 (X, Lp ⊗ E)), 0 p be any orthonormal basis of H (X, L ⊗ E) with respect to the inner product x , and z ) are Gx -invariant on U (1.3.14). In fact, in the local coordinates above, Sip (

Pp (z, z ) =

dp

Sip ( z ) ⊗ (Sip (z ))∗ ,

(5.4.12)

i=1

x representing z ∈ Ux . where we use z to denote the point in U The following analogue of Theorem 4.1.1 is true. Theorem 5.4.10. There exist smooth coefficients br (x) ∈ End(E)x which are polynomials in RT X , RE and their derivatives with order 2r − 2 at x and C0 > 0 such that for any k, l ∈ N, there exist Ck,l > 0, N ∈ N with k 1 br (x)p−r l n Pp (x, x) − p C r=0 √ √ Ck,l p−k−1 + pl/2 (1 + pd(x, Xsing ))N e− C0 p d(x,Xsing ) , (5.4.13)

for any x ∈ X, p ∈ N∗ . Moreover b0 , b1 are given in (4.1.8). Now, we explain in detail the asymptotic expansion near Xsing . Let {xi }m i=1 be points of Xsing such that the corresponding local charts xi ⊂ Cn verify (Gxi , Uxi ) with U xi , and Xsing ⊂ W := ∪m B U xi (0, 1 ε)/Gxi . B Uxi (0, 2ε) ⊂ U i=1 4


241

xg be the fixed point-set of g ∈ Gxi in U xi , and let N xi ,g be the normal Let U i xg in U xi . For each g ∈ Gxi , the exponential map N x ,g,x Y → bundle of U i i x U g i & expx (Y ) identifies a neighborhood of Uxi to Wxi ,g = {Y ∈ Nxi ,g , |Y | ε}. We identify L|W xg , E|U xg by using the parallel transport along &x ,g , E|W &x ,g to L|U i i i i the above exponential map. Then the g-action on L|W &x ,g is the multiplication i iθg g by e , and θg is locally constant on Ux . Likewise, the g-action on E| & is Wxi ,g

i

xg , End(E)), which is parallel with respect to the connection ∇E . g E ∈ C ∞ (U i xi W &xi ,g , we have ∈ U Note that there exists ε0 such that for any x −1 ∈ W &xi ,g , we write Z = (Z 1,g , Z 2,g ) with Z 1,g ∈ U xg , d(g x , x ) > ε0 . If Z i 2,g ∈ N xi ,g . Z xi as above, there exist polynomials K (Z2,g ) in Z 2,g of Theorem 5.4.11. On U r,Z1,g degree 3r, of the same parity as r, whose coefficients are polynomials in RT X , RE and their derivatives of order r − 2, and a constant C0 > 0 such that for any k, l ∈ N, there exists Ck,l > 0, N ∈ N such that k 1 Z) − −r br (Z)p (5.4.14) n Pp (Z, p r=0 2k −1 r √ −2 −iθg p E −2πp(1−g ) z2,g , z2,g − p e g (Z1,g )Kr,Z1,g ( pZ2,g )e l r=0 1=g∈Gx0 C √ N −k−1 −k+ l−1 2 Ck,l p +p (1 + pd(Z, Xsing )) exp − C0 p d(Z, Xsing ) , as in Theorem 5.4.10 and ε/2, p ∈ N∗ , with br (Z) for any |Z| K0,Z1,g (Z2,g ) = IdE ,

b0 = IdE .

(5.4.15)

Remark 5.4.12. a) The important feature in (5.4.14) is that even for the diagonal asymptotic expansion, we have the exponential decay near the singularity, which is very similar to the off-diagonal expansion. In fact, we use the off-diagonal expansion to get it. b) By the argument in Section 4.1.9, we can reduce the situation for a general metric g T X to the metric associated to ω, because X is compact. Certainly, our argument can be carried out to a certain complete orbifold version as done in Section 6.1 for a general metric on X. Proof of Theorems 5.4.10 and 5.4.11. At first, from the spectral gap property (5.4.10), we have the analogue of Proposition 4.1.5, |Pp (x, x ) − F (Dp )(x, x )|C m (X×X) Cl,m,ε p−l .

(5.4.16)

xi , and the Sobolev norm in (1.6.5) is summed To prove (5.4.16), we work on U over the Uxi .

242


Note that on an orbifold, the property of the finite propagation speed of solutions of hyperbolic equations still holds (see the proof in Appendix D.2). Thus for x, x ∈ X, F (Dp )(x, x ) = 0, if d(x, x ) ε. Likewise, given x ∈ X, F (Dp )(x, ·) only depends on the restriction of Dp to B X (x, ε). Thus the problem of the asymptotic expansion of Pp (x, ·) is local. For any compact set K ⊂ X Xsing , we get the uniform estimate (5.4.13) from Theorems 4.1.2, 4.1.18, Prop. 4.1.6, (4.1.96) and (5.4.16), as in Section 4.2, since Gx = {1} for x ∈ K. Now, working near xi , we replace X by R2n /Gxi , by the above argument. xi ⊂ R2n corresponding to be the Gxi -equivariant vector bundles on U Let L, E xi /Gxi . In particular, Gxi acts linearly and effectively on R2n . We will L, E on U add a superscript to indicate the corresponding objects on R2n . |Z | ε/2 and Z, Z ∈ R2n representing Z, Z , Now for Z, Z ∈ R2n /Gxi , |Z|, we have

F (Dp )(Z, Z ) =

Z ). p )(g −1 Z, (g, 1)F (D

(5.4.17)

g∈Gxi ∗ Here (g1 , g2 ) acts on Ep,x × Ep,x by (g1 , g2 )(ξ1 , ξ2 ) = (g1 ξ1 , g2 ξ2 ). Indeed, if s ∈ ∞ 2n xi , then s is a Gx -invariant section C (R /Gxi , Ep ) has support in B Uxi (0, ε) ⊂ U 2n p ) = F (D p )g, we have p on R . By definition and the relation g · F (D of E

(F (Dp )s)(z) = =

R2n

p )( F (D z , z )s( z )dvU x ( z) i p )(g −1 z, z )s( (g, 1)F (D z )dvU x ( z)

1 |Gxi | g∈Gxi =

i

R2n

R2n /Gxi g∈G xi

p )(g −1 z, z )s(z )dvUx (z ). (5.4.18) (g, 1)F (D i

By (4.1.12) and (4.1.27), for any l, m ∈ N, there exists Cl,m,ε > 0 such that |Z | ε/2, for p 1, |Z|, p )(Z, Z ) − P0,p (Z, Z )|C m Cl,m,ε p−l . |F (D

(5.4.19)

We use now the notation κ as in (4.1.28), Fr as in (4.1.95). By κx , Fr,x we indicate the base point x. For t = √1p , we have by (4.1.96), 0,t (Z/t, Z ) = 1 P Z /t)κ−1/2 (Z)κ−1/2 (Z ). P0,p ((Z, t2n

(5.4.20)


243

ε/2, α, α with |α | m, By (4.2.30), (4.2.36) and (5.4.20), we have for |Z| k ∂ |α| ∂ |α | 1 √ −1 √ −1 r 0,p (g −1 Z, Z)− 2,g ) P Z t F ( pg , p Z )κ ( Z α 2,g 2,g 1,g r,Z1,g Z ∂Z α pn ∂Z 2,g 1,g r=0 √ Ctk+1−m (1 + p|Z2,g |)N exp(− C0 p |Z2,g |). (5.4.21)

Especially, from Theorem 4.1.21, (4.1.68) (or (5.4.21)), we obtain for Z ∈ R2n /Gxi , ε/2, |Z| k ∂ |α| 1 −r 0,p (Z, Z) − Cp−k−1 . P sup p b ( Z) r α pn |α|m ∂ Z

(5.4.22)

r=0

ε/2, Thus from (5.4.16)–(5.4.22), we get for |Z| k ∂ |α| 1 −r P ( Z, Z) − br (Z)p sup α pn p |α|m ∂ Z r=0

r √ √ 2,g ) − p− 2 (g, 1)Fr,Z1,g ( pg −1 Z2,g , pZ2,g )κ−1 ( Z Z1,g r=0 1=g∈Gxi m−1 √ N C p−k−1 + p−k+ 2 (1 + pd(Z, Xsing )) exp − C0 p d(Z, Xsing ) . 2k

(5.4.23) By our identification, from the notation in Theorem 4.1.21 and (4.1.71), √ 2,g , √pZ2,g ) (g, 1)Fr,Z 1,g ( pg −1 Z

1,g )J (√pZ 2,g )e−πp(|Z2,g |2 −2g−1 z2,g ,z2,g ) = eiθg p g E (Z r,Z1,g 1,g )J (√pZ 2,g )e−2πp(1−g−1 )z2,g ,z2,g , (5.4.24) = eiθg p g E (Z r,Z1,g

2,g ) = IdE . where J0,Z1,g (Z

2,g ) in Z 2,g , (Z Now, in (5.4.23), if we take also the Taylor expansion for κ−1 Z 1,g

2,g . then we get (5.4.14) and the information on the polynomials Kr,Z1,g on Z The proof of Theorems 5.4.10 and 5.4.11 is complete.

Remark 5.4.13. In the same way, by Theorem 4.2.1, (5.4.17)–(5.4.19), we get the full off-diagonal expansion of the Bergman kernel on the orbifolds as in Theorem 4.2.1. Note that if x0 ∈ Xsing , then |Gx0 | > 1. Now, if in addition, L and E are usual vector bundles, i.e., Gx0 acts on both Lx0 and Ex0 as identity, then by (5.4.23), 1 (5.4.25) n Pp (x0 , x0 ) − |Gx0 |b0 (x0 ) Cp−1/2 . p

244


Thus we can never have a uniform asymptotic expansion as (4.1.7) on X if Xsing is not empty. In the spirit of (5.4.17), we study now the Bergman kernel on our model space Cn /G (compare Section 4.1.6). Lemma 5.4.14. Let G ⊂ U (n) be a finite subgroup acting C-linearly and isometric on Cn , then there exists CG,n > 0 such that for any Z ∈ Cn , −1 |G| fG (Z) := 1 + e−2π(1−g )z,z CG,n . (5.4.26) 1=g∈G

Proof. As Re(1 − g −1 )z, z 0 for any g ∈ G, z ∈ Cn , we know |fG (Z)| |G|. We recall now some notation from Remark 4.1.19. Let (L, hL ) be the trivial holomorphic line bundle on Cn with the canonical section 1, and |1|hL (z) := 2 e−π|z| (Here under our notation (4.1.71), |z|2 = 12 |Z|2 .) Then by (4.1.84), the Bergman kernel of (L, hL ) under the trivialization of L by using the unit section 2 eπ|z| 1 is

(5.4.27) P(Z, Z ) = exp −π(|z|2 + |z |2 − 2z, z ) . We define the action of G on L by g · 1 = 1 for g ∈ G. Then the Bergman kernel PCn /G of L on Cn /G is (cf. (5.4.17)), PCn /G (Z, Z ) = P(g −1 Z, Z ). (5.4.28) g∈G

Now, the L2 -holomorphic sections of L on Cn /G are H 0 (Cn , L)G , the G-invariant 2 L2 -holomorphic sections of L on Cn . Let {ϕj e−π|z| }∞ j=1 be an orthonormal basis of H 0 (Cn , L)G with G-invariant homogeneous polynomials ϕj on z and ϕ1 = 1. Thus from (5.4.27) and (5.4.28), we get for any Z ∈ Cn , ∞ 2 fG (Z) = PCn /G (Z, Z) = 1 + |ϕj (z)|2 e−2π|z| > 0.

(5.4.29)

j=2

To get the uniformly positivity of fG (Z), we will prove it by recurrence. Set CG,n = inf n fG (Z),

(5.4.30)

Z∈C

and for g ∈ G, and H ⊂ G a subgroup of G, set Vg = {z ∈ Cn ; gz = z},

VH = {z ∈ Cn ; hz = z for any h ∈ H},

(5.4.31)

and denote by Vg⊥ , VH⊥ their orthogonal complements in Cn . If n = 1, and if g does not act as identity on C, then e−2π(1−g |z| → ∞. Thus for any finite group G, we get CG,1 > 0.

−1

)z,z

→ 0 as


245

Assume that for n k, CG,n > 0 for any finite group G. For n = k + 1, let G be a finite group as above. If Vg = {0} for any g ∈ G, then for each g ∈ G, there exists an orthonormal basis {wj } of Cn such that gwj = eiθj wj and 0 < θj < 2π. Hence for z = j vj wj , 2(1 − g −1 )z, z =

(1 − e−iθj )|vj |2 .

(5.4.32)

j −1

From (5.4.32), we know that for any g ∈ G, e−2π(1−g )z,z → 0 as |z| → ∞. We get thus CG,k+1 > 0 from (5.4.26) and (5.4.29). Assume now there exists g ∈ G such that Vg = {0}. Let R be large enough such that for any g ∈ G, Z ∈ Vg⊥ , |Z| > R, we have −2π(1−g−1 )z,z e
0. Then from (5.4.26), we have Here by our recurrence hypothesis, C fG (Z) >

3 4

⊥

for Z ∈ Cn ∪g Vg × B Vg (0, R).

(5.4.34) ⊥

We will prove that fG (Z) is uniformly positive on ∪g Vg × B Vg (0, R) again by recurrence. Note that VG = {0}. If H0 is a maximal subgroup of G such that VH0 = {0}, ⊥ / H0 , Vg × B Vg (0, R) ∩ then for any g ∈ / H0 , Vg ∩ VH0 = {0}. Thus for any g ∈ ⊥ / H0 , VH0 × B VH0 (0, R) is bounded. Thus there exists R1 > 0 such that for g ∈ ⊥ Z ∈ VH0 ×B VH0 (0, R)B(0, R1 ), (5.4.33) holds. But for Z = (Z0 , Z0⊥ ) ∈ VH0 ×VH⊥0 , we have −1 e−2π(1−g )z,z = fH0 (Z0⊥ ) CH0 ,k+1−dim VH0 . (5.4.35) fH0 (Z) = 1 + 1=g∈H0 ⊥

Thus fG (Z) > 34 CH0 ,k+1−dim VH0 on VH0 × B VH0 (0, R) B(0, R1 ). From (5.4.29), ⊥

fG (Z) > 0 on B(0, R1 ), thus fG (Z) > C > 0 on VH0 × B VH0 (0, R). Now assume that H1 ⊂ G is a subgroup such that VH1 = {0} and for any subgroup H ⊂ G such that H1 ⊂ H and H1 = H, then fG (Z) > C > 0 on ⊥ VH × B VH (0, R). Then we claim that there exists C > 0 such that fG (Z) > C ⊥ on VH1 × B VH1 (0, R). In fact, for g ∈ G, g ∈ / H1 , we denote by gH1 the group generated by g and H1 . Then ⊥

⊥

⊥

VH1 × B VH1 (0, R) ∩ Vg × B Vg (0, R) = VgH1 × B VgH1 (0, R).

(5.4.36)

246

Chapter 5. Kodaira Map ⊥

⊥

Thus fG (Z) > C > 0 on VH1 × B VH1 (0, R) ∩ Vg × B Vg (0, R) by our recurrence condition. For Z = (Z0 , Z0⊥ ) ∈ VH1 × VH⊥1 , fH1 (Z) = 1 +

e−2π(1−h

−1

)z,z

= fH1 (Z0⊥ ) CH1 ,k+1−dim VH1 .

(5.4.37)

1=h∈H1 ⊥

⊥

Vg As (5.4.33) holds for Z ∈ VH1 × B VH1 (0, R) (∪g∈G,g∈H (0, R)), and / 1 Vg × B g ∈ G, g ∈ / H1 , we get from (5.4.37) that

fG (Z) > ⊥

3 CH ,k+1−dim VH1 4 1

(5.4.38)

⊥

Vg (0, R)). Therefore, we have proved on VH1 × B VH1 (0, R) (∪g∈G,g∈H / 1 Vg × B ⊥

that there exists C > 0 such that fG (Z) > C on VH1 × B VH1 (0, R). In this way, we conclude that for any g ∈ G, fG (Z) > C > 0 on Vg × Vg⊥ B (0, R). By (5.4.34), we know that CG,n > 0. The proof of Lemma 5.4.14 is complete.

5.4.4 Projective embedding theorem Recall that a ring without zero-divisors is called normal if it is integrally closed in its quotient field. Definition 5.4.15. A reduced complex space (X, OX ) is called normal in x ∈ X if the local ring OX,x is a normal ring. (X, OX ) is called normal complex space if it is normal in every point. Theorem 5.4.16 (Cartan–Serre). Let X be a complex orbifold. Then (X, OX ) is a normal complex space. OX (E) is an OX -coherent analytic sheaf for any holomorphic orbifold vector bundle E on X. To understand better Theorem 5.4.16, we explain now its local version. x ) where the fiBy Lemma 5.4.3, any x ∈ X admits a neighborhood (Gx , U x is a Gx -neighborhood of 0. Let nite group Gx acts C-linearly on Cn and U SG = C[z1 , . . . , zn ]Gx be the algebra of Gx -invariant polynomials in z1 , . . . , zn . Lemma 5.4.17 (Cartan). SG is a finite generated C-algebra. If Q1 , . . . , Qq are Gx -invariant homogeneous polynomials which generate SG , and f1 , . . . , fq are Gx x , a neighborhood of 0, such that for any invariant holomorphic functions on W ⊂ U 1 i q, fi − Qi vanishes at 0 with order > deg Qi , then the map ϕ : W → Cq , ϕ(z) = (f1 (z), . . . , fq (z)),

(5.4.39)

defines an isomorphism from (W/Gx , OW/Gx ) to a normal local model V in an open set of Cq .


247

Now let X be a compact complex orbifold with singular set Xsing , and dim X = n. Let (L, hL ) be a holomorphic Hermitian proper orbifold line bundle on X. 5.4.3. Let g T X = ω(·, J·) be We assume that (L, hL ) is positive as in Section √ −1 L the Riemannian metric on X induced by ω := 2π R . As explained in Problem 5.11, we may and we will assume that for any x ∈ X, Gx acts as identity on Lx , i.e., L is a usual line bundle on X. Theorem 5.4.18. There exists p0 ∈ N such that for p p0 , Φp : X → P(H 0 (X,Lp )∗ ) defined by (5.1.12) 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 )

(5.4.40)


hΦp O(1) (x) = Pp (x, x)−1 hL (x). p

(5.4.41)

∗

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. Near xi in Theorem 5.4.11, set b0,p (Z) =1+

e−2πp(1−g

−1

) z2,g , z2,g

,

(5.4.42)

1=g∈Gxi

2,g ). Then b0,p (Z) is real as = (Z 1,g , Z with Z b0,p (Z) =1+

e−2πp(1−g

1=g∈Gxi

=1+

−1 ) z2,g , z2,g

e−2πpz2,g ,(1−g)z2,g = b0,p (Z). (5.4.43)

1=g∈Gxi

0 | ε/2, there exists a neighborhood U such By Lemma 5.4.14, for any |Z Z0 that b0,p (Z) > C > 0 for any Z ∈ UZ0 , p > p0 . By using a partition of unity ε/2, p > p0 , argument, we know that there exist p0 ∈ N, C > 0 such that for |Z| > C. |Gxi | b0,p (Z) xi , for |Z| ε/2, By (5.4.14), as θg = 0 now, on U −n Z) − b0,p (Z) Cp−1/2 . p Pp (Z,

(5.4.44)

(5.4.45)

By using (5.4.13), (5.4.44), (5.4.45) and the same proof as for Lemma 5.1.2 and Theorem 5.1.3, we get Theorem 5.4.18.

248


For any x ∈ X, the Kodaira map Φp induces a Gx -equivariant map Ψp from x to P(H 0 (X, Lp )∗ ). Thus, if |Gx | > 1, the differential of Ψp , (dΨp )x cannot U , which contradicts |Gx | > 1). be injective (otherwise, Ψp is an embedding near x Hence the induced Fubini–Study form Φ∗p ωF S must be degenerate on the singular set Xsing . The following result tells us that we can approximate the original metric on L by using the metric induced by Φp , and we can still approximate the original K¨ ahler form by using the induced Fubini–Study form away from the singular set. Theorem 5.4.19. As p → ∞,

1 p log Pp (x, x) Φ∗ O(1) 1/p p

→ 0 in C 1,α for any 0 α < 1.

) on L converges to hL in C 0 -norm as p → ∞. Especially, the metric (h For any l ∈ N, there exists Cl > 0 such that for p p0 , 1 1 √ l (5.4.46) + p 2 e−c pd(x,Xsing ) . (Φ∗p ωF S )(x) − ω(x) l Cl p p C Proof. By (5.4.23), (5.4.41), (5.4.42) and (5.4.44), we get the first part of our theorem. By (5.4.41), the analogue equation (5.1.21) still holds here. From (5.1.21), (5.4.14) and (5.4.44), we get (5.4.46). The following result is the orbifold version of the Kodaira embedding theorem. Theorem 5.4.20 (Baily). There exists p0 ∈ N such that for p p0 , Φp (X) is an analytic subset of P(H 0 (X, Lp )∗ ). The Kodaira map Φp is injective and for any x ∈ X, Φp induces an isomorphism from OΦp (X),Φp (x) to OX,x . Thus Φp (X) is a normal algebraic variety in P(H 0 (X, Lp )∗ ) for p p0 . Before starting the proof, let us mention that we can use the asymptotic expansion of the Bergman kernel to prove Φp is injective as in Lemma 5.1.8, but it does not seem clear how to get the corresponding version of Lemma 5.1.6 by using the Bergman kernel. Thus we need to go back to the original proof of Baily which extends Kodaira’s proof of Theorem 5.1.12 to the orbifold case. Namely, we use the vanishing Theorem 5.4.9 (b) and the blow-up technique to get the precise information about Φp near the singular set Xsing such that we can apply Lemma 5.4.17. x , Gx ) is a local coordinate of x as in Lemma Proof. In what follows, for x ∈ X, (U x ⊂ Cn and 0 ∈ U x represents x. 5.4.3 such that Gx acts C-linearly on U Let Wx → Ux be the blow-up of Ux with center 0; then Gx acts naturally on Wx , and under the notation in (2.1.6), for (z, [t]) ∈ Wx , g · (z, [t]) = (g · z, [g · t]), and the exceptional divisor E defines a Gx -equivariant line bundle OWx (E) := X x = X {x} π over Wx as explained after (2.1.8). Thus we define by X induced Wx /Gx the blow-up of x, and still by OX (E) the orbifold line bundle on X α by OWx (E). (Locally, let fα be the definition function of the divisor E on U and SE the corresponding canonical section of OWx (E). Then the orbifold bundle OX (E) is induced by the action g · (z, ξ) = (g · z, ffαα(g·z) (z) ξ) on the trivialization


249

α defined by fα , and g ∈ Gxi verifies g(U α ) = U α . Moreover, SE of OWx (E) on U we always work on U x . Thus from the is g-invariant.) To obtain the result on X, proof of Proposition 2.1.7, we get an isomorphism of holomorphic orbifold line bundles KX = π ∗ (KX ) ⊗ OX ((n − 1)E).

(5.4.47)

From the proof of Proposition 2.1.8, 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 . = For (x, y) ∈ X × X, we define as in Section 5.1.3 the blow-up of π : X Xx,y → X of X with center {x, y}. Then by Theorem 5.4.9 (b) and the above arguments, we know that Lemmas 5.1.13, 5.1.14 and (5.1.34) still hold in our case. Problem 5.11 shows the existence of k0 ∈ N∗ such that |Gx | |k0 for all x ∈ X. There exists therefore a canonical section Sk0 Ex,y of OX (k0 Ex,y ) which is a usual bundle on X # (if SEx is the canonical section of OWx (Ex ) on Ux as in Definition 2.1.3, then g∈Gx g · SEx is a Gx -invariant section of OX (|Gx |Ex )). Let IEx,y be the ideal of holomorphic functions vanishing on Ex,y . Now, as in Proposition 2.1.4, the map 0 −→ OX (−mk0 Ex,y ) , IEmk x,y

f −→ f · Sk−m , 0 Ex,y

is an isomorphism. Finally, from the argument after (5.1.35), we conclude that for m ∈ N, there exists p0 ∈ N such that for all p p0 and all (x, y) ∈ X × X, β

H 0 (X, Lp ) −−−−→ H 0 (X, Lp ⊗ OX /I (x, y)mk0 )

is surjective .

(5.4.48)

For x = y, m = 1, as Gx acts trivially on Lx , we know H 0 (X, Lp ⊗ OX /I (x, y)k0 ) ⊃ Lpx ⊕ Lpy .

(5.4.49)

From (5.4.48) and (5.4.49), there exists p0 ∈ N such that Φp is injective for p p0 . Now we take x = y, then H 0 (X, Lp ⊗ OX /I (x, y)mk0 ) {Q ∈ C[z1 , . . . , zn ]Gx ; deg Q 2mk0 }. (5.4.50) xi , Gxi )}k be local coordinates of X such that U xi /Gxi is a covering of X. Let {(U i=1 j Let {Gi }j be the subgroups of Gxi and we denote by d(Gji ) the maximum degree of Q in Lemma 5.4.17 for Gji . Let N = supi,j d(Gji ), and take m > N/k0 . By (5.4.48) and (5.4.50), for any x ∈ X, let Q0,x = 1, Q1,x, . . . , Qq,x be Gx -invariant linear independent homogeneous polynomials which generate SGx ; then there exist s0x , . . . , sqx linear independent sections of H 0 (X, Lp ) such that β(slx ) = Ql,x . Then we complete them as a basis of H 0 (X, Lp ); from Lemma 5.4.17 and (5.1.17), we know that Φp induces an isomorphism from OΦp (X),Φp (x) to OX,x .

250


5.5 The asymptotic of the analytic torsion The holomorphic analytic torsion of Ray-Singer is obtained as the regularized determinant of the Kodaira Laplacian of holomorphic Hermitian vector bundles on a compact complex manifold. The Quillen metric is a natural metric on the determinant of the cohomology of a holomorphic Hermitian vector bundle, which one constructs by using the Ray-Singer analytic torsion. One of its remarkable properties, established by Bismut–Gillet–Soulé, is that the Quillen metric on the determinant of the fiberwise cohomology of a holomorphic fibration is a smooth metric and the curvature of the corresponding holomorphic Hermitian connection is given by an explicit local formula, which is compatible with the Grothendieck–Riemann–Roch theorem at the level of differential forms. The analytic torsion has also equivariant and family extensions. Especially, the analytic torsion forms of Bismut–Gillet–Soulé and Bismut–K¨ ohler are differential forms on the base manifold for a holomorphic fibration. Its 0-degree component is the analytic torsion of Ray-Singer along the fiber. The analytic torsion and its extensions were studied extensively by Bismut and his coauthors in the last two decades. The holomorphic analytic torsions have found a lot of applications, especially as the analytic counterpart of the direct image in Arakelov geometry. In this section, we will study the asymptotic of the analytic torsion when the power of the line bundle tends to ∞. In the whole theory on the analytic torsion, to get explicit local terms as in Atiyah–Singer index theory (here we need to use the secondary classes of Bott-Chern type), in other words, to apply the local index type computations, we need to assume the base metrics are Kähler. However, here we get the local term without this assumption as the power tends to ∞. This section is organized as follows. After briefly recalling the Mellin transformation in Section 5.5.1, we define the holomorphic analytic torsion in Section 5.5.2 by using the heat kernel. In Section 5.5.3, we study the dependence of the analytic torsion on a change of the metrics. In Section 5.5.4, we study the asymptotic of the analytic torsion when the power of the line bundle tends to ∞. In Section 5.5.5, we establish the corresponding version for L2 metrics by combining Sections 5.5.3 and 5.5.4. In Section 5.5.6, we study certain technical results on heat kernels. We use the notation in Sections 1.4.1, 1.6.1.

5.5.1 Mellin transformation Let Γ(z) be the Gamma function on C. Then for Re(z) > 0, we have ∞ e−u uz−1 du. Γ(z) = 0

We suppose that f ∈ C ∞ (R∗+ ) verifies the following two conditions:

(5.5.1)

5.5. The asymptotic of the analytic torsion

251

(i) There exist m ∈ N, fj ∈ C, (j −m) such that for any k ∈ N, there exists Ck > 0 such that as u → 0, k fj uj Ck uk+1 . (u) − f

(5.5.2)

j=−m

(ii) There exist c, C > 0 such that as u → +∞, |f (u)| Ce−cu .

(5.5.3)

Definition 5.5.1. The Mellin transformation of f is the function defined for Re(z) > m, ∞ 1 M [f ](z) = f (u)uz−1 du. (5.5.4) Γ(z) 0 Lemma 5.5.2. M [f ] extends to a meromorphic function on C with poles contained in the set m − N, and its possible poles are simple. M [f ] is holomorphic at 0 and M [f ](0) = f0 . (5.5.5) ∞ Proof. By (5.5.3), the function 1 f (u)uz−1 du is an entire function on z ∈ C. For any k ∈ N, we have

1

f (u)uz−1 du = 0

k j=−m

1

1

O(uk+z )du

uj+z−1 du +

fj 0

0

(5.5.6)

k

fj + R(z), = z +j j=−m where R(z) is a holomorphic function for Re(z) > −k − 1. Observe that the inverse of the Gamma function Γ(z)−1 is entire on C and Γ(z)−1 = z + O(z 2 ) near z = 0. Hence we get Lemma 5.5.2.

5.5.2 Definition of the analytic torsion For any Z2 -graded finite-dimensional vector space V = V + ⊕ V − , let τ ∈ End(V ) be defined by τ = Id on V + , and τ = − Id on V − . For any B ∈ End(V ), we define the supertrace Trs on End(V ) by Trs [B] = Tr[τ B].

(5.5.7)

Remember that [B, C] denotes the supercommutator of B and C (cf. (1.3.30)). Then by a direct check, we get for any B, C ∈ End(V ), Trs [[B, C]] = 0.

(5.5.8)

252


Let (X, J) be a compact complex manifold with complex structure J and dim X = n. Let g T X be any Riemannian metric on T X compatible with J. Let (E, hE ) be a holomorphic Hermitian vector bundle on X. Let ∇E be the holomorphic Hermitian connection on (E, hE ) with curvature RE . Recall that the operators D, E were defined in (1.4.3). The operator P is the orthogonal projection from (L2 (X, Λ(T ∗(0,1) X) ⊗ E), ·, ·) onto Ker(D). Set P ⊥ = Id −P . Let N be the number operator on Λ(T ∗(0,1) X), i.e., N acts on Λq (T ∗(0,1) X) by multiplication by q. The Z2 -grading on Λ(T ∗(0,1)X) ⊗ E and on Ω0,• (X, E) is defined by τ = (−1)N (induced by the Z-grading). The following lemma is useful in computation of the supertrace. Lemma 5.5.3. For any operators K1 , K2 on Ω0,• (X, E) with smooth kernels K1 (x, y), K2 (x, y) associated to dvX (y), we have Trs [K1 , K2 ] = 0.

(5.5.9)

Proof. In fact, if K1 or K2 preserves the Z2 -grading of Ω0,• (X, E), then by (5.5.8), Trs [K1 K2 ] =

Trs K1 (x, y)K2 (y, x)dvX (y) dvX (x) X X = Trs K2 (y, x)K1 (x, y)dvX (x) dvX (y) = Trs [K2 K1 ]. X

X

In the same way, we can verify (5.5.9) if K1 and K2 exchange the Z2 -grading of Ω0,• (X, E). By (D.1.17) and (D.1.24), we have the following expansion in the sense of (5.5.2), for any k ∈ N, k 2 u Trs N e− 2 D = cj uj + O(uk+1 ),

(5.5.10)

j=−n

when u → 0. Now let 0 = λ1 = · · · = λm < λm+1 · · · be the eigenvalues of D2 with eigenfunctions ϕλj . Then by (D.1.17), for u > 2 ∞ − u2 D2 ⊥ P )(x, x) e−λk u/2 |ϕλk (x) ⊗ ϕλk (x)∗ | (e k=m+1

C

∞

e−λk u/2 |ϕλk (x)|2 Ce−λm+1 (u−2)/2

k=m+1

∞

e−λk |ϕλk (x)|2

k=m+1

Tr[(e−D P ⊥ )(x, x)]. (5.5.11) 2 u By (D.1.17) and (5.5.11), the function Trs N e− 2 D P ⊥ verifies also (5.5.3). = Ce

−λm+1 (u−2)/2

2


In view of the above discussion and Lemma 5.5.2, set 2 u θ(z) = −M Trs [N e− 2 D P ⊥ ] (z) = − Trs [N (E )−z P ⊥ ].

253

(5.5.12)

Then θ(z) is a meromorphic function on C with poles contained in the set n − N, its possible poles are simple, and θ(z) is holomorphic at 0. Moreover, θ (0) := − 1

∞

∂θ (0) = − ∂z

0

1

0 du u Trs N exp(− D2 ) − cj u j 2 u j=−n

−1 du u cj − + Γ (1)(c0 − Trs [N P ]). (5.5.13) Trs N exp(− D2 )P ⊥ 2 u j j=−n

Definition 5.5.4. The analytic torsion of Ray–Singer for the vector bundle E is exp(− 12 θ (0)). The determinant H • (X, E) is the complex line given 0nline of theq cohomology • (−1)q by det H (X, E) = q=0 (det H (X, E)) . We define λ(E) = (det H • (X, E))−1 .

(5.5.14)

Let hH(X,E) be the L2 -metric on H • (X, E) induced by the canonical isomorphism (1.4.6) and the L2 -scalar product on Ω0,• (X, E). Let | · |λ(E) be the L2 -metric on λ(E) induced by hH(X,E) . Definition 5.5.5. The Quillen metric ·λ(E) on the complex line λ(E) is defined by 1 · λ(E) = | · |λ(E) exp(− θ (0)). 2

(5.5.15)

The Quillen metric · λ(E) depends on the choice of the metrics g T X and hE . We will study their dependence now.

5.5.3 Anomaly formula We denote by ∗E : Λ(T ∗(0,1) X) ⊗ E → Λ(T ∗ X ⊗R C) ⊗ E ∗ the Hodge ∗-operators defined by for σ, σ ∈ Λ(T ∗(0,1) X) ⊗ E (cf. (1.3.14)), σ, σ Λ0,• ⊗E dvX = (σ ∧ ∗E σ )E .

(5.5.16)

We denote ∗ the corresponding Hodge operator for E = C. Let R v → (gvT X , hE v ) be a smooth family of metrics on T X and E and TX gv compatible with J. Let ∗v , ∗E v be the Hodge ∗-operators associated to the metrics gvT X , (gvT X , hE v ). Let Dv be the operator D defined in (1.4.3) attached to the metrics (gvT X , hE v ). Let ·λ(E),v be the corresponding Quillen metric on λ(E). Set E E ∂∗v −1 ∂∗v −1 ∂hv = − ∗−1 + (hE Qv = −(∗E . (5.5.17) v) v v) ∂v ∂v ∂v

254


Theorem 5.5.6. As u → 0, for any k ∈ N, there is an asymptotic expansion k u Trs Qv exp(− Dv2 ) = Mj,v uj + O(uk+1 ), 2 j=−n

with M0,v =

∂ log · 2λ(E),v . ∂v

Proof. From (D.1.24), we get (5.5.18). Moreover, we have √ E √ E,∗ E E,∗ Dv = 2(∂ + ∂ v ), [Dv , N ] = 2(−∂ + ∂ v ).

(5.5.18)

(5.5.19)

(5.5.20)

For s ∈ Ω0,q (X, E), we have E∗

E,∗

−1 (5.5.21) ∂ v s = (−1)q (∗E ∂ ∗E v) v s. E,∗ E E E∗ q In fact, s1 , ∂ v s = ∂ s1 , s = X (∂ s1 ∧∗E (s ∧∂ ∗E v s)E = (−1) v s)E . X 1 v

v

Then by (5.5.17) and (5.5.21), E,∗ 1 ∂ ∂ E,∗ √ Dv = ∂ v = − ∂ v , Qv , ∂v 2 ∂v

∂ 2 ∂ Dv = Dv , Dv . ∂v ∂v

(5.5.22)

In what follows, we omit often the subscript v. By (D.1.19), we have u ∂ −uD2 ∂ 2 −u1 D2 −(u−u1 )D2 e = e du1 . (5.5.23) − D e ∂v ∂v 0 Observe that D2 preserves the Z-grading of Ω0,• (X, E), and thus commutes with

∂ 2 −uD2 D e is an operator with smooth kernel. By (5.5.22) and (5.5.23), N , and − ∂v we have u ∂ 2 −u1 D2 −(u−u1 )D2 ∂ −uD 2 Trs [N e D e ]= Trs Ne − du1 ∂v ∂v 0 ∂D 2 N e−uD . (5.5.24) = −u Trs D, ∂v By using (5.5.9), that D is an odd operator, (i.e., it changes the Z2 -grading 2 of Ω (X, E)), and that D commutes with e−uD , we get for 0 < u1 < u, 0,•

∂D 2 2 2 ∂D N e−uD = Trs e−(u−u1 )D D N e−u1 D Trs D ∂v ∂v 2 ∂D 2 N e−u1 D = Trs De−(u−u1 )D ∂v ∂D ∂D 2 2 2 N e−u1 D De−(u−u1 )D = − Trs N De−uD . (5.5.25) = − Trs ∂v ∂v


255

(5.5.25) entails ∂D ∂D 2 2 Trs D, N e−uD = Trs [D, N ]e−uD . ∂v ∂v E,∗

(5.5.26)

Using that ∂ commutes with e− 2 D , and relations (5.5.20), (5.5.22), (5.5.24) and (5.5.26), we get by using the same trick as in (5.5.25): E,∗ 2 ∂ u E E,∗ − u D2 Trs N e− 2 D = −u Trs Qv ∂ , −∂ + ∂ e 2 ∂v (5.5.27) 2 2 u ∂ u u Trs Qv e− 2 D . = Trs Qv D2 e− 2 D = −u 2 ∂u u

2

Let Pv be the orthogonal projection operator from Ω0,• (X, E) on Ker(Dv ) for the Hermitian product v on Ω0,• (X, E) associated with gvT X and hE v defined 2 u in (1.3.14). From (5.5.11), Trs [Qv e− 2 Dv P ⊥ ] decays exponentially when u → +∞. From (5.5.27), we obtain for Re(z) large enough, ∞ 2 u ∂ ∂ 1 θv (z) = Trs Qv e− 2 Dv du uz ∂v Γ(z) 0 ∂u (5.5.28) ∞ −z z−1 −u Dv2 ⊥ 2 = u Trs Qv e Pv du. Γ(z) 0 Using (5.5.5) and (5.5.28), we find ∂ ∂ ( θv )(0) = −M0,v + Trs [Qv Pv ]. ∂v ∂z

(5.5.29)

The line bundle λ(E) is canonically identified to λv = ⊗nq=0 (det Ker(Dv2 |Ω0,q ))(−1)

q+1

.

Under this identification, the canonical isomorphism of the line bundle φv : λ0 → λv is defined by φv (σ) = Pv (σ) for σ ∈ λ0 . If σ and σ are forms in the kernel of Dv2 , we have by definition Pv σ, Pv σ v = (Pv σ ∧ ∗E (5.5.30) v Pv σ )E . X ∂ ∂ ∂ From = Pv , we get + Pv ( ∂v Pv ) = ∂v Pv , thus the operator ∂v Pv sends Ker(Dv ) to its orthogonal complement for ·, ·v . Therefore, from (5.5.30), we get ∂ ∂ Pv σ, Pv σ v = (Pv σ ∧ ( ∗E (5.5.31) v )Pv σ )E = − Pv σ, Qv Pv σ v . ∂v ∂v X

Pv2

∂ ( ∂v Pv )Pv

Thus from (5.5.14) and (5.5.31), we get ∂ log(|φv (σ)|2λ(E),v ) = Trs [Qv Pv ]. ∂v From (5.5.15), (5.5.29) and (5.5.32), we get (5.5.19).

(5.5.32)

256


Remark 5.5.7. As the asymptotic of the heat kernel is local (as explained in (D.1.24)), M0,v is an integral of terms with local character. On the other hand, the Quillen metric · λ(E) is defined by using the spectrum of the Kodaira Laplacian, so it is a global invariant of the manifold. Even if we do not compute M0,v here, it is remarkable that (5.5.7) shows that the variation of the Quillen metrics has local character. Actually, when gvT X are K¨ ahler, Bismut–Gillet–Soulé gave a precise formula for M0,v , known as the anomaly formula for Quillen metrics.

5.5.4 The asymptotics of the analytic torsion We resume now the discussion from Section 5.5.2. Let L be a positive holomorphic line bundle on X. Let hL be a Hermitian on L such that the curvature RL √ metric L L associated to h verifies (1.5.21), i.e., −1R is a positive (1, 1)-form. We use the notation in Section 1.5.1 now, especially Dp , p were defined in (1.5.20), and Dp2 = 2p preserves the Z-grading of Ω0,• (X, Lp ⊗ E), and √ −1 L ω = 2π R . We denote by Trq exp(− up Dp2 ) the trace of exp(− up Dp2 ) acting on Ω0,q (X, Lp ⊗ E). We denote by θp (z) the function associated to Lp ⊗ E as in (5.5.12). The following result is the main result of this section. Theorem 5.5.8. As p → ∞, we have $ % pR˙ L 1 log det θp (0) = rk(E) ep ω + o(pn ). 2 2π X

(5.5.33)

Now we state two intermediate results which will be used in the proof√of TheL orem 5.5.8. Note that Theorem 5.5.9 holds without the assumption that √ −1R L is positive, but in Theorem 5.5.11 we have to assume the positivity of −1R . Let Aj ∈ C ∞ (X, End(Λ(T ∗(0,1) X))) such that as u → 0, (2π)−n

k det(R˙ L ) exp(2uωd ) Aj (x)uj + O(uk+1 ). = det(1 − exp(−2uR˙ L)) j=−n

(5.5.34)

Theorem 5.5.9. There exist Ap,j ∈ C ∞ (X, End(Λ(T ∗(0,1) X) ⊗ E)) such that for any k, l ∈ N, there exists C > 0 such that for any u ∈]0, 1], p ∈ N∗ , we have the asymptotic expansion k u −n Ap,j (x)uj Cuk . p exp − Dp2 (x, x) − p C l (X) j=−n

(5.5.35)

For any j −n, as p → ∞, we have Ap,j (x) = Aj (x) ⊗ IdE +O(p−1/2 ).

(5.5.36)


257

We will prove Theorem 5.5.9 in Section 5.5.6, and we only use (5.5.35) with C 0 -norm (i.e., l = 0) in the proof of Theorem 5.5.8. Now we establish Theorem 5.5.10 as a consequence of Theorem 5.5.9. For x ∈ X, u > 0, set Ru (x) = det

R˙ L 2π

Tr (Id − exp(uR˙ L ))−1 .

(5.5.37)

Observe that from (1.5.14), dvX = Θn /n! ,

det

R˙ L dvX (x) = ω n /n! . 2π

(5.5.38)

From (5.5.37) and (5.5.38), for any k ∈ N, we have the following asymptotic expansion for u → 0, Ru (x) =

k

j (x)uj + O(uk+1 ), A

(5.5.39)

j=−1

with j = 0 A

for j −2,

For j −n, set

n−1 −1 dvX = − Θ ω , A 2π (n − 1)!

n 0 dvX = n ω . A 2 n!

Bp,j = 2−j Trs N Ap,j (x) dvX (x), X j (x)dvX (x). A Bj =

(5.5.40)

(5.5.41)

X

Theorem 5.5.10. For any k, l ∈ N, there exists C > 0 such that for any u ∈]0, 1], p ∈ N∗ , k u −n Bp,j uj Cuk . p Trs N exp(− Dp2 ) − 2p j=−n

For any j −n, as p → ∞, we have O(p−1/2 ) for j −2, Bp,j = rk(E)Bj + O(p−1/2 ) for j −1.

(5.5.42)

(5.5.43)

Proof. By (1.5.19), we verify directly Trs [N euωd ] =

∂ det(Id −ec exp(−uR˙ L)) . ∂c c=0

(5.5.44)

258


From (5.5.37) and (5.5.44), we deduce that (2π)−n

det(R˙ L ) Trs [N euωd ] = Ru (x). det(1 − exp(−uR˙ L ))

(5.5.45)

From (5.5.34), (5.5.39) and (5.5.45), we get j (x). 2−j Trs [N Aj (x)] = A

(5.5.46)

By Theorem 5.5.9, for any k, l ∈ N, there exists C > 0 such that for any u ∈ ]0, 1], p ∈ N∗ , we have k u uj −n Trs [N Ap,j (x)] j l Cuk . (5.5.47) p Trs N exp(− Dp2 )(x, x) − 2p 2 C (X) j=−n

From (5.5.40), (5.5.41), (5.5.46) and (5.5.47), we get Theorem 5.5.10.

Theorem 5.5.11. There exist C, c, c > 0 such that for any q 1, u 1, p ∈ N, we have u (5.5.48) p−n Trq exp(− Dp2 ) C exp − (c − c /p)u . 2p Proof. By Theorem 1.5.5, for u 1, q 1, we have (u − 1) u 1 (2C0 p − CL ) Trq exp(− Dp2 ) . (5.5.49) Trq exp(− Dp2 ) exp − 2p 2p p By (1.6.4), we know that p−n Trq exp(− p1 Dp2 ) has a finite limit as p → ∞.

From (5.5.49), we get (5.5.48). Proof of Theorem 5.5.8. For p CL /C0 in Theorem 1.5.5, set u 2 −n θp (z) = −M p Trs N exp(− Dp )(1 − Pp ) (z). 2p

(5.5.50)

Clearly p−n θp (z) = p−z θp (z).

(5.5.51)

Thus from (1.4.23), (5.5.5), (5.5.42) and (5.5.51), we get for p ≥ CL /C0 p−n θp (0) = − log(p)θp (0) + θp (0), θp (0) = −Bp,0 ,

Trs [N Pp ] = 0.

(5.5.52)


259

By Lemma 5.5.2, (5.5.13) and (5.5.52), we get for p ≥ CL /C0 θp (0) = −

1

0

−

∞

1

0 du u p−n Trs N exp(− Dp2 ) − Bp,j uj 2p u j=−n

−1 du u Bp,j − + Γ (1)Bp,0 . (5.5.53) p−n Trs N exp(− Dp2 ) 2p u j j=−n

From Theorem 1.6.1, and (5.5.45), for any u > 0, lim p−n Trs [N exp(−

p→∞

u 2 D )(x, x)] = rk(E)Ru (x), 2p p

(5.5.54)

and the convergence is uniform for x ∈ X and for u varying in compact subsets of R∗+ . For z ∈ C, Re(z) > 1, set = −M ζ(z)

Ru (x)dvX (x) (z).

(5.5.55)

X

By Theorems 5.5.10, 5.5.11, (5.5.53), (5.5.54) and (5.5.55), as p → ∞, 1 du B−1 − B0 θp (0) −→ η = − rk(E) Ru (x)dvX (x) − u u X ∞ 0 du + rk(E)(B−1 + Γ (1)B0 ) − rk(E) Ru (x)dvX (x) u 1 X = rk(E)ζ (0). (5.5.56) Since R˙ L ∈ End(T (1,0) X) has positive eigenvalues, we find that for Re(z) > 1, = ζ(z)

det

R˙ L

X

Let ζ(z) =

∞

1 k=1 nz

2π

1 ∞ e−u L −z ˙ Tr[(R ) ]dvX (x) uz−1 du. Γ(z) 0 1 − e−u

(5.5.57)

be the Riemann zeta function. Classically, for Re(z) > 1, 1 ζ(z) = Γ(z)

∞

e−u du. 1 − e−u

(5.5.58)

1 ζ (0) = − log(2π). 2

(5.5.59)

0

uz−1

Moreover, 1 ζ(0) = − , 2

260


From (5.5.57), (5.5.58) and (5.5.58), ζ (0) = −

det X

R˙ L 2π

R˙ L Tr log R˙ L dvX (x) ζ(0) + n dvX (x) ζ (0) det 2π X R˙ L R˙ L 1 log det dvX (x). (5.5.60) det = 2 X 2π 2π

By (5.5.40), (5.5.43), (5.5.52), (5.5.56) and (5.5.60), we get (5.5.33).

5.5.5 Asymptotic anomaly formula for the L2 -metric Let now g0T X , g1T X be two metrics on T X compatible with J. We keep the metrics on L, E fixed. ∗ (1,0) ∗ ,0 , | · |K ∗ ,1 be the metrics on K X), the dual of the Let | · |KX X = det(T X TX TX canonical line bundle KX on X, induced by g0 , g1 respectively. Let · p,0 , · p,1 (resp. | · |p,0 , | · |p,1 ) the Quillen (resp. L2 ) metrics on p λ(L ⊗ E) induced by g0T X , g1T X respectively, and with the given metrics on L, E. Theorem 5.5.12. As p → ∞, we have $ $ 2 % % | · |K ∗ ,1 ω n n | · |2p,1 X log p + o(pn ). log = − rk(E) | · |2p,0 | · |2K ∗ ,0 n! X

(5.5.61)

X

Proof. Let θp,0 (0), θp,1 (0) be the real numbers in Theorem 5.5.8 associated to TX TX g0 , g1 respectively, and with the given metrics on L, E. By Theorem 5.5.8 and (1.5.15), as |σ|2K ∗ ,1 = |Θn1 (σ, σ)|/n!, we get

(0) θp,1

−

θp,0 (0)

1 = − rk(E) 2

$

log X

| · |2K ∗ ,1 X

| · |2K ∗ ,0

%

ωn n p + o(pn ). n!

(5.5.62)

X

Now, we choose a path of metrics gvT X for v ∈ [0, 1] connected by g0T X , g1T X . We denote the objects associated to gvT X with a subscript v. Then by Theorems 5.5.6 and 5.5.9, we have ∂ log · 2λ(Lp ⊗E),v = pn Trs Qv A0,v (x) ⊗ IdE dvX,v (x) + O(pn−1/2 ). ∂v X (5.5.63) Now, let w1 , . . . , wn be an orthonormal basis of T (1,0) X for the metric gvT X , w1 , . . . , w n the conjugate basis. Then we get Qv = − ∗−1 v

∂g T X ∂∗v = − v (wi , w j )iwi ∧ wj ∧ . ∂v ∂v

(5.5.64)

5.5. The asymptotic of the analytic torsion 0,•

In fact, let hΛ v (5.5.16) entails ∗−1 v But

and

261

be the metric on Λ• (T ∗(0,1) X) induced by gvT X . Then,

0,• 0,• ∂∗v ∂ ∂(dvX,v ) = (hΛ /(dvX,v ) . )−1 hΛ + v ∂v ∂v v ∂v

(5.5.65)

−1 ∂ T X

Λ0,• −1 ∂ Λ0,• wi , w j wj ∧ iwi , hv = − gvT X gv hv ∂v ∂v

−1 ∂ T X ∂(dvX,v ) wi , w i . /(dvX,v ) = gvT X gv ∂v ∂v TX v ∈ End(T (1,0) X) be defined by Q v wi , w j = − ∂gv (wi , wj ). From Let Q ∂v v

(5.5.64), we get ∂ det(Id −ecQv exp(−2uR˙ vL)) . Trs Qv e2uωd,v = ∂c c=0

(5.5.66)

As in (5.5.45), we obtain Trs Qv e2uωd,v v (Id − exp(2uR˙ vL ))−1 . = Tr Q det(1 − exp(−2uR˙ L ))

(5.5.67)

v

Thus from (5.5.34), (5.5.38), we get Trs [Qv A0,v ]dvX,v = X

1 2

v ] Tr[Q X

1 ωn =− n! 2

Tr |T (1,0) X

∂g T X ω n v

∂v

. n! (5.5.68)

ωn n p + O(pn−1/2 ). n!

(5.5.69)

X

Now from (5.5.63) and (5.5.68), we get $ log

· 2p,1 · 2p,0

%

1 = − rk(E) 2

$

log X

| · |2K ∗ ,1 X

|·

|2K ∗ ,0 X

%

Finally, by the definition of the Quillen metric, we get $ log

| · |2p,1 | · |2p,0

$

% (0) − θp,0 (0) + log = θp,1

From (5.5.62), (5.5.69) and (5.5.70), we get (5.5.61).

· 2p,1 · 2p,0

% .

(5.5.70)

262


5.5.6 Uniform asymptotic of the heat kernel The results in this section work without the positivity assumption on RL . Proof of Theorem 5.5.9. By (1.6.13), for any m ∈ N, there exists Cm > 0 (which depends on ε) such that for 0 < u 1, √ ε2 ). sup |a|m |Gu ( ua)| Cm exp(− 16u a∈R

(5.5.71)

Thus from (5.5.71), as in (1.6.17) and (1.6.18), we get that (1.6.17) still holds for p ∈ N∗ , 0 < u 1, especially, for any m ∈ N, ε > 0, there exists C > 0 such that for any x, x ∈ X, p ∈ N∗ , 0 < u 1, ε2 p u ). (5.5.72) G p ( u/pDp )(x, x ) m C exp(− 32u C But as explained after (1.6.18), F up (

u p Dp )(x, x )

only depends on the restriction

of Dp to B (x, ε), and is zero if d(x, x ) ε. Now, by (5.5.72), we have the analogue of Lemma 1.6.5: For any m ∈ N, there exists C > 0 such that for any p ∈ N∗ , 0 < u 1, u u ε2 p ). (5.5.73) C exp(− exp(− Dp2 )(x0 , x0 ) − exp(− Lp,x0 )(0, 0) m 2p 2p 32u C (X) X

√1 . p

By (1.6.66), we have

t u exp(− Lp,x0 )(0, 0) = pn e−uL2,x0 (0, 0). p

(5.5.74)

We denote Lt2 defined in (1.6.27) by Lt2,x0 with t =

As explained above in Theorem 1.6.11, Lt2,x0 are families of differential operators with coefficients in End(Ex0 ) = End(Λ(T ∗(0,1) X) ⊗ E)x0 , and we view t e−uL2,x0 (Z, Z ) as a smooth section of π ∗ (End(Λ(T ∗(0,1) X) ⊗ E)) on T X ×X T X. Now we need to study the asymptotic of e−uL2,x0 (0, 0) as u → 0, with pa u, G u were rameters x0 ∈ X, t ∈ [0, 1]. Recall that the holomorphic functions F defined by (1.6.26). By (1.6.14), (5.5.71), we get again the analogue of (5.5.72): ε2 ). (5.5.75) Cm exp(− Gu (uLt2,x0 )(0, 0) m 32u C (X×[0,1]) t

Here the C m -norm is for the parameters x0 ∈ X, t ∈ [0, 1]. In fact, for the C 0 norm, it is from the argument in (1.6.17). Using (5.5.71) and proceeding as in u,1,k defined on a (4.2.16), we show there exists a unique holomorphic function H u (uλ). neighborhood of Vc as in (4.2.13) which verifies the same estimates as G u,1,k (λ) → 0 as λ → +∞ and Moreover, H (k−1) (λ)/(k − 1)! = G u (uλ). H u,1,k

(5.5.76)


263

Then as in (1.6.55), we get π ∗ End(E) Gu (uLt2,x0 ) ∇U

1 = 2πi

∗

u,1,k (λ)∇π H U

Γ

End(E)

(λ − Lt2,x0 )−k dλ.

(5.5.77)

As in Theorem 1.6.11, we obtain the estimate for the C m -norm in (5.5.75). Now by the finite propagation speed, Theorem D.2.1 and (1.6.31), for t small, Tx0 X u (uLt )(0, ·) only depends on the restriction of Lt F (0, 2ε) and 2,x0 2,x0 on B u (uLt )(0, ·) ⊂ B Tx0 X (0, 2ε). supp F 2,x0

(5.5.78)

Consider a sphere bundle V = {(z, c) ∈ Tx0 X × R; |z|2 + c2 = 1} on X with fiber Vx0 for x0 ∈ X. We embed naturally B Tx0 X (0, 2ε) into Vx0 by sending z to t3,x (z, c) with c > 0, and extend the operator Lt2,x0 to a generalized Laplacian L 0 on Vx0 with values in π ∗ (End(Λ(T ∗(0,1) X) ⊗ E)). By repeating the argument as in (5.5.75), we obtain t3,x )(0, 0) exp(−uLt2,x0 )(0, 0) − exp(−uL 0

C m (X×[0,1])

C exp(−

ε2 ). (5.5.79) 32u

t3,x )(0, 0) the asymptotic expansion of But now we can apply for exp(−uL 0 the heat kernel stated in (D.1.24), as the total space is compact. By (5.5.73), (5.5.74) and (5.5.79), we get (5.5.35) and Ap,j = A∞,j +O(p−1/2 ). Moreover, (1.6.68) entails A∞,j = Aj ⊗ IdE . This completes the proof of Theorem 5.5.9. We now explain how to use the argument in Section 4.1.4 to get a proof of (5.5.35) and (D.1.24). Let ∆Tx0 X be the usual Bochner Laplacian on Tx0 X. Set (with ρ in (1.6.19)) Lt3,x0 = ρ(|Z|/ε)Lt2,x0 + (1 − ρ(|Z|/ε))∆Tx0 X .

(5.5.80)

Then by (5.5.75), as in (5.5.79), exp(−uLt2,x0 )(0, 0) − exp(−uLt3,x0 )(0, 0)

C m (X×[0,1])

C exp(−

ε2 ). (5.5.81) 32u

Set (with Sv in (1.6.27)) −1 t Lt,v 4,x0 = Sv uL3,x0 Sv

with v =

√

u.

(5.5.82)

Then as in (1.6.66), exp(−uLt3,x0 )(0, 0) = u−n exp(−Lt,v 4,x0 )(0, 0).

(5.5.83)

264


Now we use the usual Sobolev norm m on C ∞ (R2n , Ex0 ) as in (1.6.53). In Section 4.1.4, we replace ∇t,ei , t,m , L2t by ∇ei , m , Lt,v 4,x0 , and the contour ∂ ∂ , t in (4.1.45) by ∂v , v. Then we get the analogue of ∆ ∪ δ by ∂Vc in (4.2.13), ∂t Theorem 4.1.16, t,v C, (5.5.84) exp(−L4,x0 )(0, 0) m C

(X×[0,1]×[0,1])

by using the analogue of (4.1.59) exp(−Lt,v 4,x0 )

(−1)k−1 (k − 1)! = 2πi

−k e−λ (λ − Lt,v dλ. 4,x0 )

∂Vc

(5.5.85)

For k large enough, set Br,x0 ,t =

(−1)k−1 (k − 1)! 2πi r!

∂Vc

e−λ

akr Akr (λ, 0)dλ,

(5.5.86)

(k,r)∈Ik,r

with corresponding Akr (λ, 0) here. Then we get the analogue of (4.1.68): For any m ∈ N, there exists C > 0 such that if v ∈ [0, 1], k t,v Br,x0 ,t v r (0, 0) exp(−L4,x0 ) −

C m (X×[0,1])

r=0

Cv k+1 .

(5.5.87)

The analogue of Theorem 4.1.21 is Br,x0 ,t (Z, Z ) = (−1)r Br,x0 ,t (−Z, −Z ).

(5.5.88)

Tx0 X In fact, let ψ be the operation defined by (ψs)(Z) = s(−Z); as Lt,0 , we 4,x0 = ∆ get −1 −1 (λ − Lt,0 (ψs) (Z) = ψ (λ − Lt,0 s (Z). (5.5.89) 4,x0 ) 4,x0 )

By using the expansion of Lt,v 4,x0 on v as in Theorem 4.1.7 and (5.5.86), we get Br,x0 ,t (ψs) (Z) = (−1)r ψ(Br,x0 ,t s) (Z). (5.5.90) From (5.5.90), we get (5.5.88). From (5.5.81), (5.5.83), (5.5.87) and (5.5.88), we get k n B2r (0, 0)ur u exp(−uLt2,x0 )(0, 0) − r=0

C m (X×[0,1])

Cuk+1 .

From (5.5.73), (5.5.74) and (5.5.91), we get also (5.5.35). If we take t = 1 in (5.5.91), then we get also a proof of (D.1.24).

(5.5.91)


265

Problems Problem 5.1 (Grauert’s proof of the Kodaira embedding theorem [116]). Let X be a compact complex manifold and L be a Grauert positive line bundle on X (cf. Definition B.3.12). Let Z(L∗ ) ⊂ L∗ be the zero-section of L∗ . We identify Z(L∗ ) and X by means of the projection of L∗ . Let x0 ∈ Z(L∗ ). Using this identification we denote by OL∗ ,x0 , OX,x0 the rings of germs of holomorphic functions on L∗ and X, respectively, at x0 . Let e : U −→ L, e∗ : U −→ L∗ be dual holomorphic frames near x0 . Every germ s ∈ OX,x0 (Lp ) has a representative f e⊗p with f holomorphic near x0 . We make the following identifications: (i) L|U U × C, L∗ |U U × C via the frames e, e∗ , (ii) s with the polynomial f (x)z p , with x ∈ U , z ∈ C, ∞ (iii) any germ g ∈ OL,x0 with a Taylor series p=0 gp (x)z p converging in some neighborhood V × {|z| < r}, V ⊂ U . By using the identifications (ii) and (iii), define the maps Φk :

k )

OX,x0 (Lp ) → OL∗ ,x0 ,

(s0 , . . . , sk ) →

p=0

Ψk : OL∗ ,x0 →

k

fp (x)z p ,

p=0 k )

OX,x0 (Lp ) ,

p=0

∞

fp (x)z p → (s0 , . . . , sk ) .

p=0

(a) Show that Ψk ◦ Φk = Id, so Φk is injective. & the corresponding ana(b) Let F be a coherent analytic sheaf on X and F lytic inverse image sheaf on L∗ (cf. Appendix B). Show that Φk induces &|Z(L∗ ) and Φ k : F ⊗OX ⊕kp=0 OX (Lp ) → F ⊗OX OL∗ = F ∗ : morphisms Φ k q k p q ∗ &). H (X, ⊕p=0 F ⊗ L ) → H (Z(L ), F & → F ⊗O ⊕k OX (Lp ) k : F Show further that Ψk induces a morphism Ψ X

p=0

&) → H q (X, ⊕k F ⊗ Lp ). Deduce ∗ : H q (Z(L∗ ), F with induced morphism Ψ p=0 k ∗ = Id, hence Φ ∗ is injective. ∗ ◦ Φ that Ψ k k k

(c) Consider a 1-convex neighborhood M of Z(L∗ ). Let j : Z(L∗ ) → M be the inclusion and π|M : M → Z(L∗ ) the restriction of the projection π. Show that these maps induce a commutative diagram (π|

)∗

/ H q (M, F &) M &) H q (Z(L∗ ), F PPP PPP PPP j∗ PPP Id ( &) H q (Z(L∗ ), F

266


&) ∗ : H q (X, ⊕k F ⊗ Lp ) → H q (M, F Deduce that (π|M )∗ and (π|M )∗ ◦ Φ p=0 k are injective. The finiteness theorem of Andreotti–Grauert [4] shows that dim H q (M, G ) < ∞ for q 1 and any coherent analytic sheaf G (compare & is coherent, deduce Grauert’s vanishing theorem if Theorem 3.5.7). Since F L is Grauert positive, for any coherent analytic sheaf F on X, there exists p0 (F ) such that H q (X, F ⊗ Lp ) = 0 for q 1 and p p0 (F ). (d) Assume now that (L, hL ) is a positive line bundle on X. By Example B.3.10, L is Grauert positive. Conclude the Kodaira embedding theorem. Note: Grauert and Serre vanishing theorems are of course related. In Grauert’s approach, one doesn’t need the assumption that the base manifold is projective, so this yields also the embedding theorem. Once the manifold is known to be projective, Serre’s approach applies. Problem 5.2 (a modern proof of the Kodaira embedding theorem [78]). This consists in using the Nadel vanishing theorem instead of the Kodaira vanishing theorem in order to obtain a particular case of the Grauert vanishing theorem. Let (L, hL ) be a√positive line bundle over a compact complex manifold X of dimension ∗ −1 L ∗ n. Set ω = 2π R . Let hKX be a metric on KX . Fix m ∈ N. ∗ with the singular metric 1. Let (x, y) ∈ X × X. Endow the line bundle Lp ⊗ KX ∗ L p KX −ϕ 1 (h ) ⊗ h e where ϕ ∈ Lloc (X) is smooth on X {x, y} and in some local coordinates ϕ = 2(n + m) log |z − x| (resp. ϕ = 2(n + m) log |z − y|) near x (resp. y), if x = y and ϕ = 2(n + m + 1) log |z − x|, if x = y. Show that there exists p0 such that the curvature of this metric is ω (in the sense of currents) on X, for all (x, y) ∈ X × X. 2. Consider the sheaf I (x, y) introduced in the proof of the Cartan–Serre– Grothendieck lemma 5.1.11. Use Theorem B.4.7 to deduce that there exists p0 such that H 1 (X, Lp ⊗ I (x, y)m+1 ) = 0 and H 0 (X, Lp ) −→ H 0 (X, Lp ⊗ OX /I (x, y)m+1 ) is surjective for all (x, y) ∈ X × X and p p0 . Conclude the Kodaira embedding theorem.

Problem 5.3 (abstract Kodaira embedding theorem). Let L be a holomorphic line bundle over a compact complex manifold X. Show that L is ample if and only if the graded ring ⊕p0 H 0 (X, Lp ) separates any pair of points (x, y) ∈ Diag(X × X) and gives local coordinates at any point x ∈ X (See Definition 2.2.5. Hint: use the strong Noetherian property of analytic sets as in the proof of Lemma 5.1.8.) Generalize this statement to the case of compact complex spaces. Note: in this form the Kodaira embedding theorem extends to 1-concave spaces (Theorem 6.3.21 of Andreotti–Tomassini and Andreotti–Siu [5]). Problem 5.4 (Chow’s theorem). Let X be a closed analytic subset in C Pn . For any analytic set A in C Pn , we denote by IA the sheaf of holomorphic functions vanishing on A. (a) Let x ∈ C Pn X. Consider the exact sequence 0 −→ IX∪{x} ⊗ O(p) −→ IX ⊗ O(p) −→ O(p)x ⊗ OC Pn ,x /mx −→ 0


267

and apply the Grauert vanishing theorem (Problem 5.1) to deduce that for p large, there exists f ∈ H 0 (C Pn , O(p) ⊗ IX ) with f (x) = 0. (b) Show that there exist homogeneous polynomials f1 , . . . , fs ∈ C[z1 , . . . , zn+1 ] such that X = {[z] ∈ C PN : f1 (z) = · · · = fs (z) = 0}, i.e., X is a projective algebraic variety. (Hint: Consider the ideal of all homogeneous polynomials vanishing on X and apply Hilbert’s basis theorem.) Problem 5.5 (another proof of Chow’s theorem). Let A ⊂ C Pn be an irreducible closed analytic subset. We denote by C[z0 , . . . , zn ] be the graded ring of homogeneous polynomials in z0 , . . . , zn . We consider the ideal IA = {P ∈ C[z0 , . . . , zn ] : P (z) = 0 , for all [z] ∈ A} and the projective variety V = {[z] ∈ C Pn : P (z) = 0 , for all P ∈ IA } defined by IA . Clearly, A ⊂ V . (a) Show that IA is prime and V is irreducible. (b) Show that the restriction morphism MV (V ) → MA (A) is injective. (c) Knowing that Theorem 2.2.11 holds for irreducible compact complex spaces, deduce that dim V = a(V ) a(A) dim A. (d) Infer that A = V . Problem 5.6 (compact Riemann surface). Show that any compact Riemann surface X is biholomorphic to a projective algebraic curve. (Hint: any positive (1,1)-form ω is closed and its de Rham cohomology class is integral if X ω = 1.) Problem 5.7. Let ω0 be a K¨ ahler form on a compact complex manifold X such ahler that there is λ ∈ R such that λ[ω0 ] = 2πc1 (X) ∈ H 2 (X, R). Let ω be a K¨ form on X such that [ω] = [ω0 ]. Verify that there exist real functions f, ϕ on X such that √ √ ω = ω0 + −1∂∂f, Ricω0 −λω0 = −1∂∂ϕ. Verify that √ ωn Ricω − Ricω0 = − −1∂∂ log n . ω0 Prove that ω is a K¨ ahler–Einstein metric if and only if there is c ∈ R such that f verifies the Monge–Ampère equation √ (ω0 + −1∂∂f )n = ec+ϕ−λf . ω0n (Hint: Use the ∂∂-Lemma, Problem 1.7 and (1.5.8).) ∗ ∗ Problem 5.8. Refine Theorem 5.1.17 as follows. Let RΦp U∗ , ∗RE be the curvatures of the holomorphic Hermitian connections of (Lp ⊗ E, hΦp U ) and (E, hE ). Prove that as p → ∞, 1 1 Φ∗p U ∗ R − RL = O . p p

268


Suppose now that g T X is the metric associated to ω = analogue to (5.1.23), as p → ∞, 1 1 E 1 Φ∗p U ∗ L R −R − R =O . p p p2 ∗

√

−1 L 2π R ,

then write an

∗

E −1 the metric matrix (Hint: RΦp U = pRL + ∂(h−1 p ∂hp ), with hp = h ◦ Pp (x, x) for a holomorphic frame of E.) Problem 5.9. Verify Remarks 5.4.2, 5.4.5. If X is an oriented orbifold, prove that xi , then use a partition dϕ = 0 for ϕ ∈ C0∞ (X, Λ(T ∗ X)). (Hint: Work first on U X of unity.)

Problem 5.10. Here is another proof of Lemma 5.4.3. Let g T Ux be a Gx -invariant x . For g ∈ Gx , let dg ∈ End(T U x ) be the differential map induced by metric on U the action g. Then for the exponential map expx from x, we have expx (dg(u)) = x. Thus Gx acts linearly on the normal coordinate at x. g · expx (u) for u ∈ T U Problem 5.11. Let X be a compact orbifold. Verify that there are only finitely many possible values of |Gx | for x ∈ X. Let F be a proper orbifold line bundle on X. Show that there exists k ∈ N such that for any x ∈ X, Gx acts identically on Fxk and hence F k is a line bundle on X in the usual sense. Problem 5.12. Let α = (α0 , . . . , αn ) ∈ (N∗ )n+1 . The weighted projective space is P(α) = {(z0 , . . . , zn ) ∈ Cn+1 {0}}/ ∼ where (z0 , . . . , zn ) ∼ (λα0 z0 , . . . , λαn zn ), λ ∈ C∗ . Verify that P(α) is a compact complex orbifold and is naturally identified with P(mα) for any m ∈ N∗ . Prove that the universal bundle O(−1) := {([z], λz) ∈ P(α) × Cn+1 , λ ∈ C} is a proper orbifold line bundle on P(α). Verify that O(1), the dual of the universal bundle O(−1), is a positive line bundle. Problem 5.13. If a finite group G acts holomorphically on a compact complex manifold M , then X = M/G is an orbifold. If the holomorphic orbifold bundles E on M , let Pp (x, y) be L, E on X are induced by the G-equivariant bundles L, p the Bergman kernel on M associated to L ⊗ E. Show that (g, 1)Pp (g −1 x , y). Pp (x, y) = g∈G

Problem 5.14. Consider the Kodaira map Φp from Theorem 5.4.19. Assume that Xsing is finite. Show that Φ∗p (ωF S )(x) = 0 for any x ∈ Xsing . Problem 5.15. With the notation of Theorem 5.4.19, verify that as p → ∞, 1 1 2 l + p 2 e−cpd (x,Xsing ) . (Φ∗p ωF S )(x) − ω(x) l Cl p p C (Hint: Use (5.4.14) with k = l + 1, (5.4.42) and (5.4.44).)


269

5.6 Bibliographic notes Section 5.1. Estimate (5.1.20) is due to Tian [241, Theorem A] following a suggestion of Yau [258]. In [241] the case l = 2 is considered and the left-hand side √ of (5.1.20) is estimated by Cl / p. Ruan [208] proved the C ∞ convergence and improved the bound to Cl /p. Both papers use the peak section method, based on L2 -estimates for ∂ [132], [227]. Finally, Catlin and Zelditch deduced (5.1.20) from (5.1.21) and (5.1.22) [62], [261]. Bouche [48] proved that the induced Fubini–Study ∗ metric (hΦp O(1) )1/p on L converges in the C 0 topology to the initial metric hL . The Kodaira embedding theorem was published in [141] and was generalized by Grauert to compact complex spaces (the proof is outlined in Problem 5.1). Bouche [49] proposed the use of the Bergman kernel and constructed the peak sections with its help. The Noetherian property of analytic sets is an important result in analytic geometry and follows from a more general statement due to Grauert and Serre about ascending chains of coherent analytic sheaves, cf. [107, p. 76], [79, II.3.22]. The GAGA principle of Serre used in the proof of Cartan–Serre–Grothendieck lemma 5.1.11 appeared in [215]. We find a proof that i∗ F is a coherent algebraic ˇ sheaf in [214]. Cf. also [124, III.4.5] about Cech cohomology. We find (5.1.28) in [124, II.5.18]. The topic of Section 5.2 is a very large and active subject; instead of giving complete references, we only refer to a few survey papers or books [92], [111], [242], [243], [259], [23], for the interested readers. Theorem 5.2.8 was established in [262] (cf also [158], [249]), One finds Theorem 5.2.10 in [90], and it was extended by Mabuchi in [166], [168]. An introduction to various notions on stability are [240, 249] and references therein. For recent results on stability, cf. [194, 206, 205, 167, 192, 193]. For more applications of the asymptotic expansion of the Bergman kernel in this program, cf. also [91, 93, 94, 95], [231, 197, 198, 199, 21, 165, 185]. Theorems 5.2.17 and 5.2.18 were established by Wang in [248], [250]. For the further developments, cf. [154, 247, 138]. It follows from [140, Prop. IV.2.4] that the equation (5.2.34) is equivalent to the original Hermitian–Einstein equation up to a conformal change of the metric on E. By the Donaldson–Uhlenbeck–Yau theorem [246], [86], we deduce that if E is irreducible, then the solvability of (5.2.34) is equivalent to E being Mumford stable. In Section 5.3, almost all arguments are from Shiffman–Zelditch [218] where they restricted themselves to the case that g T X is induced by ω. For further developments, cf. [38, 37, 220, 221, 97, 98, 99]. Dinh-Sibony [84] obtain sharp results on the limit distribution of common zeros using the formalism of meromorphic transforms and are therefore able to treat a larger class of measures. Section 5.4. The general definition on orbifold was first given by [210] as V manifold. Most results of Section 5.4.2 and the Kodaira vanishing theorem 5.4.9 (b)

270


first appeared in [12]. Section 5.4.3 and Theorem 5.4.19 appeared in Dai–Liu–Ma [69]. If the singular set Xsing is finite points, and E = C, then Song also studied the diagonal expansion of the Bergman kernel in [230]. Theorem 5.4.20 was established in [13], and Baily also gave some applications in [14]. Theorems 5.4.16 and 5.4.17 are in [60] (cf. [61]). We have also the Atiyah–Singer index theorem for orbifolds [136, 137] (also cf. [159]). Section 5.5. The (holomorphic) analytic torsion of Ray-Singer was first defined in [201] as the complex analogue of its real version for flat vector bundles. The Quillen metric was first defined by Quillen [200] for a Riemann surface, and by Bismut, Gillet, and Soulé [32] in the general case. Theorem 5.5.6 is established in [32, Theorem 1.18] (cf. also [201]). The remainder of this Section is from Bismut– Vasserot [35]. In [36], they extend the result here by replacing the line bundle L by a positive vector bundle. In [232], we find also some arithmetic applications of Theorems 5.5.8, 5.5.12. In [89], Donaldson explains some applications of this section from his moment map picture. For a general reference on the holomorphic analytic torsion and its applications, we refer the readers to the surveys [28, 30, 31, 232, 233] or papers [142, 34, 260] and the references therein.

Chapter 6

Bergman Kernel on Non-compact Manifolds We show in Section 6.1 that the asymptotic expansion of the Bergman kernel still holds on compact sets of certain non-compact complete manifolds. In this way we can obtain another proof of some of the holomorphic Morse inequalities. As a corollary, we re-prove the Shiffman–Ji–Bonavero–Takayama criterion for Moishezon manifolds in Section 6.2. Another application, in Section 6.3, is to the compactification of manifolds with pinched negative curvature. This is possible due to the relationship between the growth of the space of holomorphic sections of the pluricanonical line bundle and the volume of the manifold, expressed in the holomorphic Morse inequalities. We obtain weak Lefschetz theorems in Section 6.4.

6.1 Expansion on non-compact manifolds In this section, we consider generalizations of Theorem 4.1.1 to complete Hermitian manifolds in Section 6.1.1 and to covering manifolds in Section 6.1.2. We use the notation from Section 1.6.1. Especially, (X, J) is a complex manifold with complex structure J and dim X = n. g T X is a Riemannian metric on T X compatible with J, and Θ is the (1, 1)-form associated to g T X . (L, hL ), (E, hE ) are holomorphic Hermitian vector bundles on X and rk(L) = 1.

6.1.1 Complete Hermitian manifolds We assume that (X, g T X ) is a complete Hermitian manifold. We denote by Rdet the ∗ curvature of the holomorphic Hermitian connection ∇det on KX = det(T (1,0) X).

272

Chapter 6. Bergman Kernel on Non-compact Manifolds

The space of holomorphic sections of Lp ⊗ E which are L2 with respect to the 0 (X, Lp ⊗E). Let Pp (x, x ), (x, x ∈ X) be norm given by (1.3.14) is denoted by H(2) the Schwartz kernel of the orthogonal projection Pp , from the space of L2 sections 0 of Lp ⊗ E onto H(2) (X, Lp ⊗ E), with respect to the Riemannian volume form dvX (x ) associated to (X, g T X ). Then by Remark 1.4.3, we know Pp (x, x ) is C ∞ . We recall at first some notation. Ω00,• (X, Lp ⊗ E) is the space of smooth compactly supported (0, •)- forms with values in Lp ⊗ E, and by L20,• (X, Lp ⊗ E) the corresponding L2 -completion with Hermitian product defined in (1.3.14) induced by hL , hE and g T X . As in (3.2.1), we denote the Dolbeault operator ∂ E E,∗ by ∂ p and by ∂ p its adjoint. Theorem 6.1.1. Suppose that there exist ε > 0 , C > 0 such that : √ √ −1RL > εΘ , −1(Rdet + RE ) > −CΘ IdE , |∂Θ|gT X < C,

Lp ⊗E

(6.1.1)

then the kernel Pp (x, x ) has a full off-diagonal asymptotic expansion analogous to Proposition 4.1.5 and Theorem 4.2.1 with Fr (Z, Z ) ∈ End(E)x0 given by (4.1.92) as p → ∞, uniformly for any x, x ∈ K, a compact set of X. Especially there exist coefficients br ∈ C ∞ (X, End(E)) , r ∈ N, such that for any compact set K ⊂ X, any k, l ∈ N, there exists Ck,l,K > 0 such that for p ∈ N∗ , k 1 br (x)p−r l Ck,l,K p−k−1 . n Pp (x, x) − p C (K) r=0

Moreover, b0 = det

R˙ L 2π

(6.1.2)

and b1 has the same form as in (4.1.9).

Let us remark that if L = det(T ∗(1,0) X) is the canonical bundle KX , the first two conditions in (6.1.1) are to be replaced by √ √ (6.1.3) hL is induced by Θ and −1Rdet < −εΘ, −1RE > −CΘ IdE . Moreover, if (X, Θ) is K¨ ahler, the condition on ∂Θ is trivially satisfied. Proof. In general, on a non-compact manifold, we define a self-adjoint extension of p by (3.1.5). By (3.2.4), the quadratic form associated to 1p p is the form Qp given by E

E,∗

Dom(Qp ) := Dom(∂ p ) ∩ Dom(∂ p ), E,∗ E E E,∗ p Qp (s1 , s2 ) = ∂ p s1 , ∂ p s2 + ∂ p s1 , ∂ p s2 , s1 , s2 ∈ Dom(Qp ) . E

E

(6.1.4)

In the previous formulas, ∂ p is the maximal extension of ∂ p to L2 forms and E,∗

∂p

is its Hilbert space adjoint.

6.1. Expansion on non-compact manifolds

273

Under hypothesis (6.1.1), there exists C1 > 0 such that for p large enough Qp (s, s) C1 s2L2 ,

for s ∈ Dom(Qp ) ∩ L20,q (X, Lp ⊗ E), q > 0.

(6.1.5)

We apply the Nakano inequality (1.4.64) to prove (6.1.5). We denote by i(Θ) = (Θ ∧ ·)∗ the interior multiplication with Θ, and T = [i(Θ), ∂Θ] is the Hermitian torsion of Θ. By (1.4.64), for s ∈ Ω00,• (X, Lp ⊗ E), we have p Qp (s, s)

2 (pRL + RE + Rdet )(wl , w k ) wk ∧ iwl s , s 3 1

∗ − T ∗ s2L2 + T s2L2 + T s2L2 , 3

(6.1.6)

where {wk } is an orthonormal frame of T (1,0) X. p Relations (6.1.1) and (6.1.6) imply (6.1.5) for s ∈ Ω0,q 0 (X, L ⊗ E) and q 0,• p 1. By the Andreotti–Vesentini density lemma 3.3.1, Ω0 (X, L ⊗ E) is dense in Dom(Qp ) with respect to the graph norm (due to the completeness of the metric g T X ). Thus (6.1.5) holds in general. E

Next, consider f ∈ Dom(p ) ∩ L20,0 (X, Lp ⊗ E) and set s = ∂ p f . It follows from (6.1.4) and (6.1.5) that E,∗

E,∗ p f 2L2 = ∂ p s , ∂ p s = p Qp (s, s) C1 ps2L2 = C1 p p f , f .

(6.1.7)

(6.1.5) and (6.1.7) imply that for p large enough, Spec(p ) ⊂ {0} ∪ [pC1 , ∞[, 0,0 0,• Ker(p ) = H(2) (X, Lp ⊗ E) = H(2) (X, Lp ⊗ E).

(6.1.8)

Certainly, under the condition (6.1.3), (6.1.5) thus (6.1.8) still holds. As the Kodaira Laplacian p = 12 Dp2 acting on sections of Lp ⊗ E has a spectral gap (6.1.5), by the argument in Section 4.1.2, we can localize the problem, and we get directly our theorem from Theorem 4.2.9, as in the proof of Theorem 4.2.1. As for b1 , the argument leading to (4.1.112)–(4.1.114) in the proof of Theorem 4.1.3 still holds locally, thus we get b1 from (4.1.9). For simplicity we consider now rk(E) = 1, with the important case E = KX 0 in mind. Choose an orthonormal basis (Sip )i1 of H(2) (X, Lp ⊗ E). For each local holomorphic frame eL and eE of L and E, we have Sip = fip e⊗p L ⊗ eE for some local holomorphic functions fip . Then p p 2 2 Pp (x, x) = |Si (x)|2 = |fi (x)|2 |e⊗p L |hLp |eE |hE i1

i1

(6.1.9)

(6.1.10)

274


is a smooth function. Observe that the quantity defined, but the current ωp =

i1

|fip (x)|2 is not globally

√ −1 ∂∂ log |fip (x)|2 2π

(6.1.11)

i1

is globally well defined on X. Indeed, by (1.5.8) and (6.1.10), we have √ √ √ 1 −1 L −1 −1 E ωp − R =− R . ∂∂ log Pp (x, x) + p 2π 2πp 2πp

(6.1.12)

0 (X, Lp ) < ∞, we have by (5.1.17) that If E is trivial of rank 1 and dim H(2) ∗ ωp = Φp (ωF S ) where Φp is defined as in (2.2.10) with H 0 (X, Lp ) replaced by 0 H(2) (X, Lp ).

Corollary 6.1.2. Assume that rk(E) = 1 and (6.1.1) or (6.1.3) holds true. Then : (a) For any compact set K ⊂ X, the restriction ωp |K is a smooth (1, 1)-form for sufficiently large p; moreover, for any l ∈ N, there exists Cl,K > 0 such that √ 1 −1 L Cl,K R l ; ωp − p 2π p C (K)

(6.1.13)

(b) The Morse inequalities hold in bidegree (0, 0) : lim inf p p−→∞

−n

0 dim H(2) (X, Lp

1 ⊗ E) n!

√ −1 L n R . 2π X

(6.1.14)

Proof. Due to (6.1.2), Pp (x, x) doesn’t vanish on any given compact set K for p sufficiently large. Thus, (a) is a consequence of (6.1.2) and (6.1.12). Part (b) follows from Fatou’s lemma, applied on X with the measure Θn /n! to

√−1 L n n −n /Θ on X. the sequence p Pp (x, x) which converges pointwise to 2π R Corollary 6.1.3. Assume that (6.1.1) holds true with E trivial and suppose that X is Andreotti pseudoconcave. Then the graded ring A (X, L) = ⊕p0 H 0 (X, Lp ) separates points and gives local coordinates on X. For any compact set K ⊂ X, the restriction of the Kodaira map Φp |K is an embedding for p large enough. Proof. Theorem 3.4.5 entails that dim H 0 (X, Lp ) < ∞, for any p 0, so we can consider the Kodaira map Φp : X Bl|H 0 (X,Lp )| → P(H 0 (X, Lp )∗ ). Fix a compact set K ⊂ X. Since Pp (x, x) doesn’t vanish on K for p sufficiently large, Φp is well defined on K. Using (6.1.13) as in Lemma 5.1.6, we see that Φp |K is an immersion. Finally, we can define the notion of peak section (Definition 5.1.7) and show that Φp |K is injective, following Lemma 5.1.8.

6.1. Expansion on non-compact manifolds

275

6.1.2 Covering manifolds Another generalization is a version of Theorem 4.1.1 for covering manifolds. Let be a paracompact complex manifold, such that there is a discrete group Γ X with a compact quotient X = X/Γ. acting holomorphically and freely on X Let T X −→ X be the projection. Let J, g , Θ, ω L, E be the pull-back of the , J, πΓ : X → X. We suppose corresponding objects in Section 1.5.1 by the projection πΓ : X that the positivity condition (1.5.21) holds for RL . p = L p ⊗E which is an essentially Let us consider the Kodaira Laplacian self-adjoint operator. It follows by the same argument as in Theorems 1.5.5 and 1.5.8 that with µ0 > 0 introduced in (1.5.26), p ) ⊂ {0}∪ ]pµ0 − 1 CL , +∞[ . Spec( 2

(6.1.15)

as in Section 4.1.1. Finally, we define the Bergman kernel Pp (x, x ) on X Theorem 6.1.4. We fix 0 < ε0 < injX := inf x∈X {injectivity radius at x}. For any p ∈ N∗ , k, l ∈ N, there exists C > 0 such that for x, x ∈ X, Pp (x, x ) − Pp (πΓ (x), πΓ (x )) l C p−k−1 , if d(x, x ) < ε0 , C (6.1.16) −k−1 P (x, x ) C p , if d(x, x ) ε0 . p l C

Especially, Pp (x, x) has the same asymptotic expansion as Pp (πΓ (x), πΓ (x)) in Theorem 4.1.1 uniformly on X. Proof. Let {ϕi } be a partition of unity subordinate to {Uxi = B X (xi , ε)} as in γ,xi } Section 1.6.2. Then {ϕ γ,i = ϕi ◦ πΓ } is a partition of unity subordinate to {U −1 where πΓ (Uxi ) = ∪γ∈Γ Uγ,xi and Uγ1 ,xi and Uγ2 ,xi are disjoint for γ1 = γ2 . The γ,xi }, since we can apply proof of Proposition 4.1.5 still holds for the pair {ϕ γ,i }, {U γ,xi . the Sobolev embedding theorems A.1.6 and A.3.1 with uniform constant on U Thus, the analogue of (4.1.12) holds uniformly on X. Using the finite propagation speed as at the end of Section 4.1.2, we conclude. Remark 6.1.5. Theorem 6.1.4 works well for coverings of non-compact manifolds. Let (X, J, Θ) be a complete Hermitian manifold, (L, hL ) be a holomorphic line → X be a Galois covering of X = X/Γ. If (X, Θ) bundle on X and let πΓ : X L hL ) and (L, h ) satisfy one of the conditions (6.1.1) or (6.1.3), (X, Θ) and (L, have the same properties. We obtain therefore as in (6.1.14) (by integrating over a fundamental domain): √ −1 L n 1 −n 0 p R lim inf p dimΓ H(2) (X, L ) , (6.1.17) p−→∞ n! X 2π 0 (X, Lp ). where dimΓ is the von Neumann dimension of the Γ-module H(2)

276


6.2 The Shiffman–Ji–Bonavero–Takayama criterion revisited Let (L, hL ) be a singular Hermitian line bundle over a compact complex manifold √ X. We assume that the curvature current −1RL is strictly positive and smooth on a non-empty Zariski open set. The main result of this section is the asymptotic expansion of the Bergman kernel of Lp on a Zariski open set endowed with the generalized Poincaré metric, which is the object of Theorem 6.2.3. One motivation is to give a new proof of Shiffman’s conjecture studied in Section 2.3. Firstly, we will prove Theorem 2.3.7 without using the Demailly approximation Theorem 2.3.10. Secondly, if we apply the approximation Theorem 2.3.10, we get a new proof of Theorem 2.3.8. We turn now to the proof of Theorem 2.3.7. In the course of the proof, we describe the Bergman kernel on the regular locus of the positive K¨ ahler current. Let X be a compact connected complex manifold of dimension n. Let Σ be a closed analytic subset of X. −→ X be a resolution of singularities (cf. Theorem 2.1.13) such Let π : X that π : X π −1 (Σ) −→ X Σ is biholomorphic and π −1 (Σ) is a divisor with normal crossings. More precisely, there exists a finite sequence of blow-ups τm−1 τ2 τm τ1 = Xm −→ X Xm−1 −→ · · · −→ X1 −→ X0 = X

(6.2.1)

such that (a) τi is the blow-up along a non-singular center Yi−1 contained in the strict transform of Σ in Xi−1 , i 1, = Xm through π = τ1 ◦ τ2 ◦ · · · ◦ τm is smooth (b) the strict transform of Σ in X and π −1 (Σ) is a divisor with normal crossings. and Θ (·, ·) = Let g0T X be an arbitrary smooth J-invariant metric on X TX g0 (J·, ·) the corresponding (1, 1)-form. The generalized Poincaré metric on X π −1 (Σ) is defined by the Hermitian form Σ=X √

Θε0 = Θ + ε0 −1 i ∂∂ log (− log(σi 2i ))2 , 0 < ε0 # 1 fixed, (6.2.2)

where π −1 (Σ) = ∪i Σi is the decomposition into irreducible components Σi of π −1 (Σ) and each Σi is non-singular; σi are holomorphic sections of the associated holomorphic line bundle OX (Σi ) which vanish to first order on Σi , and σi i is the norm for a smooth Hermitian metric · i on OX (Σi ) such that σi i < 1. Let ROX& (Σi ) be the curvature of (OX (Σi ), · i ). Lemma 6.2.1. (i) The generalized Poincaré metric (6.2.2) is a complete Hermitian metric of finite volume. Its Hermitian torsion Tε0 = [i(Θε0 ), ∂Θε0 ] and the curvature ∗ Rdet = RKX is also bounded.

6.2. The Shiffman–Ji–Bonavero–Takayama criterion revisited

277

(ii) If (E, hE ) is a holomorphic Hermitian vector bundle over X, set . / E 0 H(2) (X Σ, E) = u ∈ L20,0 (X Σ, E , Θε0 , hE ) : ∂ u = 0 ,

(6.2.3)

then 0 (X Σ, E) = H 0 (X, E). H(2)

(6.2.4)

Proof. To describe the metric more precisely, we denote by D the unit disc in C and by D∗ = D {0}. On the product (D∗ )l × Dn−l , we introduce the metric ΩP =

√ √ l n −1 dzk ∧ dz k −1 + dzk ∧ dz k . 2 |zk |2 (log |zk |2 )2 2 k=1

(6.2.5)

k=l+1

For any point x ∈ π −1 (Σ), there exists a coordinate neighborhood U of x isomor π −1 (Σ)) ∩ U = {z = (z1 , . . . , zn ) : z1 = 0, . . . , zl = 0}. phic to Dn in which (X π −1 (Σ)) ∩ U ∼ Such coordinates are called special. We endow (X = (D∗ )l × Dn−l with the metric (6.2.5). We have √ −1 ∂∂ log (− log(σi 2i ))2 √ ROX& (Σi ) ∂ log(σi 2i ) ∧ ∂ log(σi 2i ) . + = 2 −1 log(σi 2i ) (log(σi 2i ))2

(6.2.6)

Since the terms ROX& (Σi ) / log(σi 2i ) tend to zero, as we approach Σ, for ε0 small enough (cf. Definition B.2.8) ROX& (Σi ) √ > 0. Θ + 2 −1ε0 log(σi 2i ) i

(6.2.7)

√ The last term in (6.2.6) is 0, as −1∂g ∧ ∂g 0 for any real function g on X. Thus Θε0 is positive for ε0 small enough. We choose special coordinates in a neighborhood U of x0 in which Σj has the equation zj = 0 for j = 1, . . . , k and Σj , j > k, do not meet U . Then for 1 i k, σi 2i = ϕi |zi |2 for some positive smooth function ϕi on U and ∂ log(σi 2i ) ∧ ∂ log(σi 2i ) dzi ∧ dz i + ψi = (log(σi 2i ))2 |zi |2 (log(σi 2i ))2

(6.2.8)

where ψi is a smooth (1, 1)-form on U such that ψi |zi =0 = 0. Using (6.2.6) and (6.2.8), we see that the metrics (6.2.2) and (6.2.5) are equivalent for |zi | small. Therefore we have to check the first assertion of (i) for the Poincaré metric on {z ∈ C : 0 < |z| < c}, √ −1 dz ∧ dz on {z ∈ C : 0 < |z| < c < 1}. (6.2.9) ωP = 2 |z|2 (log |z|2 )2

278


The completeness follows from the fact that the length of the path {tz : t ∈]0, c]} (|z| = 1) is infinite: c c 1 dr = − log log = ∞ . r 0 0 r| log r| Moreover, the volume calculated in polar coordinates equals 2π c r dr dϑ −1 c = −2π(log r) < ∞. 2 2 0 0 0 r (log r) From this the first assertion of (i) follows. Recall that Rdet is the curvature of the holomorphic Hermitian connection on det(T (1,0) X) with respect to the Hermitian metric induced by Θε0 . We wish to show that there exists a constant C > 0 such that √ (6.2.10) −CΘε0 < −1Rdet < CΘε0 , |Tε0 |Θε0 < C where |Tε0 |Θε0 is its norm with respect to Θε0 . Since ∂Θε0 = ∂Θ by (6.2.2), ∂Θε0 and thus we get the second relation of (6.2.10). extends smoothly over X, We turn now to the first condition of (6.2.10). By (6.2.2), (6.2.6) and (6.2.8), we know that Θnε0 = #k

2k εk0 + β(z)

i=1

|zi |2 (log σi 2i )2

n n √ √ ( −1dzj ∧ dz j ) =: γ(z) ( −1dzj ∧ dz j ). j=1

j=1

(6.2.11) Here β(z) is a polynomial in the functions aiα (z)|zi |2 (log σi 2i )2 ,

biα (z)|zi |2 log σi 2i

and ciα (z),

(1 i k),

with aiα , biα smooth functions on U and ciα smooth functions on U such that ciα (z)|zi =0 = 0. Moreover, 2k εk0 + β(z) is positive on U as Θε0 is positive. Since | ∂z∂ 1 ∧ · · · ∧

∂ 2 ∂zn |Θε0

n √ ( −1dzj ∧ dz j ) = Θnε0 ,

(6.2.12)

j=1

we get from (6.2.11) and (6.2.12), k Rdet = ∂∂ log γ(z) = ∂∂ log 2k εk0 + β(z) + ∂∂ log (log σi 2i )2 . (6.2.13) i=1

By (6.2.6), the last term of (6.2.13) is bounded with respect to Θε0 . To examine the first term of the sum, we write ∂∂ log(2k εk0 + β(z)) =

∂∂β(z) ∂β(z) ∧ ∂β(z) . − 2k εk0 + β(z) (2k εk0 + β(z))2

(6.2.14)

6.2. The Shiffman–Ji–Bonavero–Takayama criterion revisited

279

Now we observe that for Wi (z) = |zi |2 (log σi 2i )2 or |zi |2 log σi 2i , the terms ∂∂Wi (z), ∂Wi (z), ∂Wi (z) are bounded with respect to the Poincaré metric (6.2.5), thus with respect to Θε0 . Combining with the form of β given after (6.2.11), this achieves the proof of (6.2.10). Let us prove (ii). First observe that Θε0 dominates the Euclidean metric in special coordinates near π −1 (Σ), being equivalent with (6.2.5). Therefore it dom We deduce inates some positive multiple of any smooth Hermitian metric on X. that, given a smooth Hermitian form Θ on X, there exists C > 0 such that 0 (X Σ, E) are L2 inteΘε0 CΘ on X Σ. It follows that elements of H(2) E grable with respect to the smooth metrics Θ and h over X, which entails that they extend holomorphically to sections of H 0 (X, E) by Lemma 2.3.22. We have 0 therefore H(2) (X Σ, E) ⊂ H 0 (X, E). The reverse of inclusion follows from the finiteness of the volume of X Σ in the Poincaré metric. Let L be a holomorphic line bundle on X. Suppose that hL is a singular Hermitian metric on curvature current RL , smooth outside the proper √ L with L analytic set Σ and −1R is a strictly positive current on X. hL ) on X which Lemma 6.2.2. There exists a singular Hermitian line bundle (L, ∗ k0 ∼ ), for some k0 ∈ N. is strictly positive and L|Xπ −1 (Σ) = π (L Proof. Let Y0 = τ1−1 (Y0 ), with Y0 = Σ in (6.2.1). By Proposition 2.1.11 (a), there exists a smooth Hermitian metric h0 on the line bundle OX1 (−Y0 ) whose curvature R(OX1 (−Y0 ), h0 ) is strictly positive along Y0 , and bounded on X1 , moreover, it vanishes outside a neighborhood of Y0 . On X1 , we consider the bundle L1 := τ1∗ (Lk1 ) ⊗ OX1 (−Y0 ) endowed with the metric hL1 = (hL )⊗k1 ⊗ h0 , for k1 ∈ N. The curvature current of (L1 , hL1 ) is √

√ √ −1RL1 = k1 τ1∗ ( −1RL ) + −1R(OX1 (−Y0 ), h0 ) .

(6.2.15)

√ The current τ1∗ ( −1RL ) is positive on X1 and strictly positive on any compact √ set disjoint from Y0 . Hence, for k1 sufficiently large, −1RL1 is a strictly positive current on X1 (cf. Proposition 2.1.8). hL ) on Continuing inductively, we construct a holomorphic line bundle (L, √ with curvature current −1RL , smooth on X τ −1 (Σ) and strictly positive X on X.

We introduce on L|XΣ the metric (hL )1/k0 whose curvature extends to a Set strictly positive (1, 1)-current on X. 1/k0 L hL (− log(σi 2i ))ε , 0 < ε # 1 , (6.2.16a) ε := (h ) i

0 H(2) (X

. / Lp Σ, L ) := u ∈ L20,0 (X Σ, Lp , Θε0 , hL u=0 . ε) : ∂ p

(6.2.16b)

280


0 The space H(2) (X Σ, Lp ) is the space of L2 -holomorphic sections relative to the

L 1/k0 metrics Θε0 on X Σ and hL is bounded away from ε on L|XΣ . Since (h ) zero (having plurisubharmonic weights), as its curvature extends a strictly positive the elements of this space are L2 integrable with respect to (1, 1)-current on X, the Poincaré metric and a smooth metric hL ∗ of L over all of X. By the proof of Lemma 6.2.1 (ii), we have 0 H(2) (X Σ, Lp ) ⊂ H 0 (X, Lp ).

(6.2.17)

(Here we cannot infer the other inclusion since hL ε might be infinity on Σ.) The 0 (X Σ, Lp ) is our space of polarized sections of Lp . space H(2) Theorem 6.2.3. Let X be a compact complex manifold with an integral K¨ ahler current ω. Let (L, hL ) be a singular polarization of [ω] with strictly positive curvature current having singular support contained in a proper analytic set Σ . Then the Bergman kernel of the space of polarized sections (6.2.16b) has the asymptotic expansion as in Theorem 6.1.1 for X Σ. Proof. We will apply Theorem 6.1.1 to the non-K¨ ahler Hermitian manifold (X Σ, Θε0 ) equipped with the Hermitian bundle (L|XΣ , hL ε ). Thus we have to show that there exist constants η > 0, C > 0 such that √ √ L −1R(L|XΣ , hε ) > ηΘε0 , −1Rdet > −CΘε0 , |Tε0 | < C .

(6.2.18)

The first one results for all ε small enough from (6.2.2), (6.2.16a) and the fact that the curvature of (hL )1/k0 extends to a strictly positive (1, 1)-current on X (dominating a small positive multiple of Θ on X). The second and third relations were proved in (6.2.10). This achieves the proof of Theorem 6.2.3. Proof of Theorem 2.3.7. Any Moishezon manifold possesses a strictly positive singular polarization (L, hL ) by Lemma 2.3.5 and 2.3.6. Conversely, suppose X has such a polarization. Then by Theorem 6.2.3, as in 0 (X Σ, Lp ) Cpn for some C > 0 and p large enough. (6.1.14), we have dim H(2) By (6.2.17), it follows that L is big and X is Moishezon. Corollary 6.2.4. Let L be a holomorphic line bundle over a compact connected complex manifold X. L is big if and only if L possesses a singular Hermitian √ metric hL with strictly positive curvature current −1RL and smooth outside a proper analytic set. Proof. By Theorem 2.3.30, any big line bundle L on a compact manifold carries a singular Hermitian metric having strictly positive curvature current with singularities along a proper analytic set. Conversely, by the above proof of Theorem 2.3.7, we know L is big.

6.3. Compactification of manifolds

281

Remark 6.2.5. The results of this section hold also for reduced compact complex spaces X possessing a holomorphic line bundle L with singular Hermitian metric hL having positive curvature current. This is just a matter of desingularizing X. 0 As a space of polarized sections, we obtain H(2) (X Σ, Lp ) where Σ is an analytic set containing the singular set of X.

6.3 Compactification of manifolds In this section, we apply the ideas developed so far about the spectral gap and the Morse inequalities to the compactification problem. We wish to find sufficient conditions under which a given non-compact complex manifold can be compactified – i.e., exhibited as a Zariski-open subset of a compact strongly pseudoconvex domain in a projective manifold. By considering such a problem, one hopes to reduce the study of certain non-compact complex manifolds to that of better understood manifolds. This section is organized as follows. In Section 6.3.1, we review the known results about the compactification of strongly pseudoconcave ends. In Section 6.3.2, we prove Theorem 6.3.24 about the compactification of a complete K¨ ahler manifold with pinched negative curvature, with a strongly pseudoconvex end and finite volume away from this end.

6.3.1 Filling strongly pseudoconcave ends We give an overview here of the basic results about filling strongly pseudoconvex ends of complex manifolds. Definition 6.3.1. For a compact subset K of a complex manifold X, an unbounded connected component of X K is called an end of X (with respect to K). If K1 ⊂ K2 are two compact subsets, the number of ends with respect to K1 is at most the number of ends with respect to K2 , so that we can define the number of ends of X. Namely, X is said to have finitely many ends if for some integer k, and for any K ⊂ X, the number of ends with respect to K is at most k. The smallest such k is called the number of ends of X, and then there exists K0 ⊂ X such that the number of ends with respect to K0 is precisely the number of ends of X. If no such k exists, we say that X has infinitely many ends. Definition 6.3.2. A manifold X with dim X 2 is said to be a strongly pseudoconcave end if there exist u ∈ R ∪ {−∞}, v ∈ R ∪ {+∞} and a proper, smooth function ϕ : X −→]u, v[ , which is strictly plurisubharmonic on {ϕ < c0 }, for some u < c0 v. If u = −∞, X is called a hyperconcave end. For u < c < v, we set Xc = {ϕ < c}. We call ϕ an exhaustion function. We say that a strongly pseudo" such concave end can be compactified or filled in if there exists a complex space X " " that X is (biholomorphic to) an open set in X and for any c < v, (X X)∪{ϕ c}

282


" (somewhat abusively) the compactification of X, is a compact set. We will call X although it is not necessarily compact. Example 6.3.3. Let X be a Stein manifold and A ⊂ X be a compact set. Assume that for any x ∈ A there exists a neighborhood Ux and a strictly plurisubharmonic function fx : Ux → [−∞, ∞[ such that A ∩ Ux = fx−1 (−∞). By a result of Colt¸oiu, there exists a neighborhood A ⊂ U and a strictly plurisubharmonic function f : U → [−∞, ∞[ such that A = f −1 (−∞). Thus X A is a hyperconcave end. Theorem 6.3.4 (Rossi–Andreotti–Siu). (a) Two normal Stein compactifications are biholomorphic by a map which is the identity on X. (b) Any strongly pseudoconcave end X can be compactified provided dim X 3. If the exhaustion function is strictly plurisubharmonic on X, the compact" can be taken to be a normal Stein space with at worst isolated ification X singularities. If dim X = 2, then the previous theorem breaks down. Example 6.3.5 (Grauert–Andreotti–Siu–Rossi). This example provides complex structures on CP2 {[1, 0, 0]}, which are not fillable. Let Qε be the family of quadrics in CP3 given in the homogeneous coordinates [w0 , w1 , w2 , w3 ] by the equation w3 (w3 + εw0 ) = w1 w2 . For ε = 0, they are nonsingular. There exists an application Φ : CP2 {[1, 0, 0]} −→ Qε A, where A is a real analytic sphere, such that Φ is a two-sheeted differentiable ramified covering. We can use Φ to pull back the complex structure of Qε on CP2 {[1, 0, 0]}, so that Φ becomes holomorphic. Then CP2 {[1, 0, 0]} with the new structure cannot be compactified. A large class of strongly pseudoconcave ends which can be compactified even in dimension 2 are the hyperconcave ends. Example 6.3.6. Let (X, g T X ) be a complete Kähler manifold. For any linearly independent U, V ∈ Tx X, we define the sectional curvature of g T X on the plane generated by U and V by K(U, V ) = R(U, V, U, V )/|U ∧ V |2 .

(6.3.1)

Let us assume that g T X has pinched negative curvature, that is, there are C1 , C2 > 0 such that for any x ∈ X, −C12 K(U, V ) −C22 < 0 ,

for any U, V ∈ Tx X, U ∧ V = 0.

(6.3.2)

A geodesic γ : [0, ∞[−→ X is called a ray if d(γ(t1 ), γ(t2 )) = |t1 − t2 | for all t1 , t2 ∈ [0, ∞[, where d(·, ·) is the distance induced by g T X . If X is non-compact, a ray emanating from any given point always exists. The Busemann function of the ray γ is defined by rγ : X → R ,

rγ (x) = lim (d(x, γ(t)) − t). t→∞

Let N be an end of X. If N has finite volume, then N is called a cusp.

(6.3.3)


283

We give now two results about the structure of complete Kähler manifolds with negative curvature. Theorem 6.3.7 (Wu). Any simply connected complete K¨ ahler manifold of nonpositive sectional curvature is Stein. Theorem 6.3.8 (Siu–Yau). Let (X, g T X ) be a complete K¨ ahler manifold of finite volume and pinched negative curvature. Then X is a hyperconcave manifold. In particular, any cusp N is a hyperconcave end. The first step in the proof is to show that, if γ is the lift of γ to the universal cover of X, then rγ is strictly plurisubharmonic. Then, using results of Margulis and Gromov, Siu and Yau show that, for c sufficiently large, the restriction of rγ to rγ < −c descends to X. Moreover, the minimum of a finite number of such functions forms an exhaustion function on X. The following result shows that hyperconcave ends can be always compactified. Theorem 6.3.9. (a) Any hyperconcave end X can be compactified, i.e., there exists a complex " such that X is (biholomorphic to) an open set in X " and for any space X " X) ∪ {ϕ c} is a compact set. More specifically, if ϕ is strictly c < v, (X " can be chosen a normal Stein space plurisubharmonic on the whole of X, X with at worst isolated singularities. (b) Assume that X can be covered by Zariski-open sets whose universal coverings " X is the union of a finite set D and an excepare Stein manifolds. Then X tional analytic set which can be blown down to a finite set D. Each connected component of Xc , for sufficiently small c, can be analytically compactified by one point from D ∪ D. If X itself has a Stein cover, D = ∅ and D consists " of the singular set of the Remmert reduction of X. A strategy to fill in a strongly pseudoconcave end is to try to fill in the CR manifold {ϕ = constant}. Let us first review the notion of CR manifold of hypersurface type. Definition 6.3.10. Let Y be a smooth orientable manifold of real dimension (2n − 1). A Cauchy-Riemann (CR ) structure on Y is an (n − 1)-dimensional complex subbundle T (1,0) Y of the complexified tangent bundle T Y ⊗R C such that T (1,0) Y ∩ T (1,0)Y = {0}, and such that T (1,0) Y is integrable as a complex subbundle of T Y ⊗R C (i.e., if U and X are sections of T (1,0) Y , the Lie bracket [U, X] is still a section of T (1,0) Y ). Let X be a smooth domain in a complex manifold W . The complex structure of W induces the CR structure T (1,0) Y = T (1,0) X ∩ T Y ⊗R C on Y = ∂X. If Y is a CR manifold, then its Levi distribution H is the real subbundle of T Y defined by H = Re{T√(1,0) Y ⊕ T (1,0)Y }. There exists on H a complex structure J given by J(U + U) = −1(U − U), with U ∈ T (1,0) Y . As Y is orientable, the

284


real line bundle H ⊥ ⊂ T ∗ Y , the annihilator of H, admits a global non-vanishing section θ. Definition 6.3.11. The CR structure is said to be strongly pseudoconvex if dθ(·, J·) defines a positive definite metric on H. The tangential Cauchy-Riemann operator ∂ b : C ∞ (Y ) → C ∞ (Y, T ∗(1,0) Y ), associates to a function f ∈ C 1 (Y ) the projection on T ∗(1,0) Y of the exterior differential df . A function f ∈ C 1 (Y ) is called a CR function if ∂ b f = 0. By a CR embedding of a manifold in a complex manifold, we mean an embedding whose components are CR functions. When we say that a CR manifold is a submanifold of a complex manifold, we understand that the inclusion is a CR embedding, that is, the CR structure is induced from the ambient manifold. If Y = ∂X, where X is a domain in a complex manifold W , then all restrictions of holomorphic functions on W to Y are CR functions. We have also the following converse which may be also seen as a form of Hartogs phenomenon for CR functions. Theorem 6.3.12 (Kohn–Rossi). Assume that X is a smooth, relatively compact connected domain in a complex manifold such that the Levi form of a defining function of X restricted to the holomorphic tangent space at ∂X has at least one positive eigenvalue everywhere. Then any CR function defined on ∂X extends to a holomorphic function in X. In particular, ∂X is connected. We also need the abstract notion of complex manifold with strongly pseudoconvex boundary. A priori, it is not a domain with boundary in a larger complex manifold. Definition 6.3.13. A complex manifold X with strongly pseudoconvex boundary is a real manifold with boundary, of real dimension 2n, satisfying the following conditions: (i) the interior Int(X) = X ∂X has an integrable complex structure and (ii) for each point x ∈ ∂X, there exist a neighborhood U in X, a strongly pseudoconvex domain M ⊂ Cn with smooth boundary, and a diffeomorphism σ from U onto a relatively open subset σ(U ) such that σ(∂U ) ⊂ ∂M and σ is biholomorphic from Int(U ) to Int(σ(U )). From this definition, we infer: Proposition 6.3.14. There exists an induced integrable Cauchy-Riemann structure on the boundary ∂X, given locally by transporting the CR structure on ∂M . Moreover, if ∂X is compact, there exists a defining function ϕ : X →] − ∞, c] such that ∂X = {ϕ = c}, with the properties: (1) its Levi form is positive definite on the holomorphic tangent space of ∂X and (2) there exists c0 < c such that ϕ is strictly plurisubharmonic on {c0 < ϕ < c}. Theorem 6.3.15 (Heunemann–Ohsawa). Let X be a compact complex manifold with strongly pseudoconvex boundary. Then X can be realized as a domain with boundary in a larger complex manifold W : there exists a strongly pseudoconvex


285

domain M ⊂ W and a diffeomorphism σ : X −→ M which is a biholomorphism between Int(X) and σ(Int(X)), σ(∂X) = ∂M . We give now the solution of the complex Plateau problem for strongly pseudoconvex CR submanifold of Cm . Theorem 6.3.16 (Harvey–Lawson). Let Y be a compact strongly pseudoconvex CR submanifold of Cm . Then there exists a normal Stein space S ⊂ Cm with boundary and at most isolated singularities, such that ∂S = Y . In this case, we say that Y bounds the Stein space S. In order to apply the Harvey– Lawson theorem, we need conditions for a strongly pseudoconvex CR manifold to be embeddable in the Euclidean space. Theorem 6.3.17 (Boutet de Monvel). Any compact CR manifold Y with dimR Y > 3 admits a CR embedding in the Euclidean space. The proof is based on the Hodge decomposition for the Kohn Laplacian ∗

∗

b = ∂ b ∂ b + ∂ b ∂ b ,

(6.3.4)

which is not an elliptic operator but has however as a parametrix a pseudodifferential operator of type 1/2. If dimR Y = 3, Boutet de Monvel’s theorem breaks down. A counterexample is given by the boundary of the strongly pseudoconcave manifold constructed in Example 6.3.5 of Grauert–Andreotti–Rossi. We are led to the following beautiful result which follows from the works of Boutet de Monvel–Sj¨ ostrand, Harvey–Lawson, Burns and Kohn. Theorem 6.3.18. Let Y be a compact complex CR manifold, dimR Y 3. The following conditions are equivalent: (a) Y is embeddable in the Euclidean space, (b) Y bounds a strongly pseudoconvex complex manifold, (c) The tangential Cauchy-Riemann operator ∂ b on functions of Y has closed range in L2 . In this context, we have even more: Theorem 6.3.19 (Lempert). Suppose a compact, strongly pseudoconvex CR manifold Y , dimR Y 3, bounds a strongly pseudoconvex Stein space (or, equivalently, a strongly pseudoconvex complex manifold ). Then Y can be realized as a smooth real hypersurface in a complex projective manifold that Y divides into a strongly pseudoconvex and a strongly pseudoconcave part. The main ingredient is the following Nash-type approximation result, which will be also useful later. Theorem 6.3.20 (Lempert). Assume a reduced Stein space X has only isolated singularities, and K ⊂ X is a compact subset. Then there are an affine algebraic variety V , and a neighborhood of K in X that is biholomorphic to an open set in V .

286


It is thus interesting to have the following version of the Kodaira embedding theorem for 1-concave manifolds. Theorem 6.3.21 (Andreotti–Tomassini). Let X be a 1-concave manifold and let L be a holomorphic line bundle on X such that the ring A (X, L) = ⊕p0 H 0 (X, Lp ) gives local coordinates and separates points everywhere on X. Then X is biholomorphic to an open set of a projective algebraic variety. We finish with a result due to Epstein–Henkin. Definition 6.3.22. A compact CR-hypersurface Y0 is called strictly CR-cobordant with at most to a compact CR-hypersurface Y1 if there exists a complex space X isolated singularities and a C ∞ -strictly plurisubharmonic function ρ with at most such that the set X = {x ∈ X : 0 < ρ(x) < 1} is a isolated critical points on X and ∂X = Y1 ∪ Y0 . relatively compact, complex subspace in X Theorem 6.3.23 (Epstein–Henkin). Let Y1 be an embeddable strictly pseudoconvex CR-hypersurface. Then any (not necessarily smooth) CR-hypersurface Y0 , strictly cobordant to Y1 , is also embeddable.

6.3.2 The compactification theorem Our goal is to prove the following result. Theorem 6.3.24. Let X be a connected complex manifold with compact strongly pseudoconvex boundary and of complex dimension n 2. Assume that Int(X) is endowed with a complete K¨ ahler metric g T X with pinched negative curvature. The following assertions are equivalent: (a) ∂X is CR embeddable in some Cm . (b) X has finite volume away from a neighborhood of ∂X. If one of the equivalent conditions (a) or (b) holds true, there exists a compact strongly pseudoconvex domain M1 in a smooth projective variety and an embedding σ : X −→ M1 which is a biholomorphism between Int(X) and σ(Int(X)), σ(∂X) = ∂M1 , and M1 σ(X) is an exceptional analytic set which can be blown down to a finite set of singular points. Proof. (a) ⇒ (b). By Proposition 6.3.14, there exists a defining function ϕ : X → ] − ∞, c] such that ∂X = {ϕ = c}, with the properties: (1) its Levi form is positive definite on the holomorphic tangent space of ∂X and (2) ϕ is strictly plurisubharmonic on {c0 < ϕ < c}. It follows that ∂X = {ϕ = c} and {ϕ = d} are strictly CR-cobordant, for d ∈]c0 , c[. By assumption, ∂X = {ϕ = c} is embeddable. We infer from Theorem 6.3.23 that also {ϕ = d} is embeddable for d ∈ ]c0 , c[. The Harvey–Lawson theorem 6.3.16 asserts there exists a normal Stein space with boundary and at most isolated singularities S ⊂ Cm , such that ∂S = {ϕ = d}. We compactify the strip {d < ϕ
0 be sufficiently small. We set W = {d − δ < ϕ c} and glue the manifolds X and (Y S) ∪ W along W . The resulting manifold will be " We have glued thus a pseudoconcave cap on the pseudoconvex denoted by X. " contains a 1-concave open set and is thus Andreotti end of X. The manifold X pseudoconcave. Hence by Theorem 3.4.5, there exists C1 > 0 such that " K p ) C1 pn , dim H 0 (X, " X

for any p ∈ N∗ .

(6.3.5)

" " which A partition of unity argument delivers a Hermitian metric g T X on X TX of Int(X) on say {ϕ < ε} with ε < c. We agrees with the original metric g endow the canonical bundle KX" with the induced metric. Since g T X has pinched negative curvature, we deduce that there exists C > 0 such that on {ϕ < ε}, √ √ √ −1RKX1 = −1RKX = − −1Rdet Cω = C ω ", (6.3.6) "

where ω, ω " are the associated (1, 1)-forms to g T X , g T X . This means that KX" is uniformly positive at infinity and we can apply the Morse inequalities (3.3.8) (or (3.3.9)) for L = KX" . We obtain that √−1 n n KX 1 " Kp ) p R dim H 0 (X, + o(pn ) , p → ∞. (6.3.7) " X n! X(1) 2π " From (6.3.5) and (6.3.7), we get √−1 n √−1 n KX 1 R RKX1 = + < +∞. 2π 2π " {ϕε}(1) X(1) The second term in the sum is an integral over a relatively compact set, therefore finite. We infer that the first term is also finite and from (6.3.6)

√ n C −n ωn −1RKX1 < +∞ . n! {ϕ 0) on X are fulfilled for ω " (here E is trivial). Corollary 6.1.3 shows that the graded ring 0 " p " ⊕p0 H (X, L ) separates points and gives local coordinates on X. " is hyper 1-concave (use again Theorem On the other hand the manifold X 6.3.8); we denote the exhaustion function by r. By Theorem 6.3.21, we find a " ⊂X as open sets. The ⊂ CPm , such that X ⊂ X smooth compact manifold X desired projective strongly pseudoconvex domain is M = X (X S ). Theorem B.3.5 shows that there exists a Stein space M and a Remmert reduction π : M → M , which blows down the exceptional set A of M to a discrete set of points. The X is a pluripolar set, namely the set where the plurisubharmonic set M X = X function r takes the value −∞. By the maximum principle for plurisubharmonic functions, A ⊂ M X. The function ϕ = r ◦ π −1 : M π(M X) →] − ∞, ∞[ is strictly plurisubharmonic and proper. By Theorem 6.3.7, the universal covering of X is Stein. We may thus apply Theorem 6.3.9 (b) for the hyperconcave end M π(M X) and deduce that the singular set Msing of M equals π(M X). Therefore M X = A, so M X is the exceptional analytic set of M and by . Actually, each end N1 , . . . , Nm blowing down this exceptional set, we obtain Msing of M can be compactified with one point of the singular set Msing = {x1 , . . . , xm }.


289

Moreover, by the uniqueness of the Stein completion from Theorem 6.3.4 (a), we see that M and S coincide. The first application of Theorem 6.3.24 is the classical theorem of Siu–Yau, which is the particular case when ∂X = ∅. This provides a geometric proof of the Satake-Baily-Borel compactification of arithmetic quotients of rank 1. Theorem 6.3.25 (Siu–Yau). Let X be a complete K¨ ahler manifold of finite volume and negative pinched sectional curvature. If dim X 2, X is biholomorphic to a quasiprojective manifold which can be compactified to a Moishezon space by adding finitely many singular points. Note that our proof of Theorem 6.3.25 is different from the original one in its second part. Siu and Yau use the Schwarz-Pick lemma of Yau, whereas we use Theorem 6.3.9, where one tool is Wermer’s theorem. As a second application, we study some quotients of the unit complex ball Bn in Cn which where considered by Burns and Napier–Ramachandran. Corollary 6.3.26. Let Γ be a torsion-free discrete group of automorphisms of the unit ball Bn in Cn , n 2, and let X = Bn /Γ. Assume that the limit set Λ is a proper subset of ∂Bn and that the quotient (∂Bn Λ)/Γ has a compact component Y . Let N be the end of X corresponding to Y . Then the following assertions are equivalent: (a) X N has finite volume. (b) Y is CR embeddable in some Cm . If one of (a) or (b) holds, X can be compactified to a strongly pseudoconvex domain in a projective variety by adding an exceptional analytic set. Proof. As is well known, the limit set Λ is the set of accumulation points of any orbit Γ · x, x ∈ Bn , and is a closed Γ-invariant subset of the sphere at infinity ∂Bn . Lemma 6.3.27. The complement ∂Bn Λ is precisely the set of points at which Γ acts properly discontinuously, and the space X ∪ (∂Bn Λ)/Γ is a manifold with boundary (∂Bn Λ)/Γ. Y is a compact subset of this boundary, hence there is a neighborhood N of Y in X which is diffeomorphic to the product Y ×]0, 1[. It follows that N is an end of X, because Y is compact and connected. Actually, N is a strongly pseudoconvex end, in the sense that its boundary Y at infinity is strictly pseudoconvex. Since X = Bn /Γ is a complete manifold with sectional curvature pinched between −4 and −1, Corollary 6.3.26 is an immediate consequence of Theorem 6.3.24. Corollary 6.3.28 (Burns, Napier–Ramachandran). Let Γ be a torsion-free discrete group of automorphisms of the unit ball Bn in Cn with n 3 and let X = Bn /Γ. Assume that the limit set Λ is a proper subset of ∂Bn and that the quotient (∂Bn Λ)/Γ has a compact component Y . Then X has only finitely many ends, all of which, except for the unique end corresponding to Y , are cusps.

290


Proof. Indeed, since Y is a strongly pseudoconvex manifold of dimension at least 5, it follows from Boutet de Monvel’s theorem 6.3.17 that Y is embeddable. By Theorem 6.3.24, X has finite volume away from Y and finitely many ends, all of which, except for the unique end corresponding to Y , have finite volume, i.e., are cusps.

6.4 Weak Lefschetz theorems Using Morse theory, Andreotti–Frankel and Bott gave the following formulation of the classical Lefschetz hyperplane theorem. Theorem 6.4.1. If Y is a smooth hypersurface in a compact projective manifold X of dimension n, such that the associated line bundle OX (Y ) is positive, then X is obtained from Y by attaching cells of real dimension n. In particular, the natural map π1 (Y ) → π1 (X) is surjective for n 2 and an isomorphism for n 3. Nori showed further: Theorem 6.4.2. If X and Y are connected projective manifolds with dim X = dim Y + 1 > 1 and ι : Y → X is a holomorphic immersion with ample normal bundle, then the image of π1 (Y ) → π1 (X) is of finite index. Napier and Ramachandran proved a generalization of Nori’s Lefschetz type theorem by removing the codimension one condition. They use the L2 estimates for ∂ on complete K¨ ahler manifolds to separate the sheets of appropriate coverings. In the sequel, we use the Bergman kernel and consider not necessarily Kähler manifolds (see also Theorem 6.1.1). First we introduce the notion of formal completion. Let Y be a complex analytic subspace of the complex manifold W and denote by IY the ideal sheaf 1 of W with respect to Y is the ringed space of Y . The formal completion W ν 1 (W , OW 1 ) = (Y, proj limν OW /IY ). If F is an analytic sheaf on W , we denote by F" the sheaf F" = proj limν F ⊗ (OW /IYν ). If F is coherent then F" is too. Lemma 6.4.3. The kernel of the canonical restriction mapping H 0 (W, F ) −→ 1 , F") consists of the sections of F which vanish on a neighborhood of Y . H 0 (W Hence for locally free F , the map is injective. Theorem 6.4.4. Let (X, Θ) be a complete connected K¨ ahler manifold and let L (L, h ) be a holomorphic Hermitian line bundle such that for some ε > 0 we have √ −1RL ε Θ. Let moreover Y ⊂ X be a connected compact complex subspace of X such that for any p, " F"p ) < ∞, dim H 0 (X,

for Fp = OX (Lp ⊗ KX ).

(6.4.1)

Let i : Y → X be the inclusion and G = i∗ (π1 (Y )) be the image of π1 (Y ) in π1 (X). Then G is of finite index in π1 (X).

6.4. Weak Lefschetz theorems

291

Proof. We follow the strategy of Napier–Ramachandran. There exists a connected G )) = G with degree k = [π1 (X) : G]. G −→ X such that π∗ (π1 (X covering π : X := π −1 (U ) is a disjoint We fix a point x ∈ X and neighborhood U of x such that U union of open sets (Ui )1ik and π|Ui : Ui → U is biholomorphic. √ = π ∗ Θ. Then −1RL εΘ on X G . As usual hL ) = π ∗ (L, hL ) and Θ Let (L, set n = dim X. By the holomorphic Morse inequalities (6.1.14) for E = KX , we have n! n,0 p) (XG , L lim inf n dim H(2) p−→∞ p

G X

√

−1 L 2π R

n

√ U

−1 L 2π R

√

=k U

n

−1 L 2π R

n .

(6.4.2)

n,0 p ) k. (XG , L Therefore, for large p, we have dim H(2)

Let us choose a small open connected neighborhood V of Y such that π1 (Y ) −→ π1 (V ) is an isomorphism; so the image of π1 (V ) in π1 (X) is G. Hence, if we denote by j the inclusion of V in X, there exists a holomorphic lifting π◦ j : V −→ X, j = j. Since j is locally biholomorphic, the pull-back map n,0 ∗ p ) −→ H n,0 (V, Lp ) is injective. On the other hand, Lemma 6.4.3 j : H(2) (XG , L delivers an injective map " F"p ). H 0 (V, Fp ) → H 0 (V" , F"p ) = H 0 (X, n,0 p) < By hypothesis, the latter space is finite-dimensional so k dim H(2) (XG , L ∞. This finishes the proof.

Remark 6.4.5. (a) By a theorem of Grothendieck, the hypothesis (6.4.1) of Theorem 6.4.4 is fulfilled if Y is locally a complete intersection with ample normal bundle NY /X (if Y is smooth cf. Definition B.3.12). (b) We can replace (6.4.1) in Theorem 6.4.4 with the requirement that Y has a fundamental system of Andreotti pseudoconcave neighborhoods {V }. Then dim H 0 (V, Fp ) is finite by Theorem 3.4.5. This happens for example if Y is a smooth hypersurface and the normal bundle NY /X of Y in X admits a Hermitian metric hNY /X such that R˙ NY /X has at least one positive eigenvalue. (c) If X contains a simply connected subvariety satisfying either (a) or (b), π1 (X) is finite. (d) We can slightly generalize Theorem 6.4.4, by assuming that Y is an analytic subset of a manifold V and there exists locally biholomorphic map ψ : 1p ) < ∞ holds for any p 1, where Fp = V → X, such that dim H 0 (V" , F ∗ p OV (ψ L ⊗ KV ). Then the image of the induced map π1 (Y ) → π1 (X) has finite index in π1 (X). The proof is the same as of Theorem 6.4.4, but we use the map ψ instead of the inclusion j.

292


Corollary 6.4.6. Let (X, Θ) be an n-dimensional complete Hermitian manifold and √ let (L, hL ) be a holomorphic Hermitian line bundle such that −1RL > CΘ, with C > 0. Let Y ⊂ X be a connected compact analytic subspace with dim Y 1. Suppose Y is locally a complete intersection and NY /X is ample. Then the image G of π1 (Y ) in π1 (X) has finite index in π1 (X). Proof. Apply Theorem 6.4.4 and Remark 6.4.5.

Corollary 6.4.7. Let X be a Zariski open set in a compact normal Moishezon space X. Let Y ⊂ Xreg be a connected compact analytic subspace with dim Y 1. Suppose Y is locally a complete intersection and NY /X is ample. Let G be the image of π1 (Y ) in π1 (X). Then G has finite index in π1 (X). Proof. Since X is normal, we have a surjective morphism π1 (Xreg ) −→ π1 (X), so we can replace X with Xreg . We can thus resolve the singularities of X and assume that it is a manifold. We consider on X a singular Hermitian positive line bundle L. We modify then the proof of√Theorem 6.2.3 in the following way. First we consider the singular support Σ of −1RL and construct the generalized −→ X such that Poincaré metric on X Σ. Then we consider a covering πΓ : X = G with degree k = [π1 (X) : G]. πΓ,∗ (π1 (X)) Then we apply the covering version of Theorem 6.2.3 on the covering X −1 −1 πΓ (Σ) of X Σ. We obtain in this way (n, 0)-forms on X πΓ (Σ) which are L2 π −1 (Σ) and a metric with respect to the pull-back of the Poincaré metric on X Γ π −1 (Σ) which is bounded below by a smooth metric on X. on πΓ−1 (L) over X Γ 2 But for (n, 0)-forms the L condition does not depend on the metric on the base manifold (cf. the proof of Theorem 3.3.5 (i)), so we can take the L2 condition with and π −1 (L). Hence these sections extend to X respect to smooth metrics on X Γ and we can apply the proof of Theorem 6.4.4.

Problems Problem 6.1 (Theorem 6.1.1 for bounded geometry). Show that if in addition hL , hE , Θ, g T X and their derivatives of order 2n+m+4 (resp. 2n+m+m +k+5) are uniformly bounded on X in the norm induced by g T X , then Proposition 4.1.5 (resp. Theorem 4.2.1) holds uniformly for x, x ∈ X (resp. x0 ∈ X) with the norm C k . (Hint: Show that the injectivity radius of X is bounded by below by a positive constant and as in in the proof of Theorem 6.1.4, the constant in the Sobolev embedding theorem is uniformly bounded on X.) Problem 6.2. Let X be a compact connected complex manifold. Assume that X admits a closed integral current ω with singular support contained in a proper n > 0. analytic set Σ such that ω is positive in a neighborhood of Σ and X(1,ω) ωac (a) Consider a singular Hermitian holomorphic line bundle (L, hL ) such that ω = c1 (L, hL ) (Lemma 2.3.5). Introduce on X Σ the metric (6.2.2) and on


293

L|XΣ the metric (6.2.16a). Using that (L, hL ) has positive curvature near Σ, show that the fundamental estimate (3.2.2) holds for (0, 1)-forms on X Σ. 0 (X Σ, Lp ) Cpn for some Apply Theorem 3.2.16 and deduce that dim H(2) 0 (X Σ, Lp ) is defined in (6.2.16b). C > 0 and p large enough, where H(2) (b) Infer that L is big and X is Moishezon. This is a proof of (6) ⇒ (1) in Theorems 2.3.28 and 2.3.30. Problem 6.3. Let X be a connected complex manifold which is pseudoconcave in the sense of Andreotti. Assume that X is embedded in a projective space C Pn . Using the same reasoning from Problem 5.5 deduce that X is contained in a projective variety of the same dimension as X.

6.5 Bibliographic notes Sections 6.1 and 6.2 appear in [161]. The generalized Poincaré metric appeared in [59, §2]. See also [113, 114]. Section 6.3. Theorem 6.3.4 is due to Rossi [207, Th. 3, p. 245]. Andreotti–Siu [5, Prop. 3.2] improved the result in different directions, e.g., they showed that it holds for normal complex spaces. The uniqueness result comes from [5, Cor. 3.2]. Example 6.3.5 appeared in Rossi [207, p. 252–256] (being attributed to Andreotti), Andreotti–Siu [5, p. 262–270] (where credit is given to Grauert) and Grauert [117, p. 273]. The study of compactifications of manifolds with negative curvature was initiated in the paper of Siu–Yau [228], where Theorem 6.3.25 appeared. For more results on the compactification of complete Kähler–Einstein manifolds of finite volume and bounded curvature we refer to Mok [178] and the references therein. Theorem 6.3.9 was proved in [172]. In this connection, note a result of Colt¸oiuTib˘ ar [68], asserting that the universal cover of a small punctured neighborhood of an isolated singularity of dimension 2 is Stein, whenever the fundamental group of the link is infinite. The result from Example 6.3.3 is proved in [67]. Kohn–Rossi’s theorem 6.3.12 stems from [145]. Theorem 6.3.15 is due to Heunemann [127, Theorem 0.2] (see also Ohsawa [188]). The Harvey–Lawson theorem 6.3.16 appeared in [125]. The embeddability theorem of Boutet de Monvel 6.3.17 comes from [51, p. 5]. A straightforward argument that the boundary of the Grauert–Andreotti–Rossi example is not embeddable can be found in Burns [57]. The author shows that the CR functions on S 3 equipped with the induced CR structure from the complex structure of Example 6.3.5 are equal at antipodal points. Therefore, CR functions cannot embed this structure in the Euclidean space. For Theorem 6.3.18, we refer to [53, 125, 57, 144]. L. Lempert introduced the idea of linking the deformations of the CR structures on Y to the deformations of the complex structure on a strongly pseudoconcave manifold Z bounding Y and proved in this context Theorems 6.3.19 and 6.3.20 in [151, 152, 153].

294


The result of Problem 6.3 was proved in [2, Théorème 6] and is a generalization of Chow’s theorem from Problems 5.4, 5.5. Proofs of Theorem 6.3.21 are in [6, Theorem 3, p.97], [5, Theorem 4.1] (a generalization for complex normal spaces) and [181, Lemma 2.1]. For Theorem 6.3.23, we refer to [104, Theorem 2]. The use of holomorphic Morse inequalities in compactification questions appears in Nadel–Tsuji [181] and Napier–Ramachandran [183]. Theorem 6.3.24 and Corollary 6.3.26 appeared in [174]. The proof of the implication (a) ⇒ (b) in Theorem 6.3.24 follows [183]. The result of Eberlein alluded to in the proof of (a) ⇒ (b) can be found in [100]. Corollary 6.3.28 is [183, Theorem 4.2]. For Lemma 6.3.27 see [101, §10]. Wu’s result (Theorem 6.3.7) was obtained in [255]. Section 6.4. The classical Lefschetz hyperplane theorem was proved by Lefschetz [150]. Andreotti–Frankel and Bott gave Morse theoretical proofs [3, 46, 176]. The weak Lefschetz theorems were studied by Nori [186], Campana [58], Kollár [147], Napier–Ramachandran [183]. The approach using the holomorphic Morse theory on covering stems from [244], from where the results of Section 6.4 are taken. Lemma 6.4.3 is [56, Prop. VI.2.7]; in [56] the reader can find out more about formal completions. For the definition of the ampleness of NY /X for nonsmooth Y , see [123, 124]. For the morphism π1 (Xreg ) −→ π1 (X), for a normal Zariski open set, see [110]. There are many references about Lefschetz theorems in algebraic geometry. We refer to Grothendieck [121], Fulton–Lazarsfeld [110] and Lazarsfeld [149].

Chapter 7

Toeplitz Operators We show in this chapter how the asymptotic expansion of the Bergman kernel implies the semi-classical properties of Toeplitz operators acting on high tensor powers of a positive line bundle over a compact manifold. In particular we obtain a construction of a star-product (a deformation quantization) using this technique. Moreover, our approach works with some modifications on non-compact and symplectic manifolds. This chapter is organized as follows: In Section 7.1, we explain the formal calculus on Cn for our model operator L . In Section 7.2, we establish the asymptotic expansion for the kernel of Toeplitz operators. In Section 7.3, we establish that the asymptotic expansion is also a sufficient condition for a family of operators to be Toeplitz. In Section 7.4, we conclude finally that the Toeplitz operators form an algebra. In Section 7.5, we extend it to the non-compact case.

7.1 Kernel calculus on Cn In this section, we state the properties of the calculus of the kernels (F P)(Z, Z ), where F ∈ C[Z, Z ] and P(Z, Z ) is the kernel of the projection on the null space of the model operator L . This calculus is the main ingredient of our approach. Let us consider the canonical coordinates (Z1 , . . . , Z2n ) on the real vector space R2n . On the complex vector space Cn we consider the complex coordinates (z √1 , . . . , zn ). The two sets of coordinates are linked by the relation zj = Z2j−1 + −1Z2j , j = 1, . . . , n. Let 0 < a1 a2 · · · an . Following (4.1.74) and (4.1.76), we define the differential operators: 1 1 ∂ ∂ + ai z i , b + bi = −2 ∂z i = 2 ∂z i + ai zi , i 2 2 + bi bi . b = (b1 , . . . , bn ), L = i

(7.1.1)

296

Chapter 7. Toeplitz Operators

1/2 On R2n , we consider the L2 -norm · L2 = | · |2 dZ where dZ is the R2n standard Euclidean volume form. L acts as a densely defined self-adjoint operator on (L2 (R2n ), · L2 ) and its spectrum was determined in Theorem 4.1.20. From there we deduced the formula for the kernel of the orthogonal projection n 1 aj |zj |2 , β ∈ Nn . (7.1.2) P : L2 (R2n ) −→ Ker(L ) = span z β exp − 4 j=1

Namely, the kernel of P with respect to dZ is (cf. (4.1.84)) P(Z, Z ) =

n 1

ai exp − ai |zi |2 + |zi |2 − 2zi z i . 2π 4 i i=1

(7.1.3)

We will repeat the use of Theorem 4.1.20 when we meet the operator P, instead of doing the direct computation by using (7.1.3). This point of view will help us to simplify a lot of computations and to understand better our operations. n 1 α β 2 As an example, if ϕ(Z) = b z exp − 4 j=1 aj |zj | with α, β ∈ Nn , then ⎧ n ⎪ ⎨ z β exp − 1 if |α| = 0, aj |zj |2 4 j=1 (Pϕ)(Z) = ⎪ ⎩ 0 if |α| > 0.

(7.1.4)

In what follows, all operators are defined by their kernels with respect to dZ. In this way, if F is a polynomial on Z, Z , then F P is an operator on L2 (R2n ) with kernel F (Z, Z )P(Z, Z ) with respect to dZ. We will add a subscript z or z when we need to specify that the operator is acting on the variables Z or Z . Lemma 7.1.1. For any polynomial F (Z, Z ) ∈ C[Z, Z ], there exist polynomials Fα ∈ C[z, Z ] and Fα,0 ∈ C[z, z ], (α ∈ Nn ) such that (F P)(Z, Z ) =

bα z (Fα P)(Z, Z ),

α

((F P) ◦ P)(Z, Z ) =

bα z Fα,0 (z, z )P(Z, Z ).

(7.1.5)

α

Moreover, |α| + deg Fα , |α| + deg Fα,0 have the same parity with the degree of F in Z, Z . In particular, F0,0 (z, z ) is a polynomial in z, z and its degree has the same parity with deg F . For any polynomials F, G ∈ C[Z, Z ] there exist polynomials K [F, G] ∈ C[Z, Z ] such that ((F P) ◦ (GP))(Z, Z ) = K [F, G](Z, Z )P(Z, Z ).

(7.1.6)

7.1. Kernel calculus on Cn

297

Proof. Note that from (7.1.1) and (7.1.2), we get bj ,z P(Z, Z ) = aj (z j − z j )P(Z, Z ), ∂ g(z, z), [g(z, z), bj ,z ] = 2 ∂zj

(7.1.7)

for any polynomial g(z, z) ∈ C[z, z]. Using repeatedly (7.1.7), we can replace z in F (Z, Z ) by a combination of bj,z and z and the first equation of (7.1.5) follows. From (7.1.4) and the first equation of (7.1.5), we get (P ◦ (F P))(Z, Z ) = (F0 P)(Z, Z ),

(7.1.8)

with F0 ∈ C[z, Z ]. When we take the adjoint of (7.1.8) for F , as P is self-adjoint, we get ((F P) ◦ P)(Z, Z ) = F (Z, z )P(Z, Z ), and F is a polynomial on Z, z . Now using again the first equation of (7.1.5), we get the second equation of (7.1.5). From (7.1.5), we get (7.1.6). As an example of how Lemma 7.1.1 works, we calculate from (7.1.7) bj ,z P(Z, Z ) + z j P(Z, Z ), aj 1 2 zi z j P(Z, Z ) = bj ,z zi P(Z, Z ) + δij P(Z, Z ) + zi z j P(Z, Z ). aj aj z j P(Z, Z ) =

(7.1.9)

We calculate some examples for K [F, G] in (7.1.6). From (7.1.4) and (7.1.9), we get K [1, z j ]P = P ◦ (z j P) = z j P,

K [1, zj ]P = P ◦ (zj P) = zj P,

K [zi , z j ]P = (zi P) ◦ (z j P) = zi P ◦ (z j P) = zi z j P, K [z i , zj ]P = (z i P) ◦ (zj P) = z i P ◦ (zj P) = z i zj P, 2 K [zi , z j ]P = (zi P) ◦ (z j P) = P ◦ (zi z j P) = δij P + zi z j P, aj 2 K [z i , zj ]P = (z i P) ◦ (zj P) = P ◦ (z i zj P) = δij P + z i zj P. aj

(7.1.10)

Thus we get K [1, z j ] = z j , zi z j ,

K [1, zj ] = zj ,

K [z i , zj ] = z i zj , 2 K [z i , zj ] = K [zj , z i ] = δij + z i zj . aj

K [zi , z j ] =

(7.1.11)

For a polynomial F ∈ C[Z, Z ], let (F P)p be the operator defined by the √ √ kernel pn (F P)( pZ, pZ ). Then by changing the variable, we get for F, G ∈ C[Z, Z ], √ √ ((F P)p ◦ (GP)p )(Z, Z ) = pn ((F P) ◦ (GP))( pZ, pZ ). (7.1.12)

298


7.2 Asymptotic expansion of Toeplitz operators We use the notation in Sections 1.6.1, 4.1.1. Let (X, J) be an n-dimensional compact complex manifold with complex structure J. Let g T X be a Riemannian metric on T X compatible with J. Let (L, hL ) and (E, hE ) be two holomorphic Hermitian vector bundles on X and rk(L) = 1. We suppose that the positivity condition (1.5.21) holds for RL . Recall that Pp is the orthogonal projection from Ω0,• (X, Lp ⊗ E) onto Ker(Dp ). As explained in Section 4.1.1, there exists p0 ∈ N such that for any p > p0 , Pp is simply the orthogonal projection from (L2 (X, Lp ⊗ E), , ) onto H 0 (X, Lp ⊗ E), where the Hermitian product , on L2 (X, Lp ⊗ E) is defined by (1.3.14) associated to g T X , hL , hE . In what follows, we always assume p > p0 . For simplicity we denote the linear operator Mg : L2 (X,Lp ⊗ E) → L2 (X,Lp ⊗ E) of multiplication with a bounded function g just by g = Mg . Definition 7.2.1. A Toeplitz operator is a family {Tp } of linear operators Tp : L2 (X, Lp ⊗ E) −→ L2 (X, Lp ⊗ E) ,

(7.2.1)

with the properties: (i) For any p ∈ N, we have Tp = Pp Tp Pp .

(7.2.2)

(ii) There exists a sequence gl ∈ C ∞ (X, End(E)) such that for all k 0 there exists Ck > 0 with k p−l gl Pp Ck p−k−1 , Tp − Pp

(7.2.3)

l=0

where · denotes the operator norm on the space of bounded operators. ∞ The full symbol of {Tp } is the formal series l=0 l gl ∈ C ∞ (X, End(E))[[]] and the principal symbol of {Tp } is g0 . If each Tp is self-adjoint, {Tp } is called selfadjoint. As our notation, we will denote (7.2.3) by Tp = Pp

k

p−l gl Pp + O(p−k−1 ).

(7.2.4)

l=0

If (7.2.3) holds for any k ∈ N, then we denote it by Tp = Pp

∞ l=0

p−l gl Pp + O(p−∞ ).

(7.2.5)

7.2. Asymptotic expansion of Toeplitz operators

299

An important particular case is when gl = 0 for all l 1. We set g0 = f . We denote then Tf,p : L2 (X, Lp ⊗ E) −→ L2 (X, Lp ⊗ E) ,

Tf,p = Pp f Pp .

The Schwartz kernel of Tf,p is given by Pp (x, x )f (x )Pp (x , x ) dvX (x ) . Tf,p (x, x ) =

(7.2.6)

(7.2.7)

X

Let us remark that if f ∈ C ∞ (X, End(E)) is self-adjoint, i.e., f (x) = f (x)∗ for all x ∈ X, then the operator of multiplication with f and therefore Tf,p are selfadjoint. The map which associates to a section f ∈ C ∞ (X, End(E)) the bounded operator Tf,p on L2 (X, Lp ⊗ E) is the famous Berezin–Toeplitz quantization. We first note that outside the diagonal of X × X the kernel of the Toeplitz operators Tf,p has the growth O(p−∞ ). Lemma 7.2.2. For any ε > 0 and any l, m ∈ N there exists Cl,m > 0 such that |Tf,p (x, x )|C m (X×X) Cl,m p−l

(7.2.8)

for all p 1 and all (x, x ) ∈ X × X with d(x, x ) > ε. Here C m -norm is induced by ∇L , ∇E and hL , hE , g T X . Proof. We know that from (4.1.12), (7.2.8) holds if we replace Tf,p by Pp . Moreover, from (4.2.1), for any m ∈ N there exist Cm > 0, Mm > 0 such that |Pp (x, x )|C m (X×X) < Cm pMm for all (x, x ) ∈ X ×X. These two facts and formula (7.2.7) imply the lemma. We concentrate next on a neighborhood of the diagonal in order to obtain the asymptotic expansion of the kernel Tf,p (x, x ). As in Section 1.6.2, let injX be the injectivity radius of X and fix 0 < ε < X inj /4. For any x ∈ X, we identify the geodesic ball B X (x, 4ε) with the ball B Tx X (0, 4ε) ⊂ Tx X by means of the exponential map. We consider trivializations of L and E as in Section 4.1.3, i.e., we trivialize L by using unit frames eL (Z) and eE (Z) which are parallel with respect to ∇L and ∇E along [0, 1] u → uZ for Z ∈ B Tx X (0, 4ε). For f ∈ C ∞ (X, End(E)), we denote it as fx0 (Z) ∈ End(Ex0 ) a family (with parameter x0 ∈ X) of functions on Z in the normal coordinate near x0 . Recall that dvT X is the Riemannian volume form on (Tx0 X, g Tx0 X ), and κx0 (Z) is the smooth positive function defined by (cf. (4.1.28)) dvX (Z) = κx0 (Z)dvT X (Z),

κx0 (0) = 1,

where we denote it by κx0 (Z) to indicate the base point x0 ∈ X.

(7.2.9)

300

Chapter 7. Toeplitz Operators (1,0)

Recall that for x0 ∈ X, we choose {wi }ni=1 as an orthonormal basis of Tx0 X, √ −1 1 √ √ such that (1.5.18) holds. Then e2j−1 = 2 (wj + w j ) and e2j = 2 (wj − 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). For the functions on the normal coordinate, we will add a subscript x0 to indicate the base point x0 ∈ X. Recall π : T X ×X T X → X is the natural projection from the fiberwise product of T X on X. Let Ξp : L2 (X, Lp ⊗E) −→ L2 (X, Lp ⊗E) be a family of linear operators with smooth kernel Ξp (x, y) with respect to dvX (y). Under our trivialization, Ξp (x, y) induces a smooth section Ξp,x0 (Z, Z ) of π ∗ (End(E)) over T X ×X T X with Z, Z ∈ Tx0 X, |Z|, |Z | < 4ε. To study the asymptotic expansion of Ξp (x, y) near the diagonal of X × X, we denote p−n Ξp,x0 (Z, Z ) ∼ =

k

k+1 r √ √ (Qr,x0 Px0 )( pZ, pZ )p− 2 + O(p− 2 ),

(7.2.10)

r=0

with Px0 in (4.1.84) (cf. (7.1.3)), if Qr,x0 ∈ End(Ex0 )[Z, Z ] are a smooth family on x0 ∈ X of polynomials on Z, Z with values in End(Ex0 ), and there exist 0 < ε ≤ 4ε, C0 > 0 such that for any l ∈ N, there exist Cl > 0, M > 0 such that for any Z, Z ∈ Tx0 X, |Z|, |Z | < ε , p > p0 , we have k r √ √ −n 1/2 (Qr,x0 Px0 )( pZ, pZ )p− 2 p Ξp,x0 (Z, Z )κ1/2 x0 (Z)κx0 (Z ) − r=0

C l (X)

k+1 √ √ Ck, l p− 2 (1 + p |Z| + p |Z |)M exp(− C0 p |Z − Z |) + O(p−∞ ). (7.2.11) Recall that C l (X) is the C l -norm for the parameter x0 ∈ X, and 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 . Recall that Jr (Z, Z ) ∈ End(Λ(T ∗(0,1) X) ⊗ E)x0 , (r ∈ N) defined in Theorem 4.1.21. By Theorem 4.1.21 and (4.1.98), we know Jr,x0 ∈ End(Ex0 )[Z, Z ] are polynomials on Z, Z with the same parity as r and smooth on x0 ∈ X, and J0,x0 = IdE .

(7.2.12)

Lemma 7.2.3. For any k ∈ N, Z, Z ∈ Tx0 X, |Z|, |Z | < 2ε, we have p−n Pp,x0 (Z, Z ) ∼ =

k r=0

k+1 r √ √ (Jr,x0 Px0 )( pZ, pZ )p− 2 + O(p− 2 ).

(7.2.13)

7.2. Asymptotic expansion of Toeplitz operators

301

Proof. By Theorem 4.2.1 we get: for any k, m ∈ N, there exist M ∈ N, C > 0 such that for Z, Z ∈ Tx0 X, |Z|, |Z | 2ε, k 1 1 r √ √ −n (Jr,x0 Px0 )( pZ, pZ )p− 2 p Pp,x0 (Z, Z )κx20 (Z)κx20 (Z ) − r=0

C m (X)

√ √ √ Cp−(k+1)/2 (1 + p |Z| + p |Z |)M exp(− C µ0 p |Z − Z |) + O(p−∞ ). (7.2.14)

From (7.2.14), we get (7.2.13). ∞

Lemma 7.2.4. Let f ∈ C (X, End(E)). There exists a family of polynomials {Qr,x0 (f ) ∈ End(Ex0 )[Z, Z ]}x0 ∈X with the same parity as r, and smooth in x0 ∈ X such that for any k ∈ N, Z, Z ∈ Tx0 X, |Z|, |Z | < ε/2, p−n Tf,p,x0 (Z, Z ) ∼ =

k

k+1 r √ √ (Qr,x0 (f )Px0 )( pZ, pZ )p− 2 + O(p− 2 ).

(7.2.15)

r=0

Moreover, under the notation in (7.1.6) and Jr,x0 in (7.2.13), we have ∂ α fx0 Zα J K Jr1 ,x0 , (0) , Qr,x0 (f ) = r ,x 2 0 ∂Z α α!

(7.2.16)

r1 +r2 +|α|=r

especially, Q0,x0 (f ) = f (x0 ).

(7.2.17)

Proof. From (7.2.7) and (7.2.8), we know that Tf,p,x0 (Z, Z ) for |Z|, |Z | < ε/2, is determined up to terms of order O(p−∞ ) by the behavior of f in B X (x0 , ε). For |Z|, |Z | < ε/2, with ρ in (1.6.19), we get Pp,x0 (Z, Z )ρ(2|Z |/ε)fx0 (Z )Pp,x0 (Z , Z ) Tf,p,x0 (Z, Z ) = Tx0 X

× κx0 (Z ) dvT X (Z ) + O(p−∞ ). (7.2.18) We consider the Taylor expansion of fx0 : fx0 (Z) =

∂ α fx Zα 0 + O(|Z|k+1 ) (0) α ∂Z α! |α|k √ k+1 ( pZ)α ∂ α fx0 √ + p− 2 O(| pZ|k+1 ). (7.2.19) = p−α/2 (0) α ∂Z α! |α|k

We multiply now the expansions given in (7.2.19) and (7.2.14) and obtain the expansion of 1/2 κ1/2 x0 (Z)Pp,x0 (Z, Z )(κx0 fx0 )(Z )Pp,x0 (Z , Z )κx0 (Z )

302


which we substitute in (7.2.18). We integrate then on Tx0 X by using the change of √ variable pZ = W and conclude (7.2.15) and (7.2.16) by using formulas (7.1.6) and (7.1.12). From (7.2.12) and (7.2.16), we get Q0,x0 (f ) = K [1, fx0 (0)] = fx0 (0) = f (x0 ). The proof of Lemma 7.2.4 is complete.

(7.2.20)

As an example, we compute Q1,x0 (f ). Lemma 7.2.5. For Q1,x0 (f ) in (7.2.15), We have ∂fx0 Q1,x0 (f ) = f (x0 )J1,x0 + K J0,x0 , (0)Zj J0,x0 . ∂Zj

(7.2.21)

Proof. At first, by taking f = 1 in (7.2.16), we get J1,x0 = K [J0,x0 , J1,x0 ] + K [J1,x0 , J0,x0 ].

(7.2.22)

From (1.2.30) and (4.1.34), we know that O1 defined in (4.1.31) (considered as a differential operator with coefficients in End(Λ(T ∗(1,0) X) ⊗ E)x0 ) acts as identity on the E-component and preserves the Z-grading on Λ(T ∗(1,0) X). (Note that, if g T X = ω(·, J·), its restriction on C⊗E is zero by (4.1.100), thus J1,x0 = 0.) Thus from (4.1.93) and (7.1.6), we obtain K [J1,x0 , f (x0 )J0,x0 ] = f (x0 )K [J1,x0 , J0,x0 ]. From (7.2.16), (7.2.22) and (7.2.23), we get (7.2.21).

(7.2.23)

7.3 A criterion for Toeplitz operators We will prove next a useful criterion which ensures that a given family is a Toeplitz operator. Theorem 7.3.1. Let {Tp : L2 (X, Lp ⊗E) −→ L2 (X, Lp ⊗E)} be a family of bounded linear operators which satisfies the following three conditions: (i) For any p ∈ N, Pp Tp Pp = Tp . (ii) For any ε0 > 0 and any l, m ∈ N, there exists Cl,m > 0 such that for all p 1 and all (x, x ) ∈ X × X with d(x, x ) > ε0 , |Tp (x, x )| Cl,m p−l .

(7.3.1)

(iii) There exists a family of polynomials {Qr,x0 ∈ End(Ex0 )[Z, Z ]}x0 ∈X such that: (a) each Qr,x0 has the same parity as r, (b) the family is smooth in

7.3. A criterion for Toeplitz operators

303

x0 ∈ X and (c) there exists 0 < ε < ε such that for any x0 ∈ X and Z, Z ∈ Tx0 X, |Z|, |Z | < ε , in the sense of (7.2.10) and (7.2.11), we have p−n Tp,x0 (Z, Z ) ∼ =

k

k+1 r √ √ (Qr,x0 Px0 )( pZ, pZ )p− 2 + O(p− 2 ).

(7.3.2)

r=0

Then {Tp } is a Toeplitz operator. We start the proof of Theorem 7.3.1. Let Tp∗ be the adjoint of Tp . By writing Tp =

√ 1 1 (Tp + Tp∗ ) + −1 √ (Tp − Tp∗ ), 2 2 −1

we may and we will assume from now on that Tp is self-adjoint. We will define inductively the sequence (gl )l0 , gl ∈ C ∞ (X, End(E)) such that for any m 1, Tp =

m

Pp gl p−l Pp + O(p−m−1 ).

(7.3.3)

l=0

Moreover, we can make these gl ’s to be self-adjoint. For x0 ∈ X, we set

We will show that

g0 (x0 ) = Q0,x0 (0, 0) .

(7.3.4)

Tp = Pp g0 Pp + O(p−1 ).

(7.3.5)

Proposition 7.3.2. In the conditions of Theorem 7.3.1 we have Q0,x0 (Z, Z ) = Q0,x0 (0, 0) for all x0 ∈ X and all Z, Z ∈ Tx0 X. Proof. Our first observation is as follows. Lemma 7.3.3. Q0,x0 is a polynomial in z, z . Proof. Indeed, by (7.3.2) √ √ p−n Tp,x0 (Z, Z ) ∼ = (Q0,x0 Px0 )( pZ, pZ ) + O(p−1/2 ).

(7.3.6)

By (7.2.12) and (7.2.13), we have √ √ p−n (Pp Tp Pp )x0 (Z, Z ) ∼ = (P ◦ (Q0 P) ◦ P)x0 ( pZ, pZ ) + O(p−1/2 ). (7.3.7) Since Pp Tp Pp = Tp , we deduce from (7.3.6) and (7.3.7) that Q0,x0 Px0 = Px0 ◦ (Q0,x0 Px0 ) ◦ Px0 , hence Q0,x0 ∈ End(Ex0 )[z, z ] by (7.1.5) and (7.1.8).

(7.3.8)

304


For simplicity we denote in the rest of the proof Fx = Q0,x . Let Fx = (i) (i) i0 Fx be the decomposition of Fx in homogeneous polynomials Fx of degree

(i)

i. We will show, that Fx vanish identically for i > 0, that is, Fx(i) (z, z ) = 0 for all i > 0 and z, z ∈ Cn .

(7.3.9)

The first step is to prove Fx(i) (0, z ) = 0

for all i > 0 and all z ∈ Cn .

(7.3.10)

Let us remark that since Tp are self-adjoint we have Fx(i) (z, z ) = (Fx(i) (z , z))∗ .

(7.3.11)

2n For y = expX Tx0 X as explained above (7.2.10), set x (Z ), Z ∈ R

F (i) (x, y) = Fx(i) (0, z ) ∈ End(Ex ), F (i) (x, y) = (F (i) (y, x))∗ ∈ End(Ey ).

(7.3.12)

F (i) and F (i) define smooth sections on a neighborhood of the diagonal of X × X. Clearly, F (i) (x, y)’s need not be polynomials of z and z . Since we wish to define global operators induced by these kernels, we use a cut-off function in the neighborhood of the diagonal. Pick a smooth function η ∈ C ∞ (R), such that η(u) = 1 for |u| ε /2 and η(u) = 0 for |u| ε . We denote by F (i) Pp and Pp F(i) the operators defined by the kernels η(d(x, y))F (i) (x, y)Pp (x, y)

and

η(d(x, y))Pp (x, y)F(i) (x, y)

with respect to dvX (y). Set Tp = Tp −

(F (i) Pp ) pi/2 .

(7.3.13)

ideg F

The operators Tp extend naturally to bounded operators on L2 (X, Lp ⊗ E). From (7.3.2) and (7.3.13), in the sense of (7.2.11), we deduce that for any k 1 and |Z | ε , we have the following expansion in the normal coordinates around x0 ∈ X: p−n Tp,x0 (0, Z ) ∼ =

k

√ (Rr,x0 Px0 )(0, pZ )p−r/2 + O(p−(k+1)/2 ),

(7.3.14)

r=1

for some polynomials Rr,x0 of the same parity as r. For simplicity we denote by Rr,p the operator defined as in (7.3.12) by the kernel √ Rr,p (x, y) = pn (Rr,x Px )(0, pZ )κ−1/2 (Z )η(d(x, y)) , (7.3.15) x where y = expX x (Z ).


305

Lemma 7.3.4. There exists C > 0 such that for any p > p0 , s ∈ L2 (X, Lp ⊗ E), Tp sL2 Cp−1/2 sL2 ,

(7.3.16)

Tp∗ sL2

(7.3.17)

Cp

−1/2

sL2 .

Proof. In order to use (7.3.14) we write k p−r/2 Rr,p ) s Tp sL2 (Tp −

L2

r=1

k + p−r/2 Rr,p s 2 . r=1

(7.3.18)

L

By the Cauchy-Schwarz inequality we have k

Tp − p−r/2 Rr,p s2L2 r=1

k

p−r/2 Rr,p (x, y)dvX (y) Tp − X

X

r=1

k

× p−r/2 Rr,p (x, y)|s(y)|2 dvX (y) dvX (x). Tp − X

(7.3.19)

r=1

We split then the inner integrals into integrals over B X (x, ε ) and X B X (x, ε ) k and use the fact that the kernel of Tp − r=1 p−r/2 Rr,p has the growth O(p−∞ ) outside the diagonal, by (7.3.1), the definition of the operators F (i) Pp in (7.3.13) (using the cut-off function η), and the definition (7.3.15) of Rr,p (which involves P). We get for example, uniformly in x ∈ X, k

p−r/2 Rr,p (x, y)|s(y)|2 dvX (y) Tp − X

r=1

= B X (x,ε )

k p−r/2 Rr,p (x, y)|s(y)|2 dvX (y) (Tp − r=1

+ O(p−∞ )

XB X (x,ε )

|s(y)|2 dvX (y). (7.3.20)

By (7.2.11), (7.3.14) for k sufficiently large, which we fix from now on, we obtain B X (x,ε )

k

T − p−r/2 Rr,p (x, y)|s(y)|2 dvX (y) p r=1

= O(p−1 )

B X (x,ε )

|s(y)|2 dvX (y). (7.3.21)

In the same vein we obtain k

p−r/2 Rr,p (x, y)dvX (y) = O(p−1 ) + O(p−∞ ). Tp − X

r=1

(7.3.22)

306


Using (7.3.19)–(7.3.22) finally gives k

p−r/2 Rr,p s Tp −

L2

r=1

C p−1/2 sL2 ,

s ∈ L2 (X, Lp ⊗ E) .

A similar proof as for (7.3.23) delivers for s ∈ L2 (X, Lp ⊗ E), Rr,p s 2 CsL2 , L

(7.3.23)

(7.3.24)

which implies k p−r/2 Rr,p s

L2

r=1

C p−1/2 sL2 ,

for s ∈ L2 (X, Lp ⊗ E) ,

(7.3.25)

for some constant C > 0. Relations (7.3.23) and (7.3.25) entail (7.3.16), which is equivalent to (7.3.17), by taking the adjoint. Let us consider the Taylor development of F (i) in normal coordinates around x with y = expX x (Z ): F (i) (x, y) =

√ ∂ α F (i) ( pZ )α −|α|/2 p (x, 0) + O(|Z |k+1 ). ∂Z α α!

(7.3.26)

|α|k

The next step in the proof of Proposition 7.3.2 is the following. Lemma 7.3.5. For any j > 0, we have ∂ α F(i) (x, 0) = 0 , ∂Z α

for i − |α| ≥ j > 0.

(7.3.27)

Proof. The definition (7.3.13) of Tp shows Tp∗ = Tp −

pi/2 (Pp F (i) ) .

(7.3.28)

ideg F

Let us develop the sum in the right-hand side. Combining the Taylor development (7.3.26) with the expansion (7.2.13) of the Bergman kernel we obtain: p−n

(Pp F (i) )x0 (0, Z )pi/2

i

∼ =

i

|α|,rk

√ ( pZ )α (i−|α|−r)/2 √ ∂ α F (i) p (Jr,x0 Px0 ) (0, pZ ) (x0 , 0) ∂Z α α! + O(p(deg F −k−1)/2 ), (7.3.29)


307

where k deg F +1. Having in mind (7.3.17), this is only possible if the coefficients of pj/2 , j > 0 in the right-hand side of (7.3.29) vanish. Thus we get for any j > 0, deg F

l=j |α|+r=l−j

√ ( pZ )α √ ∂ α F (l) = 0. Jr,x0 (0, pZ ) (x0 , 0) ∂Z α α!

(7.3.30)

From (7.3.30), we will prove by recurrence that for any j > 0, (7.3.27) holds. As the first step of the recurrence, let us take j = deg F in (7.3.30). Since J0,x0 = IdE (see (7.2.12)), we get F(deg F ) (x0 , 0) = 0, thus (7.3.27) holds for j = deg F . Assume that (7.3.27) holds for j > j0 > 0. Then for j = j0 , the coefficient with r > 0 in (7.3.30) is zero, thus by J0,x0 = IdE , (7.3.30) reads ∂ α F (j0 +|α|) α

∂Z α

(x0 , 0)

√ ( pZ )α = 0. α!

(7.3.31)

From (7.3.31), we get (7.3.27) for j = j0 . The proof of (7.3.27) is complete.

Lemma 7.3.6. We have (i)

∂ α Fx (0, 0) = 0 , ∂z α

|α| i .

(7.3.32)

Therefore Fx (0, z ) = 0 for all i > 0 and z ∈ Cn , i.e., (7.3.10) holds true. Moreover, (7.3.33) Fx(i) (z, 0) = 0 for all i > 0 and all z ∈ Cn . (i)

Proof. Let us start with some preliminary observations. By (7.3.17), (7.3.27) and (7.3.29), by comparing the coefficient of p0 in (7.3.6) and (7.3.28), we get F(i) (x, Z ) = Fx(i) (0, z ) + O(|Z |i+1 ).

(7.3.34)

By the definition (7.3.12) of F , we take the adjoint of (7.3.34) and get F (i) (Z , x) = (Fx(i) (0, z ))∗ + O(|Z |i+1 )

(7.3.35)

which implies ∂ α ∗ ∂ α (i) (i) (0, z ) , F (·, x)|x = α Fx α ∂z ∂z

for |α| i ,

(7.3.36)

so in order to prove the lemma it suffices to show that ∂ α (i) F (·, x)|x = 0 , ∂z α

for |α| i .

(7.3.37)

We prove this by induction over |α|. For |α| = 0 it is obvious that F (i) (0, x) = 0 since we have a homogeneous polynomial of degree i > 0. For the induction step

308


let jX : X → X × X be the diagonal injection. By Lemma 7.3.3 and the definition (7.3.12) of F (i) (x, y), ∂ (i) F (x, y) = 0 , ∂zj

near jX (X),

(7.3.38)

n where y = expX x (Z ). Assume now that α ∈ N and (7.3.37) holds for |α| − 1. Consider j with αj > 0 and set α = (α1 , . . . , αj − 1, . . . , αn ). Taking the derivative of (7.3.12) and using the induction hypothesis and (7.3.38) we have ∂ α (i) ∂ ∗ ∂ α (i) ∂ α ∂ (i) F (·, x)| = j F − F (·, ·) = 0. x ∂z α ∂zj X ∂z α ∂z α ∂zj x 0,0

(7.3.39)

Thus (7.3.32) is proved and this is equivalent to (7.3.10). (7.3.33) follows from (7.3.10) and (7.3.11). This finishes the proof of Lemma 7.3.6. Lemma 7.3.7. We have Fx (z, z ) = 0 for all i > 0 and z, z ∈ Cn . (i)

Proof. Let us consider the operator 1 Lp ⊗E √ Pp ∇η(d(x,y))( ∂ + ∂ )x Tp Pp . ∂zj ∂z j p

(7.3.40)

The leading term of its asymptotic expansion as in (7.3.2) is ∂ √ √ √ √ Fx0 ( p z, p z )Px0 ( p Z, p Z ) . ∂zj

(7.3.41)

By (7.3.10) and (7.3.33), ( ∂z∂ j Fx0 )(z, z ) is an even degree polynomial on z,z whose constant term vanishes. We reiterate the arguments from (7.3.13)–(7.3.36) by replacing the operator Tp with the operator (7.3.40); we get for i > 0, ∂ (i) F (0, z ) = 0. ∂zj x

(7.3.42)

∂ (i) F (z, 0) = 0. ∂z j x

(7.3.43)

By (7.3.11) and (7.3.42),

By continuing this process, we show that for all i > 0, α ∈ Zn , z, z ∈ Cn , ∂ α (i) ∂ α (i) Fx (0, z ) = F (z, 0) = 0. α ∂z ∂z α x Thus the lemma is proved and (7.3.9) holds true. Lemma 7.3.7 finishes the proof of Proposition 7.3.2.

(7.3.44)


309

Proposition 7.3.8. In the sense of notation (7.2.4), we have (7.3.5). Proof. Let us compare the asymptotic expansion of Tp and Tg0 ,p = Pp g0 Pp . In the notation (7.2.10), the expansion (7.2.15) (for k = 1) reads ∼ (g0 (x0 )Px + Q1,x (g0 )Px p−1/2 )(√pZ, √pZ ) + O(p−1 ) , p−n Tg0 ,p,x0 (Z, Z ) = 0 0 0 (7.3.45) since Q0,x0 (g0 ) = g0 (x0 ) by (7.2.17). The expansion (7.3.2) (also for k = 1) takes the form √ √ (7.3.46) p−n Tp,x0 ∼ = (g0 (x0 )Px0 + Q1,x0 Px0 p−1/2 )( pZ, pZ ) + O(p−1 ) where we have used Proposition 7.3.2 and the definition (7.3.4) of g0 . Thus, subtracting (7.3.45) from (7.3.46) we obtain √

√ p−n (Tp − Tg0 ,p )x0 (Z, Z ) ∼ = (Q1,x0 − Q1,x0 (g0 ))Px0 ( pZ, pZ ) p−1/2 + O(p−1 ) . (7.3.47) Thus it suffices to prove: Lemma 7.3.9. F1,x := Q1,x − Q1,x (g0 ) ≡ 0 .

(7.3.48)

Proof. As in Lemma 7.3.3 we show that F1,x is an odd polynomial in z, z . Let (i) (i) F1,x = i0 F1,x be the decomposition of F1,x in homogeneous polynomials F1,x of degree i. We will show that F1,x (z, z ) = 0 for all i > 0 and z, z ∈ Cn . (i)

(7.3.49)

(7.3.49) together with the fact that F1,x is an odd polynomial, hence with vanishing constant term, will prove (7.3.48). The proof of (7.3.49) is similar to that of (7.3.9). (i) (i) Namely, we define as in (7.3.12) the operator F1 , by replacing Fx (0, z ) by (i) F1,x (0, z ), and we set (analogously to (7.3.13)) (i) (F1 Pp ) pi/2 . (7.3.50) Tp,1 = Tp − Pp g0 Pp − ideg F1

r,x0 ∈ C[Z, Z ] such Then by (7.2.15) and (7.3.2), there exist odd polynomials R that the following expansion in the normal coordinates around x0 ∈ X holds: p−n Tp,1,x0 (0, Z ) ∼ =

k

r,x0 Px0 )(0, √pZ )p− r2 + O(p−(k+1)/2 ), (R

(7.3.51)

r=2

for k 2 and |Z | ε /2. This is the analogue of (7.3.14). Now we can repeat (i) with obvious modifications the proof of (7.3.9) and obtain (7.3.9) for F1 . This achieves the proof of Lemma 7.3.9 and of Proposition 7.3.8. By Lemma 7.2.4 and Proposition 7.3.8, p(Tp − Pp g0 Pp ) verifies the condition of Theorem 7.3.1, thus we can continue our process to get (7.3.3), thus the proof of Theorem 7.3.1 is complete.

310


7.4 Algebra of Toeplitz operators The Poisson bracket { , } on (X, 2πω) is defined by: for f, g ∈ C ∞ (X), if ξf is the Hamiltonian vector field generated by f which is defined by 2πiξf ω = df , then {f, g} = ξf (dg).

(7.4.1)

One of our main goals is to show that the set of Toeplitz operators is closed under the composition of operators, so forms an algebra. Theorem 7.4.1. Let f, g ∈ C ∞ (X, End(E)). The product of the Toeplitz operators Tf,p and Tg,p is a Toeplitz operator; more precisely, it admits the asymptotic expansion in the sense of (7.2.5), Tf,p Tg,p =

∞

p−r TCr (f,g),p + O(p−∞ ),

(7.4.2)

r=0

where Cr are bidifferential operators. In particular C0 (f, g) = f g. If f, g ∈ C ∞ (X), we have √ C1 (f, g) − C1 (g, f ) = −1{f, g}, and therefore

√ −1 T{f,g},p + O(p−2 ). [Tf,p , Tg,p ] = p

(7.4.3)

(7.4.4)

Proof. Firstly, it is obvious that Pp Tf,p Tg,p Pp = Tf,p Tg,p . By Lemmas 7.2.2 and 7.2.4, we know Tf,p Tg,p verifies (7.3.1). Now as in (7.2.18), for Z, Z ∈ Tx0 X, |Z|, |Z | < ε/4, we have (Tf,p Tg,p )x0 (Z, Z ) =

Tf,p,x0 (Z, Z )ρ(4|Z |/ε)Tg,p,x0 (Z , Z ) Tx0 X

× κx0 (Z ) dvT X (Z ) + O(p−∞ ).

(7.4.5)

By Lemma 7.2.4 and (7.4.5), we know as in the proof of Lemma 7.2.4, for Z, Z ∈ Tx0 X, |Z|, |Z | < ε/4, we have p−n (Tf,p Tg,p )x0 (Z, Z ) ∼ =

k

k+1 r √ √ (Qr,x0 (f, g)Px0 )( pZ, pZ )p− 2 + O(p− 2 ),

r=0

(7.4.6) and with the notation in (7.1.6), Qr,x0 (f, g) =

r1 +r2 =r

K [Qr1 ,x0 (f ), Qr2 ,x0 (g)].

(7.4.7)

7.4. Algebra of Toeplitz operators

311

Thus Tf,p Tg,p is a Toeplitz operator from Theorem 7.3.1. Moreover, it follows from the proofs of Lemma 7.2.4 and Theorem 7.3.1 that gl = Cl (f, g), where Cl are bidifferential operators. From (7.1.6), (7.2.17) and (7.4.7), we get C0 (f, g)(x) = Q0,x (f, g) = K [Q0,x (f ), Q0,x (g)] = f (x)g(x).

(7.4.8)

By the proof of Theorem 7.3.1 (cf. Proposition 7.3.2, Lemma 7.3.9 and (7.3.4)), we get Q1,x (f, g) = Q1,x (C0 (f, g)),

(7.4.9)

C1 (f, g) = (Q2,x (f, g) − Q2,x (C0 (f, g)))(0, 0). Moreover, by (7.2.17) and (7.4.7), we get Q2,x (f, g) = K [f (x), Q2,x (g)] + K [Q1,x (f ), Q1,x (g)] + K [Q2,x (f ), g(x)].

(7.4.10)

Now Tf,p Pp = Pp Tf,p implies Qr,x (f, 1) = Qr,x (1, f ), we get from (7.4.10), K [J0,x , Q2,x (f )] − K [Q2,x (f ), J0,x ] = K [Q1,x (f ), J1,x ] − K [J1,x , Q1,x (f )] + K [f (x)J0,x , J2,x ] − K [J2,x , f (x)J0,x ]. (7.4.11) Assume now that f, g ∈ C ∞ (X), by (7.4.9), (7.4.10) and (7.4.11), we get C1 (f, g)(x) − C1 (g, f )(x) = K [Q1,x (f ), Q1,x (g)] − K [Q1,x (g), Q1,x (f )] + f (x) K [Q1,x (g), J1,x ] − K [J1,x , Q1,x (g)] − g(x) K [Q1,x (f ), J1,x ] − K [J1,x , Q1,x (f )] . (7.4.12) From (7.2.12), (7.2.17), (7.2.21) and (7.4.12), we get ∂fx ∂gx C1 (f, g)(x) − C1 (g, f )(x) = K K [1, ∂Z (0)Z ], K [1, (0)Z ] j j ∂Zj j ∂gx ∂fx (0)Z ], K [1, (0)Z ] . (7.4.13) − K K [1, ∂Z j j ∂Zj j From (7.1.11), we obtain ∂fx K 1, ∂Z (0)Z = j j

∂fx ∂zi (0)zi

+

∂fx ∂z i (0)z i .

(7.4.14)

Plugging (7.1.11), (7.4.14) into (7.4.13), n 2 ∂fx ∂gx ∂z i (0) ∂zi (0) − a i i=1 √ = −1{f, g} IdEx .

C1 (f, g)(x) − C1 (g, f )(x) =

This finishes the proof of Theorem 7.4.1.

∂fx ∂gx ∂zi (0) ∂z i (0)

IdEx

(7.4.15)

312


The next result and Theorem 7.4.1 show that the Berezin–Toeplitz quantization has the correct semi-classical behavior. Theorem 7.4.2. For f ∈ C ∞ (X, End(E)), the norm of Tf,p satisfies lim Tf,p = f ∞ :=

p→∞

sup

|f (x)(u)|hE /|u|hE .

(7.4.16)

0=u∈Ex ,x∈X

Proof. Take a point x0 ∈ X and u0 ∈ Ex0 with |u0 |hE = 1 such that |f (x0 )(u0 )| = f ∞ . Recall that in Section 7.2, we trivialize the bundles L, E in our normal coordinates near x0 , and eL is the unit frame of L which trivializes L. Moreover, in these normal coordinates, u0 is a trivial section of E. Considering the sequence of sections Sxp0 = p−n/2 Pp (e⊗p L ⊗ u0 ), we have by (4.2.1), Tf,p Sxp0 − f (x0 )Sxp0 L2

p C √ 2 p Sx0 L .

(7.4.17)

The proof of (7.4.16) is complete.

∞ Remark 7.4.3. If E = C, (X), we associated by Theorem 7.4.1 thenl to f, g ∈ C ∞ C (f, g) ∈ C (X)[[]], where Cl are bidifferential a formal power series ∞ l l=0 operators. Therefore, we have constructed in a canonical way an associative star ∞ product f ∗ g = l=0 l Cl (f, g).

7.5 Toeplitz operators on non-compact manifolds We assume that (X, Θ), (L, hL ) and (E, hE ) satisfy the same hypothesis as in Theorem 6.1.1 or (6.1.3). Especially, (X, Θ) is now a complete Hermitian manifold. ∞ Let Cconst (X, End(E)) denote the algebra of smooth sections of X which are ∞ a constant map outside a compact set. For any f ∈ Cconst (X, End(E)), we consider the Toeplitz operator (Tf,p )p1 as in (7.2.6) Tf,p : L2 (X, Lp ⊗ E) −→ L2 (X, Lp ⊗ E) ,

Tf,p = Pp f Pp .

(7.5.1)

The following result generalizes Theorem 7.4.1 to non-compact manifolds. ∞ Theorem 7.5.1. Let f, g ∈ Cconst (X, End(E)). The product of the two corresponding Toeplitz operators admits the asymptotic expansion (7.4.2) in the sense of (7.2.3). Where Cr are bidifferential operators, especially,

supp(Cr (f, g)) ⊂ supp(f ) ∩ supp(g),

and C0 (f, g) = f g.

∞ ∞ (X), then (7.4.4) holds. Theorem 7.4.2 also holds for f ∈ Cconst (X). If f, g ∈ Cconst

Proof. The most important observation here is that for p large enough, by the 0 spectral gap property, Theorem 1.5.5, as in (4.1.14), for any s ∈ H(2) (X, Lp ⊗ E), we have F (Dp )s = Pp s,

F (Dp ) − Pp = O(p−∞ ),

(7.5.2)

7.5. Toeplitz operators on non-compact manifolds

313

moreover, by the proof of Proposition 4.1.5, for any compact set K, l, m ∈ N, ε > 0, there exists Cl,m,ε > 0 such that for p ≥ 1, x, x ∈ K, |F (Dp )(x, x ) − Pp (x, x )|C m (K×K) Cl,m,ε p−l .

(7.5.3)

As explained in Section 4.1.2, F (Dp )(x, ·) only depends on the restriction of Dp to B X (x, ε) and is zero outside B X (x, ε). For g ∈ C0∞ (X, End(E)), let (F (Dp )gF (Dp ))(x, x ) be the smooth kernel of F (Dp )gF (Dp ) with respect to dvX (x ). Then for any relative compact open set U in X such that supp(g) ⊂ U , we have from (7.5.2) and (7.5.3), Tp,g − F (Dp )gF (Dp ) = O(p−∞ ), Tp,g (x, x ) − (F (Dp )gF (Dp ))(x, x ) = O(p−∞ )

on U × U.

(7.5.4)

Now we fix f, g ∈ C0∞ (X, End(E)). Let U ⊂ W be relative compact open sets in X such that supp(f ) ∪ supp(g) ⊂ U and d(x, y) > 2ε for any x ∈ supp(f ) ∪ supp(g), y ∈ X U . From (7.5.2), we have Tf,p Tg,p = Pp F (Dp )f Pp gF (Dp )Pp .

(7.5.5)

Let (F (Dp )f Pp gF (Dp ))(x, x ), be the smooth kernel of F (Dp )f Pp gF (Dp ) with respect to dvX (x ). Then the support of (F (Dp )f Pp gF (Dp ))(·, ·) is contained in U × U . If we fix x0 ∈ U , the kernel of F (Dp )f Pp gF (Dp ) has exactly the same asymptotic expansion as in the compact case by (7.5.3). More precisely, as in (7.4.6), we have p−n (F (Dp )f Pp gF (Dp ))x0 (Z, Z ) ∼ =

k k+1 r √ √ (Qr,x0 (f, g)Px0 )( pZ, pZ )p− 2 + O(p− 2 ),

(7.5.6)

r=0

with the same local formula for Qr,x0 (f, g) given in (7.4.7). But since all formal computation is local, we have studied well Qr,x0 (f, g), which is a polynomial with coefficients as bidifferential operators acting on f and g. Thus we know from (7.5.4) that there exist (gl )l0 , gl ∈ C0∞ (X, End(E)), supp(gl ) ⊂ supp(f ) ∩ supp(g) such that for any m 1, s ∈ L2 (X, Lp ⊗ E), k F (Dp )Pp gl p−l Pp F (Dp )s F (Dp )f Pp gF (Dp )s −

L2

l=0

C sL2 . pk+1

(7.5.7)

(7.5.5) and (7.5.7) imply that k T − Pp gl p−l Pp Ck p−k−1 . Tf,p g,p

(7.5.8)

l=0

For the last part of Theorem 7.5.1, we repeat the proof of Theorem 7.4.2. We conclude thus Theorem 7.5.1.

314


Remark 7.5.2. By Sections 7.2–7.4, we know we can associate to any f, g ∈ ∞ that l C (f, g) ∈ C ∞ (X, End(E))[[]], C ∞ (X, End(E)) a formal power series l l=0 where Cl are bidifferential operators. This follows from the fact that the construction in Section 7.4 is local. However, the problem here is which Hilbert space they act on. Theorem 7.5.1 claims that the space of holomorphic L2 -sections 0 H(2) (X, Lp ⊗ E) of Lp ⊗ E, is a suitable Hilbert space which allows the Berezin– ∞ Toeplitz quantization of the algebra Cconst (X, End(E)).

Problems Problem 7.1. Using (4.1.4) and (4.1.7), deduce that |Pp (x, x )|C 0 (X×X) Cpn , for some constant C > 0. Problem 7.2. If f ∈ C ∞ (X), verify that in (7.2.16), ∂fx Q2,x (f ) = f (x)J2,x + K J1,x , ∂Z (0)Z j j ∂fx + K 1, ∂Zj (0)Zj J1,x +

∂ α fx Zα ∂Z α (0) α!

.

|α|=2 √

Assume moreover that ω =

−1 L 2π R

is the K¨ ahler form of (T X, g T X ), then

1 (∆f )(x) pn−1 + O(pn−2 ), Tf,p (x, x) = f (x)pn + b1 (x)f (x) − 4π with b1 in (4.1.8). (Hint: by (4.1.111), J1,x = 0 and J2,x (0, 0) = b1 (x).)

7.6 Bibliographic notes The Berezin–Toeplitz quantization was studied by Bordemann–Meinrenken–Schlichenmaier [42], [211] (cf. the references therein for earlier special cases). They √ −1 L TX is the K¨ ahler metric associated to ω = 2π R and the consider the case when g twisting bundle E is trivial. Under these assumptions, they established Theorems 7.4.1 and 7.4.2 (cf. also [135], [63]). We also have the Berezin–Toeplitz quantization for a pseudoconvex domain by Englis [102, 103]. All the above works are based on the Boutet de Monvel–Sj¨ ostrand parametrix for the Szeg¨ o kernel [53], [105], and the theory of Toeplitz operators of Boutet de Monvel–Guillemin [52]. The approach used here is taken from [162]. In contrast to the previous approaches, we establish directly the results as a consequence of our full asymptotic expansion of Bergman kernel (Theorem 4.2.1). The present approach and the results from Section 5.4.3 also imply the Berezin–Toeplitz quantization for complex orbifolds.

Chapter 8

Bergman Kernels on Symplectic Manifolds In this chapter, we study the asymptotic expansion of the Bergman kernel associated to modified Dirac operators and renormalized Bochner Laplacians on symplectic manifolds. We will also explain some applications of the asymptotic expansion in the symplectic case. One is, for example, the extension of the Berezin– Toeplitz quantization studied in Chapter 7. We also find Donaldson’s Hermitian scalar curvature as the second coefficient of the expansion. This chapter is organized as follows: In Section 8.1, we study the asymptotic expansion of the Bergman kernel associated to modified Dirac operators and the corresponding Toeplitz operators. In Section 8.2, we obtain the corresponding results when we allow line bundles with mixed curvature (In Section 8.1, we suppose that the curvature of the line bundle is positive). In Section 8.3, we establish the asymptotic expansion for the renormalized Bochner Laplacian.

8.1 Bergman kernels 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 ) with curvatures RE , RL . Let g T X be any Riemannian metric on T X compatible with J. We assume that the positivity condition (1.5.21) holds for RL . This section is organized as follows: In Section 8.1.1, we establish the asymptotic expansion of Bergman kernels of modified Dirac operators by adapting the arguments in Chapter 4; again the spectral gap property plays an essential role

316

Chapter 8. Bergman Kernels on Symplectic Manifolds

here. In Section 8.1.2, we establish the theory on Toeplitz operators associated to the modified Dirac operators in the symplectic case. We use the notation from Section 1.5.2 and from (1.5.14)–(1.5.19).

8.1.1 Asymptotic expansion of the Bergman kernel Let ∇det be a Hermitian connection on det(T ∗(0,1) X) with curvature Rdet . We denote by Dpc the spinc Dirac operator associated to Lp ⊗E and ∇det as in (1.3.15). Following (1.6.1), we denote Ej = Λj (T ∗(0,1) X) ⊗ E, E = E+ ⊕ E− ,

E− := ⊕j E2j+1 ,

Ep− = E− ⊗ Lp ,

E+ := ⊕j E2j ,

Ep+ = E+ ⊗ Lp ,

Ep = Ep+ ⊕ Ep− .

(8.1.1)

For A ∈ Λ3 (T ∗ X), let ∇A be the Hermitian connection on Ep induced by p ∇ , A and ∇L , ∇E as in (1.3.33). Let Dpc,A be the modified Dirac operator on X defined in (1.5.27). Cl

Definition 8.1.1. The Bergman kernel PpA (x, x ), (x, x ∈ X), is the smooth kernel of the orthogonal projection PpA , from Ω0,• (X, Lp ⊗ E) onto Ker(Dpc,A ), with respect to the Riemannian volume form dvX (x ). Hence, PpA (x, x ) ∈ (Ep )x ⊗ (Ep )∗x , and PpA (x, x) ∈ End(Λ(T ∗(0,1) X) ⊗ E)x . By using the spectral gap property (Theorem 1.5.8), and the same proof of Proposition 4.1.5, we obtain: Theorem 8.1.2. For any l, m ∈ N, ε > 0, there exists Cl,m,ε > 0 such that for any p 1 the following estimate holds: |PpA (x, x )|C m Cl,m,ε p−l

for any x, x ∈ X with d(x, x ) ε.

(8.1.2)

The C m norm used here is induced by ∇T X , ∇L , ∇E and hL , hE , g T X . We denote by IC⊗E the projection from Λ(T ∗(0,1) X) ⊗ E onto C ⊗ E corresponding to the decomposition Λ(T ∗(0,1) X) = C ⊕ Λ>0 (T ∗(0,1) X). Theorem 8.1.3. There exist smooth coefficients br (x) ∈ End(Λ(T ∗(0,1) X) ⊗ E)x where b0 = det(R˙ L /(2π))IC⊗E ,

(8.1.3)

with the following property: for any k, l ∈ N, there exists Ck,l > 0 such that k A br (x)pn−r Pp (x, x) − r=0

for any x ∈ X, p ∈ N∗ .

Cl

Ck,l pn−k−1 .

(8.1.4)

8.1. Bergman kernels of modified Dirac operators

317

The coefficients br (x) are polynomials in A, RT X , Rdet , RE (and RL ) and their derivatives with order 2r − 1 (resp. 2r) and reciprocals of linear combinations of eigenvalues of R˙ L at x. Moreover, the expansion (8.1.4) is uniform in the following sense : for any fixed k, l ∈ N, assume that the derivatives of g T X , hL , ∇L , hE , ∇E , J, with 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 on the parameters. Let x0 ∈ X. As in Section 4.1.3, we trivialize Ep on B X (x0 ,4ε) B Tx0 X (0,4ε) by using the parallel transport with respect to the connection ∇A along the curve [0, 1] u → uZ. Under this trivialization, we have PpA (Z, Z ) ∈ End(Ep,x0 ) = A (Z, Z ) := PpA (Z, Z ), (Z, Z ∈ End(Ex0 ) for Z, Z ∈ Tx0 X. Thus we can view Pp,x 0 ∗ Tx0 X, |Z|, |Z | 2ε), as a smooth section of π (End(E)) over T X ×X T X as in Section 4.1.5, and we denote by | · |C m (X) the C m -norm with respect to the parameter x0 ∈ X. Recall that µ0 and κ were defined in (1.5.26), (4.1.28). Recall also that P(Z, Z ) is the smooth kernel of the orthogonal projection from L2 (R2n ) onto Ker(L ) defined in (4.1.84). The following off-diagonal expansion of the Bergman kernel, which is an analogue of Theorem 4.2.1, is the main result of this section. Theorem 8.1.4. 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 polynomials in A, RT X , Rdet , RE ( and RL ) and their derivatives of order r − 1 ( resp. 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 ) = IC⊗E ,

(8.1.5)

and such that there exists C > 0 such that for k, m, m ∈ N, there exist C > 0, N ∈ N such that if p 1, Z, Z ∈ Tx0 X, |Z|, |Z | 2ε, α, α ∈ N2n , |α| + |α | m, then $ % k ∂ |α|+|α | √ √ −r/2 1 A 1/2 1/2 P (Z, Z )κ (Z)κ (Z ) − Fr ( pZ, pZ )p α α ∂Z ∂Z m pn p r=0 C (X) √ √ √ N −(k+1−m)/2 −∞ Cp (1 + | pZ| + | pZ |) exp(− C µ0 p|Z − Z |) + O(p ). (8.1.6) Remark 8.1.5. By Theorem 1.5.7 and because (Dpc,A )2 preserves the Z2 -grading of Ω0,• (X, Lp ⊗E), PpA is the orthogonal projection from C ∞ (X, Ep+ ) onto Ker(Dpc,A ) for p large enough. Thus PpA (x, x), br (x) ∈ End(E+ )x and Jr (Z, Z ) ∈ End(E+ )x0 .

318


As in Theorem 4.2.3, the following result relates the coefficients of the expansion of the Bergman kernel and the heat kernel. Theorem 8.1.6. There exist smooth sections br,u of End(Λ(T ∗(0,1) X) ⊗ E) on X which are polynomials in A, RT X , Rdet , RE (and RL ) and their derivatives with order 2r − 1 (resp. 2r) and functions on the eigenvalues of R˙ L at x, and b0,u in (4.2.3) such that for each u > 0 fixed, we have the asymptotic expansion in the sense of (8.1.4) as p → ∞, k u c,A 2 exp(− (Dp ) )(x, x) = br,u (x)pn−r + O(pn−k−1 ). p r=0

(8.1.7)

Moreover, as u → +∞, with br in (8.1.4), we have br,u (x) = br (x) + O(e− 8 µ0 u ). 1

(8.1.8)

Proof of Theorems 8.1.3–8.1.6. Observe that in the construction from Section 4.1.3, we only need to replace the holomorphic connection ∇det on det(T (1,0) X) by the Hermitian connection ∇det considered here, and − 14 Tas by A. Then we get the modified Dirac operator Dpc,A0 on R2n . Now in Theorem 4.1.7, we only need 0,• to replace RB,Λ , − 41 Tas therein by RCl and A. Then the arguments in Section 4.1 go through until (4.1.96). In particular, by using (1.3.11), we obtain Fr from (8.1.5) in the same way as in Theorem 4.1.21. Of course, here Fr will not preserve the Z-grading on Ex0 = (Λ(T ∗(0,1) X) ⊗ E)x0 . Since (Dpc,A )2 preserves the Z2 -grading on C ∞ (X, Ep ) and is invertible on C ∞ (X, Ep− ) for p large enough, then in view of our construction, L2t preserves the Z2 -grading on L2 (R2n , Ex0 ) and is invertible on L2 (R2n , E− x0 ) for t small enough. Consequently, we know that for p large enough, PpA (Z, Z ) and Fr (Z, Z ) from (8.1.6) are zero when we restrict to E− x0 . We run now the arguments from Section 4.2 without any change, concluding the proof of Theorems 8.1.4 and 8.1.6. Remark 8.1.7. As in Remark 4.1.4, the formula (8.1.7) holds without the assumption (1.5.21) and b0,u is computed in Theorem 1.6.1.

8.1.2 Toeplitz operators on symplectic manifolds We use the same notation as in Chapter 7. In view of Definition 7.3.1, it is natural to introduce the Toeplitz operator on symplectic manifolds. Definition 8.1.8. A Toeplitz operator is a family {Tp } of linear operators Tp : L2 (X, Ep ) −→ L2 (X, Ep ) , with the properties:

(8.1.9)

8.1. Bergman kernels of modified Dirac operators

(i) For any p ∈ N, we have

Tp = PpA Tp PpA .

(ii) There exist a sequence gl ∈ C exists Ck > 0 with

∞

319

(8.1.10)

(X, End(E)) such that for all k 0 there

k p−l gl PpA Ck p−k−1 , Tp − PpA

(8.1.11)

l=0

where · denotes the operator norm on the space of bounded operators, and gl acts on Λ(T ∗(0,1) X) ⊗ Lp ⊗ E as IdΛ(T ∗(0,1) X)⊗Lp ⊗gl . ∞ The full symbol of {Tp } is the formal series l=0 l gl ∈ C ∞ (X, End(E))[[]] and the principal symbol of {Tp } is g0 . If each Tp is self-adjoint, {Tp } is called selfadjoint. Certainly, for f ∈ C ∞ (X, End(E)), Lemma 7.2.2 still holds for the kernel Tf,p (x, x ) of the Toeplitz operator Tf,p by Theorems 8.1.2 and 8.1.4. Now let Ξp : L2 (X, Ep ) −→ L2 (X, Ep ) be a family of linear operator with smooth kernel Ξp (x, y). We use the same notation as in (7.2.10) except that here Qr,x0 ∈ End(Λ(T ∗(0,1) X) ⊗ E)x0 [Z, Z ]. Then Lemmas 7.2.3 and 7.2.4 still hold and Jr,x0 , Qr,x0 (f ) ∈ End(Λ(T ∗(0,1) X) ⊗ E)x0 [Z, Z ].

(8.1.12)

Now (7.2.12) and (7.2.17) need to be replaced by J0,x0 = IC⊗E ,

Q0,x0 (f ) = f (x0 )IC⊗E .

(8.1.13)

We have still the following criterion for Toeplitz operators. Theorem 8.1.9. If Tp : L2 (X, Ep ) −→ L2 (X, Ep ) be a family of bounded linear operators verifying i), ii), iii) of Theorem 7.3.1 by replacing Pp therein by PpA , and with {Qr,x0 ∈ End(Λ(T ∗(0,1) X) ⊗ E)x0 [Z, Z ]}x0 ∈X ; then {Tp } is a Toeplitz operator. Proof. We claim first Q0,x0 ∈ End(Ex0 ) ◦ IC⊗E [Z, Z ], and Q0,x0 is a polynomial in z, z . In fact, by (7.2.13) and (8.1.13), analogous to (7.3.7), we have p−n (PpA Tp PpA )x0 (Z, Z ) √ √ ∼ = ((PJ0 ) ◦ (Q0 P) ◦ (PJ0 ))x0 ( pZ, pZ ) + O(p−1/2 ). (8.1.14) Since PpA Tp PpA = Tp , we deduce from (7.3.6), (8.1.13) and (8.1.14) that Q0,x0 Px0 = IC⊗E Px0 ◦ (Q0,x0 Px0 ) ◦ Px0 IC⊗E ,

(8.1.15)

hence Q0,x0 ∈ End(Ex0 ) ◦ IC⊗E [z, z ] by (7.1.5) and (7.1.8). By the same proof as of Proposition 7.3.2, we get Q0,x0 (Z, Z ) = Q0,x0 (0, 0) ∈ End(Ex0 ) ◦ IC⊗E .

(8.1.16)

320


Now we set g0 (x0 ) = Q0,x0 (0, 0)|C⊗E ∈ End(Ex0 ).

(8.1.17)

Then as in the proof of Proposition 7.3.8, we get the analogue of (7.3.5): Tp = PpA g0 PpA + O(p−1 ).

(8.1.18)

Since p(Tp − PpA g0 PpA ) verifies the condition of Theorem 8.1.9, we can continue the process to generate a sequence (gl )l0 , gl ∈ C ∞ (X, End(E)), such that (8.1.11) holds for any k 1. This completes the proof of Theorem 8.1.9. As in the holomorphic case, the set of Toeplitz operators is closed under the composition of operators, so forms an algebra. Theorem 8.1.10. Let f, g ∈ C ∞ (X, End(E)). The product of the Toeplitz operators Tf,p and Tg,p is a Toeplitz operator; more precisely, it admits the asymptotic expansion (7.4.2) where Cr are bidifferential operators, and Cr (f, g) ∈ C ∞ (X, End(E)). In particular C0 (f, g) = f g. If f, g ∈ C ∞ (X), then (7.4.4) holds. The norm of Tf,p satisfies (7.4.16), for any f ∈ C ∞ (X, End(E)). Remark 7.4.3 about star-products still holds if E = C. Proof. The same proof as in Section 7.4 delivers Theorem 8.1.10.

8.2 Bergman kernel: mixed curvature case In this section, we do not suppose that the almost complex structure polarizes the curvature of the line bundle L, that is we allow line bundles with mixed curvature (negative and positive eigenvalues). We will extend the results from Section 8.1 to this situation. This section is organized as follows: In Section 8.2.1, we establish the spectral gap property for the modified Dirac operator in the mixed curvature case, by simply repeating the argument in the proof of Theorem 1.5.7. In Section 8.2.2 we explain that the asymptotic expansion from Section 8.1 still holds, and compute the first coefficients in the expansion. We use the same notation as in Section 8.1. We choose the almost complex structure J such that ω is J-invariant, i.e., ω(·, ·) = ω(J·, J·). But we only suppose that ω is non-degenerate, and we do not suppose that ω(·, J·) is positive. This is the difference comparing with the assumption in Section 8.1.

8.2.1 Spectral gap Under our assumption, R˙ L ∈ End(T (1,0) X) is still well defined by (1.5.15), and for any x ∈ X, we can diagonalize R˙ xL , i.e., find an orthonormal basis {wj }nj=1 of T (1,0) X such that R˙ xL wj = aj (x)wj with aj (x) ∈ R. As ω is non-degenerate,

8.2. Bergman kernel: mixed curvature case

321 (1,0)

the number of negative eigenvalues of R˙ xL ∈ End(Tx X) does not depend on x, and we denote it by q. (In Section 8.1, we suppose that q = 0.) From now on, we assume that R˙ xL wj = aj (x) wj ,

aj (x) < 0 for j q and aj (x) > 0 for j > q.

(8.2.1)

Then the vectors {wj }qj=1 span a sub-bundle W of T (1,0) X. Set ω d (x) = −

n

aj (x)w j ∧ iwj +

j=1

=

q

τ(x) =

aj (x)

j=1

aj (x)iwj ∧ wj −

j=1 n

q

n

aj (x)w j ∧ iwj , (8.2.2)

j=q+1

|aj (x)| = −

j=1

q j=1

aj (x) +

n

aj (x) ,

j=q+1

µ0 = inf |aj (x)| . x∈X, j

The following result extends Theorem 1.5.7 to the current situation. We denote Ω0,=q (X, Lp ⊗ E) := k=q Ω0,k (X, Lp ⊗ E). Theorem 8.2.1. There exists C > 0 such that for any p ∈ N, Dpc,A s2L2 (2pµ0 − C)s2L2 ,

for s ∈ Ω0,=q (X, Lp ⊗ E).

(8.2.3)

Proof. We consider now s ∈ C ∞ (X, Lp ⊗ E), where E = Λ(T ∗(0,1) X) ⊗ E. We will apply (1.5.30) to the almost complex structure J ∈ End(T X) defined by √ √ J wj = − −1wj for j q and J wj = −1wj for j > q, then τ is τ associated to J in (1.5.16). By (1.5.30), there exists C > 0 such that for any p ∈ N, s ∈ C ∞ (X, Lp ⊗ E), we have Lp ⊗E 2 ∇ sL2 − p τ (x)s, s −Cs2L2 .

(8.2.4)

By (1.3.35) and (8.2.2), comparing to (1.5.19) and (1.5.34), we have Dpc,A s2 =∇A s2 − p τ s, s − 2p ωd s, s X r 1 + c (RE + Rdet) + c (dA) − 2|A|2 s, s . + 4 2

(8.2.5)

If s ∈ Ω0,=q (X, Lp ⊗ E), the third term of (8.2.5), −2p ωd s, s is bounded below by 2pµ0 s2L2 , while the fourth term of (8.2.5) is O(s2L2 ). The proof of (8.2.3) is completed.

322


From Theorem 8.2.1, the same proof as in Theorem 1.5.8 gives the following spectral gap property. Theorem 8.2.2. There exists CL > 0 such that for p ∈ N, Spec((Dpc,A )2 ) ⊂ {0} ∪ [2pµ0 − CL , +∞[.

(8.2.6)

Set oq = − if q is even; oq = + if q is odd, then for p large enough, we have ) = {0}. Ker(Doc,A q ,p

(8.2.7)

8.2.2 Asymptotic expansion of the Bergman kernel The existence of the spectral gap expressed in Theorem 8.2.2 allows us to obtain immediately the analogue of (4.1.12). Namely, for any l, m ∈ N and ε > 0, there exists Cl,m,ε > 0 such that for p 1, x, x ∈ X, d(x, x ) > ε, |PpA (x, x )|C m Cl,m,ε p−l .

(8.2.8)

Recall that W is the subbundle of T (1,0) X spanned by the eigenvectors of ∗ negative eigenvalues of R˙ L ∈ End(T (1,0) X). We denote by (det(W ))⊥ the or∗ thogonal complement of det(W ) in Λ(T ∗(0,1) X). We denote by Idet(W ∗ )⊗E the ∗

orthogonal projection from E = Λ(T ∗(0,1) X) ⊗ E onto det(W ) ⊗ E. We use the trivialization of Ep on the normal coordinate B X (x0 , 4ε) Tx0 X A B (0, 4ε) as in Section 8.1. We view Pp,x (Z, Z ) := PpA (Z, Z ), (Z, Z ∈ Tx0 X, 0 ∗ |Z|, |Z | 2ε), as a smooth section of π (End(E)) over T X ×X T X. We still use the notation in Section 4.1.6. Then with ω d,x0 , τx0 introduced in (8.2.2), we get that the limit operator L20 is &= − L

q n (∇0,ei )2 − τx0 = b+ b + bj b+ j j j , i

j=1

j=q+1

(8.2.9)

&− 2 ωd,x0 . L20 = L Note that by (8.2.2), we have for ω d,x0 ∈ End(Λ(T ∗(0,1) X))x0 , ∗

Ker( ωd,x0 ) = det(W )x0 , ω d,x0 −µ0

∗

on (det(W ))⊥ x0 .

(8.2.10)

Set z = (z 1 , . . . , z q , zq+1 , . . . , zn ), b = (b+ , . . . , b+ , bq+1 , . . . , bn ). 1

q

(8.2.11)

8.2. Bergman kernel: mixed curvature case

323

Then by the same argument as the proof of Theorem 4.1.20, we have: & to L2 (R2n ) is given by Theorem 8.2.3. The spectrum of the restriction of L n & Spec(L |L2 (R2n ) ) = 2 αi |ai | : α = (α1 , . . . , αn ) ∈ Nn

(8.2.12)

i=1

and an orthogonal basis of the eigenspace of 2

n i=1

bα zβ exp − 1 |ai | |zi |2 , 4 i

αi |ai | is given by

with β ∈ Nn .

(8.2.13)

& from & Z ) be the smooth kernel of the orthogonal projection P Let P(Z, 2n & (L (R ), ·L2 ) onto Ker(L ), calculated with respect to dvT X (Z ). From (8.2.13) we get 2

& Z ) = P(Z,

n 1

exp − |ai | |zi |2 + |zi |2 2π 4 i=1

n |ai | i=1

q n 1 |ai | z i zi + |ai | zi z i . (8.2.14) − 4 i=1 i=q+1

Let P N be the orthogonal projection from (L2 (R2n , Ex0 ), · L2 ) onto N = Ker(L20 ), and P N (Z, Z ) its smooth kernel with respect to dvT X (Z ). Hence (8.2.10) yields & Z )I ∗ P N (Z, Z ) = P(Z, (8.2.15) det(W )⊗E . From the above discussion we deduce: Theorem 8.2.4. Theorems 8.1.2, 8.1.3 and 8.1.6 hold with b0,u given by (4.2.3) and to b0 , J0 from (8.1.3), (8.1.5) correspond now to (8.2.16) b0 = det(R˙ L /(2π)) Idet(W ∗ )⊗E , J0 (Z, Z ) = Idet(W ∗ )⊗E . & Z ) is given by (8.2.14). The corresponding P(Z, The Toeplitz operators are introduced as in Definition 8.1.8 by using the more general meaning of PpA considered in this section. Theorem 8.2.5. Theorem 8.1.10 holds under the assumption of this section. Proof. Observe that the kernel calculus for P presented in Section 7.1 has the & (cf. (8.2.14)). Basically, we only need to replace b, z corresponding version for P therein by b, z. If we denote by K&[F, G] ∈ C[Z, Z ] the operation associated to

324


& as in (7.1.6), then by exactly the same computation, we get the analogue of P (7.4.13) by replacing K by K&: for f, g ∈ C ∞ (X), ∂fx &[1, ∂gx (0)Zj ] C1 (f, g)(x) − C1 (g, f )(x) = K& K&[1, ∂Z (0)Z ], K j ∂Z j j ∂gx ∂fx & & & (0)Z ] . (8.2.17) − K K [1, ∂Zj (0)Zj ], K [1, ∂Z j j Now our theorem follows from Problem 8.2.

Remark 8.2.6. We specify now the preceding results to the complex case and we return to the notation from Section 1.5.1. Then our assumption here is that the curvature RL is non-degenerate as a 2-form on X. Dp in (1.5.20) is a modified Dirac operator by (1.4.17). Thus all results in this section apply. As Dp2 = 2p preserves the Z-grading on Ω0,• (X, Lp ⊗ E), we get from (8.2.3), that for p large enough, H k (X, Lp ⊗ E) = 0 for any k = q. (8.2.18) Moreover, for p large enough, the Bergman kernel Pp (x, x ) is in this case the kernel of the orthogonal projection from Ω0,q (X, Lp ⊗ E) on Ker(Dp2 ) H q (X, Lp ⊗ E).

8.3 Generalized Bergman kernel 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 , RE be the curvature of ∇L , ∇E . Let g T X be any Riemannian metric on T X compatible with J. In this section, we assume that the positivity condition (1.5.21) holds for RL . We still use the notation in Sections 1.5.2 and 8.1.1. This section is organized as follows: In Section 8.3.1, we establish the spectral gap property for the renormalized Bochner Laplacian. In Section 8.3.2, we state the asymptotic expansion for the generalized Bergman kernel. In Section 8.3.3, we study the near diagonal asymptotic expansion. In particular, we obtain Theorem 8.3.3 as a special case of Theorem 8.3.8. In Section 8.3.4 we prove Theorem 8.3.4 and calculate the second coefficient of the expansion; one of the summands thereof is the Hermitian scalar curvature. In Section 8.3.5, we establish a symplectic version of the Kodaira embedding Theorem. Especially, this gives another analytic proof of the classical Kodaira embedding Theorem 5.1.12.

8.3.1 Spectral gap Let ∆L ⊗E be the Bochner Laplacian acting on C ∞ (X, Lp ⊗ E) associated to ∇L , ∇E . We fix a smooth Hermitian section Φ of End(E) on X, i.e., Φ∗ = Φ. The p

8.3. Generalized Bergman kernel

325

renormalized Bochner Laplacian ∆p,Φ on Lp ⊗ E is defined by ∆p,Φ = ∆L

p

⊗E

− pτ + Φ.

(8.3.1)

Recall that µ0 was defined in (1.5.26). L , CL > 0 such that for p ∈ N, Theorem 8.3.1. There exist C L , C L ] ∪ [2pµ0 − CL , +∞[ . Spec(∆p,Φ ) ⊂ [−C

(8.3.2)

L , C L ]. Then for p Let Hp be the eigenspace of ∆p,Φ with the eigenvalues in [−C large enough, Td(T (1,0) X) ch(Lp ⊗ E). (8.3.3) dim Hp = X

Proof. Let Ip : Ω0,• (X, Lp ⊗ E) −→ C ∞ (X, Lp ⊗ E) be the orthogonal projection. For s ∈ Ω0,• (X, Lp ⊗ E), we will denote by s0 = Ip s its 0-degree component. c is the Recall that Dpc is the spinc Dirac operator as in Section 8.1, and D+,p c ∞ + c restriction of Dp on C (X, Ep ). We will estimate ∆p,Φ on Ip (Ker(D+,p )) and c ))⊥ ∩ C ∞ (X, Lp ⊗ E). (Ker(D+,p In the sequel we denote with C all positive constants independent of p, although there may be different constants for different estimates. From (1.5.31) and (1.5.32) by taking ∇det = ∇det1 , A = 0, there exists C > 0 such that for s ∈ C ∞ (X, Lp ⊗ E), c 2 D s 2 − ∆p,Φ s, s Cs2 2 . (8.3.4) p L L Theorem 1.5.8 and (8.3.4) show that there exists C > 0 such that for p ∈ N, ∆p,Φ s, s (2pµ0 − C)s2L2 ,

c for s ∈ C ∞ (X, Lp ⊗ E) ∩ (Ker(D+,p ))⊥ . (8.3.5)

c We focus now on elements from Ip (Ker(D+,p )), and assume s ∈ Ker(Dpc ). Set 0,>0 p 0,q p s = s − s0 ∈ Ω (X, L ⊗ E) = ⊕q1 Ω (X, L ⊗ E). By (1.5.30) and (1.5.34),

−2pωds, s Cs2L2 .

(8.3.6)

We obtain thus from (1.5.19), (1.5.21) and (8.3.6), for p 1, s L2 Cp−1/2 s0 L2 .

(8.3.7)

In view of (1.5.34) and (8.3.7), ∇Cl s2L2 − pτ (x)s0 , s0 Cs0 2L2 .

(8.3.8)

By (1.3.8) with ∇det = ∇det1 , ∇Cl s = ∇L

p

⊗E

s0 + A2 s2 + α ,

(8.3.9)

326


where s2 is the component of degree 2 of s, A2 is a contraction operator coming from the middle term of (1.3.8), and α ∈ Ω0,>0 (X, Lp ⊗ E). By (8.3.8) and (8.3.9), we know Lp ⊗E 2 ∇ s0 + A2 s2 L2 − pτ (x)s0 , s0 Cs0 2L2 , (8.3.10) and by (8.3.7) and (8.3.10), Lp ⊗E 2 ∇ s0 L2 Cps0 2L2 . By (8.3.7) and (8.3.11), we get Lp ⊗E 2 2 p p ∇ s0 + A2 s2 L2 ∇L ⊗E s0 L2 − 2∇L ⊗E s0 L2 A2 s2 L2 2 p ∇L ⊗E s0 L2 − Cs0 2L2 .

(8.3.11)

(8.3.12)

Thus, (8.3.10) and (8.3.12) yield Lp ⊗E 2 ∇ s0 L2 − pτ (x)s0 , s0 Cs0 2L2 .

(8.3.13)

L > 0 such that By (1.5.30) and (8.3.13), there exists a constant C

c ∆p,Φ s, s C L s2 2 , s ∈ Ip (Ker(D+,p )) . L

(8.3.14)

c c By (8.3.7), we know that Ip : Ker(D+,p ) −→ Ip (Ker(D+,p )) is bijective for p large enough, and c c C ∞ (X, Lp ⊗ E) = Ip (Ker(D+,p )) ⊕ (Ker(D+,p ))⊥ ∩ C ∞ (X, Lp ⊗ E) .

(8.3.15)

The proof is now reduced to a direct application of the minimax principle for the operator ∆p,Φ . It is clear that (8.3.5) and (8.3.14) still hold for elements in the Sobolev space H 1 (X, Lp ⊗ E), which is the domain of the quadratic form p 2 Qp (s) = ∇L ⊗E sL2 − pτ (x)s, s + Φs, s associated to ∆p,Φ . Let µp1 µp2 · · · µpj · · · , (j ∈ N), be the eigenvalues of ∆p,Φ . By the minimax principle (C.3.3), min max Qp (s) (8.3.16) µpj = V ⊂Dom(Qp ) s∈V , sL2 =1

where V runs over the subspaces of dimension j of Dom(Qp ). c L , for j dim Ker(D+,p ). Moreover, By (8.3.14) and (8.3.16), we know µpj C c any subspace V ⊂ Dom(Qp ) with dim V dim Ker(D+,p ) + 1 contains an element c ))⊥ . By (8.3.5), (8.3.16), we obtain µpj 2pµ0 − CL , for 0 = s ∈ V ∩ (Ker(D+,p c j dim Ker(D+,p ) + 1. Thus we get (8.3.2) and c dim Hp = dim Ker(D+,p ).

(8.3.17)

By Theorem 1.3.9 and Theorem 1.5.7, we obtain (8.3.3). The proof of our theorem is completed.


327

8.3.2 Generalized Bergman kernel Definition 8.3.2. The smooth kernel of PHp with respect to the Riemannian volume form dvX (x ) is denoted by PHp (x, x ) (x, x ∈ X), and is called a generalized Bergman kernel of ∆p,Φ . Especially, PHp (x, x ) ∈ (Lp ⊗ E)x ⊗ (Lp ⊗ E)∗x . Recall that R˙ L ∈ End(T (1,0) X) is defined by (1.5.15) associated to RL . Condition (1.5.21) means aj (x) µ0 > 0 in (1.5.18). For ψ a tensor on X, set ∇X ψ the covariant derivative of ψ induced by the Levi–Civita connection ∇T X . A corollary of Theorem 8.3.8 is one of our main results: Theorem 8.3.3. There exist smooth coefficients br (x) ∈ End(E)x where b0 = (2π)−n det(R˙ L ) IdE ,

(8.3.18)

with the following property: for any k, l ∈ N, there exists Ck, l > 0 such that k 1 br (x)p−r Ck, l p−k−1 , n PHp (x, x) − p Cl r=0

(8.3.19)

for any x ∈ X, p ∈ N∗ . The coefficients br (x) are polynomials in RT X , RE , Φ (and RL ), their derivatives of order 2r − 2 (resp. 2r), and reciprocals of linear combinations of eigenvalues of R˙ L at x. Moreover, the expansion is uniform in the following sense: for any fixed k, l ∈ N, assume that the derivatives of g T X , hL , ∇L , hE , ∇E , J and Φ with order 2n + 2k + l + 5 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 the data; and the C l -norm in (8.3.19) includes also the derivatives on the parameters. The following result will be established in Section 8.3.4. Theorem 8.3.4. If R˙ L = 2π Id, we have b1 =

√ 1 X 1 X 2 r + |∇ J| + 4 −1Λω (RE ) , 8π 4

(8.3.20)

2 TX where |∇X J|2 = ij |(∇X ) and ei J)ej | , {ej } is an orthonormal basis of (T X, g E Λω (R ) was introduced in (4.1.5). Recall that ∇T X induces by projection a Hermitian connection ∇1,0 on T (1,0) X (cf. Section 1.3.1). By (1.3.45), the Chern–Weil representative of c1 (T (1,0) X) is √ −1 (1,0) 1,0 Tr |T (1,0) X (∇1,0 )2 . c1 (T X, ∇ ) = (8.3.21) 2π

328


In Problem 8.4, we verify that if R˙ L = 2π Id, then 1 X 1 X 2 c1 (T (1,0) X, ∇1,0 ), ω = r + |∇ J| . 4π 4

(8.3.22)

Therefore, by integrating over X the expansion (8.3.19) for k = 1 we obtain (8.3.3), so (8.3.20) is compatible with (8.3.3). The term rX + 14 |∇X J|2 in (8.3.20) is called the Hermitian scalar curvature of (X, J, ω) in the literature and is a natural substitute for the Riemannian scalar curvature in the almost-K¨ ahler case. It was used by Donaldson to define the moment map on the space of compatible almost-complex structures.

8.3.3 Near diagonal asymptotic expansion We will use in the sequel the function F defined in (4.1.11). Let F be the holomorphic function on C such that F (a2 ) = F (a). The restriction of F to R lies in the Schwartz space S(R). Then there exists {cj }∞ j=1 such that for any k ∈ N, the function Fk (a) = F(a) −

k

cj aj F (a),

(8.3.23)

j=1

verifies (i)

Fk (0) = 0

for any 0 < i k.

(8.3.24)

Proposition 8.3.5. Let 0 < θ < 1. For any k, m ∈ N, there exists Ck,m > 0 such that for p 1, −θ Fk p ∆p,Φ (x, x ) − Pp (x, x ) m Ck,m p−kθ+3m+8n+8 . (8.3.25) C (X×X) Here the C m norm is induced by ∇L , ∇E , hL , hE and g T X . > 0 such that Proof. By (4.1.11), for any m ∈ N, there exists Ck,m sup |a|m |Fk (a)| Ck,m .

(8.3.26)

a∈R

Set Gk,p (a) = 1[p1−θ µ0 ,+∞[ (a)Fk (a),

Hk,p (a) = 1[0,C L p−θ ] (|a|)Fk (a).

In view of (8.3.2), we have for p large enough,

Fk p−θ ∆p,Φ = Gk,p p−θ ∆p,Φ + Hk,p p−θ ∆p,Φ .

(8.3.27)

(8.3.28)

As in Section 1.6.2, we trivialize now L, E on each Uxi by parallel transport with respect to the connections ∇L , ∇E . We define a Sobolev norm on the lth


329

Sobolev space H l (X, Lp ⊗ E) as in (1.6.5). Then as ∇L analogous to (1.6.6), we have sH 2m+2 (p) Cm p4m+4

m+1

p

⊗E

= ∇· + p ΓL + ΓE ,

p−4j ∆jp,Φ sL2 .

(8.3.29)

j=0

On Uxi × Uxj , we use the Sobolev inequality and we know for any l, m ∈ N, there exists C > 0 such that for p > 1,

C p−l , Gk,p p−θ ∆p,Φ (x, x ) m C (X×X) (8.3.30)

C p3m+8n+8−kθ . Hk,p p−θ ∆p,Φ − PHp (x, x ) m C

(X×X)

By (8.3.28) and (8.3.30), we get our Proposition 8.3.5.

Using (4.1.11), (8.3.23)

and the finite propagation speed, Theorem D.2.1, it is clear that for x, x ∈ X, Fk p−θ ∆p,Φ (x, ·) only depends on the restriction of ∆p,Φ

to B X (x, εp−θ/2 ), and Fk p−θ ∆p,Φ (x, x ) = 0, if d(x, x ) εp−θ/2 . This means that the asymptotic of PHp (x, ·) when p → +∞, modulo O(p−∞ ) (i.e., terms whose C m -norm is O(p−l ) for any l, m ∈ N), only depends on the restriction of ∆p,Φ to B X (x, εp−θ/2 ). Now, we are in a much simpler situation than in Section 4.1.3. For x0 ∈ X, we proceed step by step as there, but we just omit Λ(T ∗(0,1) X) therein. For example, Φ0 = Φp ◦ ϕε is a smooth Hermitian section of End(E0 ) on X0 Tx0 X. L0 ⊗E0 0 − pτ0 + Φ0 be the renormalized Bochner Laplacian on X0 Let ∆X p,Φ0 = ∆ associated to the above data, as in (8.3.1). Observe that RL0 is uniformly positive on R2n . By the argument in the proof 0 of Theorem 8.3.1, we see that the spectral gap property (8.3.2) still holds for ∆X p,Φ0 . L0 > 0 such that Specifically, there exists C 8 0 Spec(∆X p,Φ0 ) ⊂ [−CL0 , CL0 ] ∪ [ p µ0 − CL0 , +∞[ . 5

(8.3.31)

0 We note that ∆X p,Φ0 has not necessarily discrete spectrum. Let SL be a unit vector of Lx0 . Using SL and the above discussion, we get 0 an isometry E0 ⊗ Lp0 Ex0 . Let P0,Hp be the spectral projection of ∆X p,Φ0 from p 2 2 L0 , C L0 ], and L (X0 , L0 ⊗ E0 ) L (X0 , Ex0 ) corresponding to the interval [−C let P0,Hp (x, x ) be the smooth kernel of P0,Hp with respect to the Riemannian volume form dvX0 (x ). The following proposition shows that PHp and P0,Hp are asymptotically close on B Tx0 X (0, ε) in the C ∞ -topology, as p → ∞. Using (8.3.28) and (8.3.31) as in the proof of Proposition 4.1.6 we obtain:

Proposition 8.3.6. For any l, m ∈ N, there exists Cl,m > 0 such that for x, x ∈ B Tx0 X (0, ε), (P0,Hp − PHp )(x, x ) m Cl,m p−l . (8.3.32) C

330


Recall that R, κ were defined in (4.1.18) and (4.1.28). Denote t = s ∈ C ∞ (R2n , Ex0 ) and Z ∈ R2n , as in (1.6.27), set p

∇t =tSt−1 κ 2 ∇L0 ⊗E0 κ− 2 St , 1

√1 . p

1

−2 0 Lt =St−1 t2 κ 2 ∆X St . p,Φ0 κ 1

1

(8.3.33)

Recall that the operators ∇0,· and L were defined in (4.1.72). Set with ∂ α := I(Z)(·) =

(∂ α RL )x0

|α|=2

For

Zα (R, ·). α!

∂α ∂Z α ,

(8.3.34)

The operator Lt is the rescaled operator, which we now develop in Taylor series as in Theorem 4.1.7. Theorem 8.3.7. There exist polynomials Ai,j,r (resp. Bi,r , Cr ) (r ∈ N, i, j ∈ {1, . . . , 2n}) in Z with the following properties: (i) their coefficients are polynomials in RT X (resp. RT X , RE , Φ, RL ) and their derivatives at x0 up to order r − 2 (resp. r − 2, r − 2, r − 2, 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 with r − 1 (resp. r) , (iii) if we denote O r as in (4.1.30), then in the sense of (4.1.31), we have Lt = L +

m

tr O r + O(tm+1 ).

(8.3.35)

r=1

Moreover 2 1 O 1 (Z) = − (∂j RL )x0 (R, ei )Zj ∇0,ei − (∂i RL )x0 (R, ei ) − (∇R τ )x0 , 3 3

1 TX Rx0 (R, ei )R, ej x0 ∇0,ei ∇0,ej O 2 (Z) = (8.3.36) 3 1

1 + RxT0X (R, ej )ej , ei x0 − I(Z)(ei ) − RxE0 (R, ei ) ∇0,ei 3 2 2 1 1 − ∇ei (I(Z)(ei )) − (∂j RL )x0 (R, ei )Zj 4 9 i j −

rxX0 Zα − + Φx 0 . (∂ α τ )x0 6 α! |α|=2

Proof. As in Lemma 1.2.3, set gij (Z) = g T X (ei , ej )(Z) = ei , ej Z and let (g ij (Z)) be the inverse of the matrix (gij (Z)). Let ∇TeiX ej = Γlij (Z)el . Now by (8.3.1), p

∆p,Φ = −g ij (∇eLi

⊗E

L ⊗E ⊗E ∇L − ∇∇ T X e ) − pτ + Φ. ej j p

p

ei

(8.3.37)


331

From (8.3.33) and (8.3.37) we infer the expression Lt = −g ij (tZ) ∇t,ei ∇t,ej − tΓlij (t·)∇t,el (Z) − τ (tZ) + t2 Φ(tZ).

(8.3.38)

Let ΓE , ΓL be the connection forms of ∇E and ∇L with respect to any fixed frames for E, L which are parallel along the curve γZ : [0, 1] u → uZ under our trivializations on B Tx0 X (0, 4ε). (8.3.33) yields, on B Tx0 X (0, 2ε/t), 1 1 1 ∇t,ei |Z = κ 2 (tZ) ∇ei + ΓL (ei )(tZ) + tΓE (ei )(tZ) κ− 2 (tZ). t

(8.3.39)

By (1.2.30), (4.1.101) and (8.3.39), we get t ∇t,ei |Z = ∇0,ei + (∂k RL )x0 (R, ei )Zk + 3 t2 t2 + I(Z)(ei ) + 4 6

t2 E R (R, ei ) 2 x0

TX Rx0 (R, ek )ek , ei + O(t3 ). (8.3.40)

Relations (1.2.19), (4.1.102), (8.3.38) and (8.3.40), show that the first coefficient of the expansion (8.3.35) is L and prove (8.3.36). Lemmas 1.2.3, 1.2.4, (8.3.38) and (8.3.39) settle the rest of our theorem. We denote by · , · 0,L2 and · 0,L2 the scalar product and the L2 norm on C ∞ (X0 , Ex0 ) induced by g T X0 , hE0 as in (1.3.14) and in Section 1.6.4. For s ∈ C ∞ (X0 , Ex0 ), set |s(Z)|2hEx0 dvT X (Z), (8.3.41) s2t,0 = s20 = s2t,m =

m

R2n 2n

∇t,ei1 · · · ∇t,eil s2t,0 .

l=0 i1 ,··· ,il =1

Applying the same proof in Section 4.1.4, by using (8.3.31) and the corresponding Sobolev norms above, we get Theorems 4.1.9–4.1.14 for Lt defined in (8.3.38). Now we view PHp ,x0 (Z, Z ) := PHp (Z, Z ) (Z, Z ∈ Tx0 X, |Z|, |Z | < 2ε) as a smooth section of π ∗ (End(E)) over T X ×X T X as in Section 4.1.5. The next step is to convert the estimates for the resolvent into estimates for the spectral projection P0,t : (L2 (X0 , Ex0 ), · 0 ) → (L2 (X0 , Ex0 ), · 0 ) of Lt corresponding L0 t2 ]. By the arguments in Sections 4.1.5 and 4.1.7, we L0 t2 , C to the interval [−C finally get with P in (4.1.84): Theorem 8.3.8. There exist Jr (Z, Z ) polynomials in Z, Z with the same parity as r and deg Jr (Z, Z ) 3r, whose coefficients are polynomials in RT X , RE , Φ

332


( and RL ) and their derivatives of order r − 2 ( resp. r), and reciprocals of linear combinations of eigenvalues of R˙ L at x0 , such that if we define Fr (Z, Z ) = Jr (Z, Z )P(Z, Z ),

J0 = IdE ,

(8.3.42)

then for 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 PHp (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

. (8.3.43)

Moreover, F1 , F2 were given by (4.1.93) by replacing L20 , Or therein by L , O r here. Proof of Theorem 8.3.3. As in Section 4.1.7, from Theorem 8.3.8 with Z = Z = 0, we get the first part of Theorem 8.3.3 and br (x0 ) = F2r (0, 0).

(8.3.44)

Now as the coefficients of Lt are functions of g T X , ∇L , ∇E their derivatives with order 1, from the argument at the end of Section 4.1.7, we get the last part of Theorem 8.3.3. Theorem 8.3.9 below gives a version of (4.1.94) in the current situation. We use the notation in Section 4.1.6 now. As in Section 1.2.2, we denote by · , · the C-bilinear form on T X ⊗R C induced by g T X . Theorem 8.3.9. We have the relation with O 1 in (8.3.36), PO 1 P = 0.

(8.3.45)

Proof. We define J ∈ End(T X) by: for U, V, W ∈ T X, J V, W = RL (V, W ).

(8.3.46)

L X (∇X U R )(V, W ) = (∇U J )V, W , √ −1 Tr |T X [∇X ∇U τ = − U (JJ )]. 2

(8.3.47)

By (1.5.19) and (8.3.46),

From (8.3.47), we obtain X X L (∇X U J )V, W + (∇V J )W, U + (∇W J )U, V = dR (U, V, W ) = 0. (8.3.48)


333

X As J, J ∈ End(T X) are skew-adjoint and commute, ∇X U J, ∇U J are skewX 2 adjoint and ∇U (JJ ) is symmetric. From J = − Id, we know that

J(∇X J) + (∇X J)J = 0,

(8.3.49)

(1,0) thus ∇X X and T (0,1) X. From (8.3.47) and (8.3.48), we have U J exchanges T √ ∂ ∂ ∂ ∂ = 2 (∇X (∇R τ )x0 = −2 −1 (∇X R (JJ )) ∂zi , ∂z i R J ) ∂zi , ∂z i , ∂ (∂i RL )x0 (R, ei ) = 2 (∇X∂ J )R, ∂z∂ i + 2 (∇X∂ J )R, ∂z (8.3.50) i ∂zi ∂z i ∂ ∂ = 4 (∇X∂ J )R, ∂z∂ i − 2 (∇X R J ) ∂zi , ∂z i . ∂zi

From (4.1.75), (8.3.36), (8.3.47) and (8.3.50), we infer 2 X + X ∂ ∂ O1 = − b (∇R J )R, ∂z − (∇ J )R, R i ∂z i bi i 3 ∂ ∂ + 2 (∇X∂ J )R, ∂z∂ i + 2 (∇X J ) , R ∂zi ∂z i =−

2 3

∂zi

+ ∂ (∇X R J )R, ∂zi bi − bi

(8.3.51)

∂ (∇X . J )R, R ∂z i

Note that by (4.1.74) and (4.1.84), (b+ i P)(Z, Z ) = 0 ,

(bi P)(Z, Z ) = ai (z i − z i )P(Z, Z ).

(8.3.52)

By Theorem 4.1.20 and (7.1.5), any polynomial g(z, z) in z, z satisfies Pbα g(z, z)P = 0 ,

for |α| > 0,

and relations (8.3.51)–(8.3.53) yield the desired relation (8.3.45).

(8.3.53)

8.3.4 The second coefficient b1 In the rest of this section we assume that g T X = ω(·, J·). A very useful observation is that (8.3.48), (8.3.49) imply √ J = −2π −1J and ai = 2π in (1.5.18), τ = 2πn. ∇X U J is skew-adjoint (8.3.54) X

∗(1,0) ⊗3 and the tensor (∇· J)·, · is of the type (T X) ⊕ (T ∗(0,1) X)⊗3 . Before computing b1 , we establish the relation between the scalar curvature rX and |∇X J|2 . Lemma 8.3.10. The following identity holds, 1 X 2 ∂ ∂ ∂ ∂ rX = 8 RT X ( ∂z , ) , ∂z i − 4 |∇ J| . i ∂z j ∂zj

(8.3.55)

334


Proof. By (8.3.54), |∇X J|2 = 4 (∇X∂ J)ej , (∇X∂ J)ej = 8 (∇X∂ J) ∂z∂ j , (∇X∂ J) ∂z∂ j . (8.3.56) ∂zi

∂z i

∂zi

∂z i

By (4.1.71), (8.3.48) and (8.3.54), X X X ∂ ∂ ∂ ∂ ∂ ∂ (∇X∂ J) ∂z = 2 (∇ (∇ , (∇ J) J) , J) , ∂ ∂ ∂ ∂z j ∂zi ∂zk ∂z j ∂z k i ∂zj

∂z i

∂zj

∂z i

X ∂ ∂ ∂ ∂ (∇ = 2 (∇X∂ J) ∂z∂k − (∇X∂ J) ∂z , J) , ∂ ∂z k ∂z j i ∂zj ∂z

∂z

∂z

i i k X ∂ ∂ . (8.3.57) , (∇ J) = (∇X∂ J) ∂z∂k , (∇X∂ J) ∂z∂ k − (∇X∂ J) ∂z ∂ ∂z i k

∂zi

∂z i

∂zk

∂z i

By (8.3.56) and (8.3.57), 1 X 2 X ∂ ∂ (∇X∂ J) ∂z |∇ J| . = , (∇ J) ∂ ∂z i j 16 ∂zj ∂z i

(8.3.58)

The definition of ∇X ∇X J, RT X and (8.3.48) imply, for U, V, W, Y ∈ T X, (∇X ∇X J)(U,V ) − (∇X ∇X J)(V,U) = [RT X (U, V ), J],

X X (∇ ∇ J)(Y,U) V, W + (∇X ∇X J)(Y,V ) W, U

+ (∇X ∇X J)(Y,W ) U, V = 0.

(8.3.59)

Now, from (8.3.49), we get X (∇X ∇X J)(U,V ) J + (∇X U J) ◦ (∇V J) X X X + (∇X V J) ◦ (∇U J) + J(∇ ∇ J)(U,V ) = 0. (8.3.60)

Note that ∇X ∇X J is a two tensor with values in the bundle of anti-symmetric endomorphisms of T X. Thus from (8.3.54), (8.3.59) and (8.3.60), for u1 , u2 , u3 ∈ T (1,0)X, v 1 , v 2 ∈ T (0,1) X, (∇X ∇X J)(u1 ,u2 ) u3 , (∇X ∇X J)(v 1 ,v2 ) u3 ∈ T (0,1) X, (∇X ∇X J)(u1 ,v2 ) u3 ∈ T (1,0) X, √

X 2 −1 (∇X ∇X J)(u1 ,v1 ) u2 , v 2 = (∇X u1 J)u2 , (∇v 1 J)v 2 .

(8.3.61)

Formulas (8.3.59) and (8.3.61) yield

(∇X ∇X J)(u1 ,u2 ) v 1 , v 2

= − (∇X ∇X J)(u1 ,v1 ) v 2 , u2 − (∇X ∇X J)(u1 ,v2 ) u2 , v 1

1 X X = √ (∇u1 J)u2 , (∇X v 1 J)v 2 − (∇v 2 J)v 1 . (8.3.62) 2 −1


335

From (8.3.59), (8.3.58) and (8.3.62), we deduce R

TX

√ −1 T X ∂ ∂ [R ( ∂zi , ∂zj ), J] ∂z∂ i , ∂z∂ j = 2 √ −1 X X ∂ ∂ (∇ ∇ J) ∂ ∂ − (∇X ∇X J) ∂ ∂ = , ( ∂z , ∂z ) ( ∂z , ∂z ) ∂z i ∂z j 2 i j j i 1 X 2 1 X X ∂ ∂ (∇ ∂ J) ∂zj , (∇ ∂ J) ∂zj = |∇ J| . (8.3.63) = 4 32 ∂zi ∂z i

∂ ( ∂z , ∂ ) ∂ , ∂ i ∂zj ∂z i ∂z j

The scalar curvature rX of (X, g T X ) is

rX = − RT X (ei , ej )ei , ej = −4 RT X ( ∂z∂ i , ej ) ∂z∂ i , ej ∂ ∂ = − 8 RT X ( ∂z , ∂ ) ∂ , ∂ − 8 RT X ( ∂z , ∂ ) ∂ , ∂ . i ∂zj ∂z i ∂z j i ∂z j ∂z i ∂zj In conclusion, relations (8.3.63) and (8.3.64) imply (8.3.55).

(8.3.64)

Proof of Theorem 8.3.4. Recall that we do all our computations on C ∞ (R2n , Ex0 ). Thus here we still use P to denote the orthogonal projection from L2 (R2n , Ex0 ) onto Ker(L ) and P ⊥ = 1 − P. Recall that b1 (x0 ) = F2 (0, 0). By (4.1.93) and (8.3.45), we get F2 =L −1 P ⊥ O 1 L −1 P ⊥ O 1 P − L −1 P ⊥ O 2 P + PO 1 L −1 P ⊥ O 1 L −1 P ⊥ − PO 2 L −1 P ⊥ ⊥

+P L

−1

O 1 PO 1 L

−1

⊥

P − PO 1 L

−2

(8.3.65)

⊥

P O 1 P.

First, by (8.3.51), (8.3.52) and (8.3.54), we obtain 2 X (O 1 P)(Z, Z ) = bi (∇z J )z, ∂z∂ i P (Z, Z ) √ 3 4 −1π bi bj X ∂ (∇ ∂ J)z + bi (∇X =− z J)z , ∂z i P (Z, Z ). (8.3.66) 3 2π ∂z j Thus by Theorem 4.1.20 and aj = 2π, we get (L

−1

√ 4 −1π bi bj X ∂ (∇ P O 1 P)(Z, Z ) = − J)z , ∂ ∂z i 3 16π 2 ∂z j bi ∂ P (Z, Z ). (8.3.67) J)z , + (∇X z ∂z i 4π ⊥

From (8.3.52), (8.3.54), (8.3.66) and (8.3.67), we have (O 1 P)(Z, 0) = (L −1 P ⊥ O 1 P)(0, Z ) = 0.

(8.3.68)

336


In view of (8.3.68), the first and last two terms in (8.3.65) are zero at (0, 0). Thus we only need to compute −(L −1 P ⊥ O 2 P)(0, 0), since the third and fourth terms in (8.3.65) are adjoint of the first two terms by the same remark as Remark 4.1.8. ∂ Note that for ψ a 1-form, ψ(ej )∇0,ej = ψ( ∂z∂ j )b+ j − ψ( ∂z j )bj . By (4.1.76), (8.3.36), (8.3.47) and (8.3.52), we get 4π 2

4π T X RxT0X (z, z)z, z − R (R, ∂z∂ j )R, ∂z∂ j 3 3

2π T X 1 − Rx0 (R, ej )ej , z + πI(Z)(z) − ∇el I(Z)(el ) 3 4 X 2 r 4π x 2 + 2πRE (z, z) − 0 + |(∇X R J)R| + Φx0 P (Z, 0). (8.3.69) 6 9

(O 2 P)(Z, 0) =

In normal coordinates, (∇TeiX ej )x0 = 0, so from (4.1.102), we have at x0 , X TX TX TX TX ∇ej ∇ei Jek , el = (∇X ej ∇ei J)ek + J(∇ej ∇ei ek ), el + Jek , ∇ej ∇ei el 1

X RT X (ej , ei )ek + RT X (ej , ek )ei , Jel = (∇X ej ∇ei J)ek , el − 3

1 TX R (ej , ei )el + RT X (ej , el )ei , Jek . (8.3.70) + 3 From (8.3.34), (8.3.47), (8.3.54) and (8.3.70), we obtain √ I(Z)(el ) = − −1π(∇ej ∇ei Jek , el )x0 Zi Zj Zk √

2π T X

R (R, el )z, z . (8.3.71) = − −1π (∇X ∇X J)(R,R) R, el − 3 From (8.3.59) and (8.3.61), we have

X X (∇ ∇ J)(R,R) z, z = 2 (∇X ∇X J)(z,z) z, z + [RT X (z, z), J]z, z 2 √ = − −1 (∇X z J)z . (8.3.72) From (8.3.71), (8.3.72) and (∇X ∇X J)(Y,U) is skew-adjoint, we get 2 2π T X

I(Z)(z) = −π (∇X R (z, z)z, z . z J)z − 3 Note that J, (∇X ∇X J)(Y,U) are skew-adjoint, thus, X X X X ∂ ∂ ∂ ∂ (∇ + (∇ ∇ J) R, ∇ J) R, (R,R) (R,R) ∂zi ∂z i ∂z i ∂zi X X = (∇X ∇X J) R, ∂z∂ i ∂ R + (∇ ∇ J) ∂ (R, ∂z ) ( ∂z ,R) i i X X X X ∂ + (∇ ∇ J) R + (∇ ∇ J) R, , ∂ ∂ ∂zi (R, ∂z ) i

( ∂z ,R) i

(8.3.73)

(8.3.74)

8.3. Generalized Bergman kernel ∂ ∂zi

337

∂ RT X (R, ∂z∂ i )z, z + ∂z∂ i RT X (R, ∂z )z, z i TX TX ∂ ∂ ∂ = R (R, ∂zi ) ∂zi , z + R (R, ∂zi )z, ∂z∂ i TX ∂ ∂ ∂ = RT X (z, ∂z∂ i ) ∂z . , z + R (z, )z, ∂z ∂z i i i

Observe that for a polynomial H on z, z, from (8.3.52), the contribution of P ⊥ HP at (0, 0) consists of the terms whose degree of z is the same as the degree of z. Thus for G a polynomial of degree 2 on z, z, we get b ∂G j P (Z, 0), (P ⊥ GP)(Z, 0) = 2π ∂z j (8.3.75) b ∂G 1 ∂ 2G j −1 ⊥ (L P GP)(0, 0) = P (0, 0) = − 2 . 8π 2 ∂z j 4π ∂zj ∂z j Thus from (8.3.62), (8.3.74) and (8.3.75), 1 −1 ⊥ L P (∇el I(Z)(el ))P (0, 0) (8.3.76) 4 √ −1 X X = 0, Re (∇ ∇ J) ∂ ∂ ∂z∂ j + (∇X ∇X J) ∂ ∂ ∂z∂ j , ∂z∂ i = ( ∂z , ∂z ) ( ∂z , ∂z ) 4π j i i j

L −1 P ⊥ 2 RT X (R, ∂z∂ j )R, ∂z∂ j + RxT0X (R, ej )ej , z P (0, 0) = 0. Let fij (z), (i, j = 1, . . . , n) be arbitrary polynomials in z. By (4.1.75) and (8.3.52), we have bb i j (fij z i z j P)(Z, 0) = fij 2 P (Z, 0) 4π b b bj ∂fij bi ∂fij 1 ∂ 2 fij i j P (Z, 0). (8.3.77) = f + + + ij 4π 2 2π 2 ∂zi 2π 2 ∂zj π 2 ∂zi ∂zj Thus from Theorem 4.1.20, (4.1.74) and (8.3.77), we get (L −1 P ⊥ fij z i z j P)(0, 0) =

b b bj ∂fij bi ∂fij i j P (0, 0) f + + ij 32π 3 8π 3 ∂zi 8π 3 ∂zj 3 ∂ 2 fij =− 3 . (8.3.78) 8π ∂zi ∂zj

By (8.3.63) and (8.3.78),

(L −1 P ⊥ RxT0X (z, z)z, z P)(0, 0) 3 TX ∂ ∂ ∂ ∂ ∂ ∂ ∂ , ) + R ( , ) , = − 3 RxT0X ( ∂z x 0 ∂zj ∂z j ∂zi ∂z i i ∂z j ∂zj 8π 3 TX ∂ ∂ ∂ ∂ 1 = − 3 2 Rx0 ( ∂zi , ∂zj ) ∂zj , ∂zi + 5 |∇X J|2 , (8.3.79) 8π 2

338


and from (8.3.56), (8.3.58) and (8.3.78), 2 L −1 P ⊥ |(∇X z J)z| P (0, 0) 9 3 X ∂ ∂ = − 7 3 |∇X J|2 . (8.3.80) , (∇ J) = − 3 (∇X∂ J) ∂z∂ j + (∇X∂ J) ∂z ∂ ∂z i j 8π 2 π ∂zi ∂zj ∂z i Thus by (8.3.73), (8.3.79) and (8.3.80), L

−1

4π 2 6

4π 2 TX X 2 Rx0 (z, z)z, z + πI(Z)(z) + |(∇R J)R| P (0, 0) P 3 9 2π 2 6

π2 2 = L −1 P ⊥ P (0, 0) |(∇X J)z| RxT0X (z, z)z, z − z 3 9 1 TX ∂ ∂ ∂ ∂ =− R ( ∂zi , ∂zj ) ∂zj , ∂zi . (8.3.81) 2π ⊥

Owing to (8.3.69), (8.3.75), (8.3.76) and (8.3.81), we get 1 E ∂ ∂ 1 TX ∂ ∂ ∂ ∂ − L −1 P ⊥ O 2 P (0, 0) = R ( ∂zi , ∂zj ) ∂zj , ∂zi + R ( ∂zi , ∂zi ). 2π 2π (8.3.82) Formulas (8.3.44), (8.3.55), (8.3.68) and (8.3.82), and the discussion after (8.3.68) yield finally ∗ b1 (x0 ) = − L −1 P ⊥ O 2 P (0, 0) − L −1 P ⊥ O 2 P (0, 0) √ 1 X 1 rx0 + |∇X J|2x0 + 4 −1Λω (RE ) . = 8π 4 The proof of Theorem 8.3.4 is complete.

(8.3.83)

8.3.5 Symplectic Kodaira embedding theorem Recall that (X, ω) is a compact symplectic manifold of real dimension 2n and √ −1 L R , and g T X is a (L, ∇L , hL ) is a Hermitian line bundle on X such that ω = 2π Riemannian metric on X as at the beginning of Section 8.3. p Let Hp ⊂ C ∞ (X, Lp ) be the span of those eigensections of ∆p = ∆L − τ p L ]. Note that Hp is endowed with the L , C corresponding to eigenvalues from [−C 2 induced L product (1.3.14) so there is a well-defined Fubini–Study form ωF S on P(Hp∗ ).


339

We have the following analogue of Theorem 5.1.3 in the symplectic case. Theorem 8.3.11. There exists p0 ∈ N such that for p > p0 , Φp : X −→ P(Hp∗ ) defined by Φp (x) = {s ∈ Hp : s(x) = 0} is well defined on X and the map Ψp : Φ∗p O(1) → Lp , Ψp ((Φ∗p σs )(x)) = s(x),

for any s ∈ Hp

(8.3.84)


hΦp O(1) (x) = PHp (x, x)−1 hL (x). p

(8.3.85)

Proof. By Theorem 8.3.4, as in Lemma 5.1.2, we know that there exists p0 > 0 such that for p > p0 , Φp is defined on X. The remainder is the same proof of Theorem 5.1.3. Theorem 8.3.12. (i) The induced Fubini–Study metric p1 Φ∗p (ωF S ) converges in the C ∞ topology to ω ; for any l 0 there exists Cl > 0 such that 1 Cl . (8.3.86) Φ∗p (ωF S ) − ω l p p C (ii) For large p the Kodaira maps Φp are embeddings. Proof. (i) Let us fix x0 ∈ X. We identify a small geodesic ball B X (x0 , ε) to B Tx0 X (0, ε) by means of the exponential map and consider a trivialization of L as in Section 8.3.3, i.e., we trivialize L by using a unit frame eL (Z) which is parallel with respect to ∇L along [0, 1] u → uZ for Z ∈ B Tx0 X (0, ε). Let dp w2 = j=1 |wi |2 . We can express the Fubini–Study metric in the homogeneous coordinate [w1 , . . . , wdp ] ∈ P(Hp∗ ) as (cf. Problem 1.8) √ √ dp dp

1 −1 −1 1 2 . ∂∂ log w = dw ∧ dw − w w dw ∧ dw j j j k j k 2π 2π w2 j=1 w4 j,k=1

Let {sj } be an orthonormal basis of Hp , and {sj } be its dual basis. We write sj = fj e⊗p L , then by (5.1.17), √

dp dp 1 −1 1 df ∧ df − fj fk dfj ∧ dfk (x0 ) j j p 2 p 4 2π |f | j=1 |f | j,k=1 √ (8.3.87) −1 p −2 p p −4 p p |f (x0 )| dx dy f (x, y) − |f (x0 )| dx f (x, y) ∧ dy f (x, y) |x=y=x0 , = 2π dp fi (x)fi (y) and |f p (x)|2 = f p (x, x). where f p (x, y) = i=1

Φ∗p (ωF S )(x0 ) =

340


Since

PHp (x, y) = f p (x, y)epL (x) ⊗ epL (y)∗ ,

(8.3.88)

thus PHp (x, y) is f p (x, y) under our trivialization of L. By (4.1.101), Theorem 8.3.8, we obtain √ 1 ∗ −1 1 1 Φp (ωF S )(x0 ) = dx dy F0 − 2 dx F0 ∧ dy F0 (0, 0) p 2π F0 F0 √ 1 1 −1 1 − (dx F1 ∧ dy F0 + dx F0 ∧ dy F1 ) (0, 0) + O . (8.3.89) √ 2 2π p F0 p Using again (4.1.84), (8.3.42), we obtain √ n

1

1 −1 1 ∗ Φp (ωF S )(x0 ) = = ω(x0 ) + O , (8.3.90) aj (x0 )dzj ∧ dz j + O p 4π j=1 p p and the convergence takes place in the C ∞ topology with respect to x0 ∈ X. (ii) Since X is compact, we have to prove two things for p sufficiently large: (a) Φp are immersions and (b) Φp are injective. We note that (a) follows immediately from (8.3.86). To prove (b) let us assume the contrary, namely that there exists a sequence of distinct points xp = yp such that Φp (xp ) = Φp (yp ). Since Φp (xp ) = Φp (yp ) we can construct as in (5.1.24) the peak section Sxpp = Sypp as the unit norm generator of the orthogonal complement of Φp (xp ) = Φp (yp ). We fix in the sequel such a section which peaks at both xp and yp . We consider the distance d(xp , yp ) between the two points xp and yp . By √ passing to a subsequence we have two possibilities: either pd(xp , yp ) → ∞ as √ p → ∞ or there exists a constant C > 0 such that d(xp , yp ) C/ p for all p. Assume that the first possibility is true. For large p, we learn from relation (5.1.25) that the mass of Sxpp = Sypp (which is 1) concentrates both in neighborhoods B(xp , rp ) and B(yp , rp ) with rp = d(xp , yp )/2 and approaches therefore 2 if p → ∞. This is a contradiction which rules out the first possibility. We identify as usual B X (xp , ε) to B Txp X (0, ε) so the point yp gets identified √ to Zp / p where Zp ∈ B Txp X (0, C). We define then fp : [0, 1] −→ R ,

√ |Sxpp (tZp / p)|2 fp (t) = √ √ . PHp (tZp / p, tZp / p)

(8.3.91)

We have fp (0) = fp (1) = 1 (again because Sxpp = Sypp ) and fp (t) 1 by the definition of the generalized Bergman kernel. We deduce the existence of a point tp ∈]0, 1[ such that fp (tp ) = 0. Equations (5.1.24), (8.3.43), (8.3.91) imply the estimate 2

t2 √ fp (t) = e− 4 j aj |zp,j | 1 + gp (tZp )/ p (8.3.92)


341

Txp X and the C 2 norm of gp over (0, C) is uniformly bounded in p. From (8.3.92), B √ 1 2 we infer that |Zp |0 := 4 j aj |zp,j |2 = O(1/ p). Using a limited expansion ex = 1 + x + x2 ϕ(x) for x = t2 |Zp |20 in (8.3.92) and taking derivatives, we obtain

√ √ fp (t) = −2|Zp |20 + O(|Zp |40 ) + O(|Zp |20 / p) = (−2 + O(1/ p))|Zp |20 .

(8.3.93)

√ Evaluating at tp we get 0 = fp (tp ) = (−2+O(1/ p))|Zp |20 , which is a contradiction since by assumption Zp = 0. Remark 8.3.13. The proof of Theorem 8.3.12 gives another analytic proof of the classical Kodaira embedding theorem, Theorem 5.1.12. Certainly, we can use also Ker(Dpc,A ) in Section 8.1 instead Hp here to get again an analogue of Theorem 8.3.12.

Problems Problem 8.1. Under the assumptions of Section 8.1.1, let G be a compact connected Lie group with Lie algebra g and dim G = n0 . Suppose that G acts on X and its action on X lifts on L and E. Moreover, we assume the G-action preserves the above connections and metrics on T X, L, E and J. Verify that Dpc,A commutes with the G-action. Thus Ker(Dpc,A ) is a representation of G. Denote by Ker(Dpc,A )G the G-trivial component of Ker(Dpc,A ). ∂ −tK e x|t=0 the corresponding vector field For K ∈ g, we denote by KxX = ∂t ∗ on X. Let µ : X → g be defined by √ 2π −1µ(K) := ∇L K X − LK , K ∈ g. √

Verify that µ is the moment map, associated to ω =

−1 L 2π R ,

i.e., for any K ∈ g,

dµ(K) = iK X ω. Let PpG be the orthogonal projection from (Ω0,• (X, Lp ⊗ E), ) on Ker(Dpc,A )G . The G-invariant Bergman kernel is PpG (x, x ) (x, x ∈ X), the smooth kernel of PpG with respect to the Riemannian volume form dvX (x ). Assume that G acts freely on µ−1 (0). Denote by vol(Gx), (x ∈ µ−1 (0)) the volume of the orbit Gx equipped with the metric induced by g T X . Prove that there exists an asymptotic expansion in the sense of (8.1.4) for x0 ∈ µ−1 (0), k ∈ N, p

−n+

n0 2

vol(Gx0 )PpG (x0 , x0 )

=

k

−r bG + O(p−k−1 ). r (x0 )p

r=0 G n0 /2 If g T X = ω(·, J·), then b0 (x0 ) = 2−1 IC⊗E . (Hint: For dg a Haar measure on G, G we have Pp (x, x) = G (g, 1) · Pp (g x, x)dg. Then use Theorem 8.1.4.)

342


Problem 8.2. In Theorem 8.2.5, verify that (7.1.11) holds for K& with i, j > q. While for l, m q, we have K&[1, zl ] = zl , 2 K&[z l , zm ] = K&[zm , z l ] = − δlm + z l zm , al K&[zl , z m ] = zl z m , K&[z l , zm ] = z l z .

K&[1, z l ] = z l ,

m

Especially, analogue to (7.4.14), we have ∂fx ∂fx ∂fx ∂fx K& 1, ∂Z (0)Zj = ∂zi (0)zi + ∂z i (0)z i + ∂zl (0)zl + j i>q

∂fx ∂z l (0)z l

.

lq

Problem 8.3. Let (X, J, g T X ) be a Kähler manifold and ∇L , ∇E be the holomorphic Hermitian connections on the holomorphic Hermitian bundles (L, hL ), (E, hE ). By using (1.4.31), (1.5.20) and (8.3.1), show that ∆p,0 = 2p − RE (wj , wj ) ,

on C ∞ (X, Lp ⊗ E) .

Problem 8.4. Verify that in (8.3.21), 1 (∇1,0 )2 = P 1,0 RT X − (∇X J) ∧ (∇X J) P 1,0 . 4 Verify (8.3.22). (Hint: Use (8.3.49), (8.3.56), (8.3.63) and (8.3.64).) Problem 8.5. We denote by Or the coefficients of the Taylor expansion associated to the rescaled operator from (Dpc,A )2 introduced in Section 8.1.1. As in (4.1.31), verify that O1 = O 1 + (∇R τ )x0 +

1 X (∇R J )x0 el , em c(el ) c(em ). 2

with O 1 , J defined in (8.3.36), (8.3.46). Verify (4.1.94), i.e., P N O1 P N = 0. Finally prove that Tr |Λ(T ∗(0,1) X) [b1 (x)] =

√ 1 X 1 X 2 r + |∇ J| + 4 −1Λω (RE ) . 8π 4

(Hint: Use (1.3.35) and compare the proof of Theorems 8.3.7, 8.3.9.)


343

8.4 Bibliographic notes Our book is already quite long, thus we will not try to report exhaustively on results about the Bergman kernel on symplectic manifolds here. Section 8.1. If A = 0, i.e., for the spinc Dirac operator Dpc , the results in Section 8.1.1 was obtained by Dai–Liu–Ma [69] by using the heat kernel as in Section 4.2. Section 8.1.2 is from [162]. Section 8.2 is taken from [163]. In the holomorphic case, i.e., in the situation of Remark 8.2.6, (8.2.18) is Andreotti–Grauert’s coarse vanishing theorem [4, §23] (where it is proved by using the cohomology finiteness theorem for the disc bundle of L∗ ), the asymptotic expansion of Pp (x, x ) was studied independently by Berman and Sj¨ ostrand [19]. Section 8.3 is from [160, 161]. In the case E is a trivial line bundle, (8.3.2) without knowing precise µ0 is the main result of Guillemin and Uribe [122, Theorem 2], they apply the analysis of Toeplitz structures of Boutet de Monvel– Guillemin [52], cf. also [43, 45] for related topic. The Hermitian scalar curvature was used by Donaldson [88, 111] to define the moment map on the space of compatible almost-complex structures. This section is also related to Donaldson’s work [87]. The proof of Theorem 8.3.12 is inspired by [219]. Borthwick and Uribe [44, Th. 1.1], Shiffman and Zelditch [219, Th. 2, 3] prove a different symplectic version of Theorem 8.3.12 when g T X = ω(·, J·). Instead of Hp , they use the space HJ0 (X, Lp ) := Im(Πp ) (cf. [44, p. 601], [219, §2.3]) of ‘almost holomorphic sections’ proposed by Boutet de Monvel and Guillemin [52, 53] of a first order pseudodifferential operator Db on the circle bundle of L∗ . The associated Szegö kernels are well-defined modulo smooth operators on the associated circle bundle, even though Db is neither canonically defined nor unique (Indeed, Boutet de Monvel– Guillemin define the Szeg¨ o kernels first, and construct the operator Db from the Szegö kernels.) Problem 8.1 is from Ma-Zhang [164, §6.3] where the full asymptotic expansion ahler, of PpG is obtained. Related results in the special case when (X, J, Θ) is K¨ Θ = ω, E = C, and ∇L is the Chern connection can be found in [64, 190, 191]. Note, however, that the value bG 0 (x0 ) = 1 in the main result of [191] is wrong. Problem 8.5 is from [163, §2].

Appendix A

Sobolev Spaces In this appendix, we explain the Sobolev embedding theorem and the basic elliptic estimates. A good reference for the matters discussed here is [237, Chap. 4, 5].

A.1 Sobolev spaces on Rn We denote by B(x, r) the ball with center x and radius r in Rn . For W ⊂ Rn , we denote by W its closure in Rn . Let U ⊂ Rn be an open set. For k ∈ N, let C k (U ) be the space of complexvalued functions on U whose partial derivatives of order k exist and are continuous on U . The support supp(f ) of a function f ∈ C k (U ) is defined as the closure in U of the set {x ∈ U : f (x) = 0}. Set C0k (U ) := {f ∈ C k (U ) : supp(f ) is compact}. Set C ∞ (U ) = {f ; f ∈ C k (U ) for any k ∈ N}, C0∞ (U ) = {f ; f ∈ C0k (U ) for any k ∈ N}.

(A.1.1)

The space C ∞ (U ) is called the space of smooth functions on U , and the space C0∞ (U ) is called the space of smooth functions with compact support in U . For f ∈ C k (U ), we define its C k -norm by ∂α (A.1.2) |f |C k (U) = sup α f (x). x∈U ∂x |α|k

α

|α|

∂ α1 := ∂∂xα := ( ∂x ) · · · ( ∂x∂ n )αn for α = (α1 , . . . , αn ) ∈ Nn . 1 The L2 -space L2 (U ) is the space of square integrable functions on U , i.e., for a measurable function f : U → C, f ∈ L2 (U ) if and only if 1/2 f L2 = |f |2 dx < +∞, (A.1.3)

Here

∂ ∂xα

U

where dx = dx1 · · · dxn is the canonical Euclidean volume form on Rn .

346

Appendix A. Sobolev Spaces

Then · L2 defines a norm on L2 (U ). The space L2 (U ) is the completion of with respect to the L2 -norm · L2 . For f, g ∈ L2 (U ), set f · g dx. (A.1.4) f, g =

C0∞ (U )

U

Then (L (U ), ·, ·) is a Hilbert space with inner product ·, ·. 2

Theorem A.1.1 (Riesz representation theorem). If a linear map φ : L2 (U ) → C is continuous, i.e., there exists C > 0 such that for any f ∈ L2 (U ), |φ(f )| Cf L2 ,

(A.1.5)

then there exists u ∈ L2 (U ) such that for any f ∈ L2 (U ), φ(f ) = f, u . Definition A.1.2. For u ∈ L2 (U ), α ∈ Nn , we say that C > 0 such that for any f ∈ C0∞ (U ), ∂ α f, u Cf L2 . α ∂x

(A.1.6) ∂α ∂xα u

∈ L2 (U ) if there exists

(A.1.7) α

∂ 2 If (A.1.7) holds, by Theorem A.1.1, we can define ∂x α u ∈ L (U ) as follows: ∞ for any f ∈ C0 (U ), ∂α ∂α f, α u = (−1)|α| f, u . (A.1.8) ∂x ∂xα

Definition A.1.3. For k ∈ N, the Sobolev space H k (U ) is defined as H k (U ) = {u ∈ L2 (U );

∂α u ∈ L2 (U ) for any |α| k}. ∂xα

The Sobolev norm · k on H k (U ) is defined by: for u ∈ H k (U ), ∂ α 2 u2k = α u 2 . ∂x L

(A.1.9)

(A.1.10)

|α|k

Definition A.1.4. For k ∈ N∗ , the Sobolev space H −k (Rn ) is the dual space of H k (Rn ), i.e., the continuous linear maps from H k (Rn ) to C. Definition A.1.5. The Schwartz space S(Rn ) of rapidly decreasing functions is defined by S(Rn ) := {u ∈ C ∞ (Rn ); xβ

∂α u is bounded on Rn for all α, β ∈ Nn }. ∂xα (A.1.11)

A.1. Sobolev spaces on Rn

347

Certainly C0∞ (Rn ) ⊂ S(Rn ) ⊂ H k (Rn ) for any k ∈ N. Theorem A.1.6 (Sobolev embedding theorem). If m, k ∈ N, satisfy m > then H m (Rn ) → C k (Rn ).

n 2

+ k,

(A.1.12)

More precisely, there is C > 0 such that if f ∈ H m (Rn ), then f ∈ C k (Rn ) and |f |C k (Rn ) Cf m .

(A.1.13)

Let U1 ⊂ U be a bounded open set such that U1 ⊂ U . Let χ ∈ C0∞ (U ) such that χ = 1 on U1 . Then there exists C > 0 such that for any u ∈ H m (U ), we have χu ∈ H m (Rn ) and χum Cum .

(A.1.14)

Assume m > n2 + k. Theorem A.1.6 and (A.1.14) imply the existence of a constant CU1 > 0 such that for any u ∈ H m (U ), we have u ∈ C k (U ), and |u|C k (U1 ) CU1 um .

(A.1.15)

In our book, we always apply (A.1.15) for U = B(x, 1) ⊂ Rn and U1 = B(x, 12 ). We will denote by C k (Rn , Cl ), · · · , the corresponding spaces of Cl -valued functions. Certainly, all above results hold for Cl -valued functions. Let aα ∈ C ∞ (Rn , End(Cl )) with α ∈ Nn such that m = sup{|α| : aα ≡ 0} < ∞. We define a mth order differential operator H on Rn by H=

|α|m

aα (x)

∂α . ∂xα

(A.1.16)

√ The principal symbol of H is σ(H)(x, ξ) = |α|=m aα (x)( −1ξ)α with ξ ∈ Rn . If σ(H)(x, ξ) = a(x, ξ) IdCl with a(x, ξ) ∈ C ∞ (Rn ), we say that H has scalar principal symbol. We suppose that H is elliptic on Rn , that is, for any x ∈ Rn and ξ ∈ Rn \ {0}, σ(H) (x, ξ) ∈ End(Cl ) is invertible. Theorem A.1.7 (Basic elliptic estimate). Let U ⊂ Rn be a bounded open set. Then for any k ∈ N, there exist C1 , C2 > 0 such that for any u ∈ C0∞ (U, Cl ), we have uk+m C1 Huk + C2 uL2 .

(A.1.17)

348


A.2 Sobolev spaces on Rn+ Let Rn+ = {x = (x , xn ) ∈ Rn ; xn > 0} be the half space with closure Rn+ . Let S(Rn+ ) be the space of restrictions to Rn+ of elements of S(Rn ). For k ∈ N, let C k (Rn+ ), C0k (Rn+ ) be the space of restrictions to Rn+ of elements of C k (Rn ), C0k (Rn ). Let | · |C k (Rn ) be the C k -norm on C k (Rn+ ) induced by (A.1.2). +

For U ⊂ Rn , we define C k (Rn+ ∩ U ), etc, in a similar way. Now fix an integer N > 0, and for u ∈ S(Rn+ ), set u(x) , for xn 0, (Ψu)(x) = N for xn < 0 . j=1 cj u(x , −jxn ) ,

(A.2.1)

Theorem A.2.1. The space S(Rn+ ) is dense in H k (Rn+ ) for any k ∈ N. For any N ∈ N∗ , there exist {c1 , . . . , cN } such that the map Ψ has a unique continuous extension to Ψ : H k (Rn+ ) → H k (Rn ), for k N − 1.

(A.2.2)

Especially, each u ∈ H k (Rn+ ) is the restriction to Rn+ of an element of H k (Rn ). From Theorems A.1.6 and A.2.1, we get Theorem A.2.2 (Sobolev embedding theorem). If m, k ∈ N, satisfy m > then H m (Rn+ ) → C k (Rn+ ).

n 2

+k

(A.2.3)

More precisely, there is C > 0 such that if f ∈ H m (Rn+ ), then f ∈ C k (Rn+ ) and |f |C k (Rn ) Cf m . +

(A.2.4)

By using a cut-off function χ as in (A.1.14), we show that, if m > n2 + k, then there exists C > 0 such that if u ∈ H m (Rn+ ∩ B(0, 1)), then u ∈ C k (Rn+ ∩ B(0, 1)), and |u|C k (Rn ∩B(0, 1 )) Cum . +

(A.2.5)

2

A.3 Sobolev spaces on manifolds Now let X be a Riemannian manifold without boundary endowed with the Riemannian metric g T X on the tangent bundle T X, and dimR X = n. Let ∇T X be the Levi–Civita connection on (T X, g T X ). Let (E, hE ) be a Hermitian vector bundle on X with Hermitian connection ∇E .

A.3. Sobolev spaces on manifolds

349

For k ∈ N∪{∞}, we denote by C k (X, E), (resp. C0k (X, E)), the spaces of C k (resp. C k with compact support) sections of E on X. For a compact set K ⊂ X we denote C0k (K, E) := {s ∈ C k (X, E) : supp(s) ⊂ K}. For l ∈ N, we denote by ∇E the connection induced on (T ∗ X)⊗l ⊗ E by ∇T X and ∇E ; and by | · | the pointwise norm induced by g T X and hE . Let · L2 be the L2 -norm on C0∞ (X, (T ∗ X)⊗l ⊗ E) introduced as in (1.3.14). For k ∈ N, the C k (X)-norm | · |C k and Sobolev norm · k are defined for s ∈ C0∞ (X, E) by |s|C k (X) =

k l=0

s2k

=

k l=0

sup | 7∇E ·89 · · ∇E: s|, X

l times

7∇ ·89 ··∇ E

(A.3.1)

2 : sL2 .

E

l times

Let K ⊂ X be a compact set. The Sobolev space H k0 (K, E) is the completion respect to · k . If K ⊂ Y , where Y is a compact manifold, {s ∈ H k (Y, E) : supp(s) ⊂ K}. Assume moreover that K = U, where U is a relatively open set; then H k0 (K, E) coincides with the closure of C0∞ (U, E) with respect to · k , and we denote it by H k0 (U, E). For these statements we refer to [237, p. 291]. Let L2loc (X, E) be the space of locally L2 -integrable sections of E on X. We also define H k (X, E, loc) := {s ∈ L2loc (X, E) : χs ∈ H k0 (supp(χ), E) for any χ ∈ C0∞ (X)}. If X is a compact manifold, we set H k (X, E) := H k0 (X, E). of C0∞ (K, E) with then H k0 (K, E) =

Now, we state the corresponding version of Theorems A.1.6 and A.1.7. Theorem A.3.1. Let K ⊂ X be compact. a) (Rellich’s theorem) For any k ∈ N, the inclusion map j : H k+1 (K, E) → H k0 (K, E) 0

is compact,

i.e., j sends any bounded subset of H k+1 (K, E) to a relatively compact subset 0 of H k0 (K, E), equivalently, a set with compact closure. b) (Sobolev embedding theorem) If m, k ∈ N satisfy m > k such that if s ∈ H m 0 (K, E), then s ∈ C0 (K, E) and |s|C k (X) Csm .

n 2

+ k, there is C > 0 (A.3.2)

We call a generalized Laplacian associated to ∇E an operator H of the form H = ∆E + Q

(A.3.3)

where Q is a Hermitian section of End(E) on X, (i.e., Q(x)∗ = Q(x) for x ∈ X), and ∆E is the Bochner Laplacian associated to ∇E as in (1.3.19). We consider in the sequel a generalized Laplacian H.

350


Theorem A.3.2 (Basic elliptic estimate). Let K ⊂ X be compact. For any k ∈ N, (K, E), we have there exist C1 , C2 > 0 such that for any s ∈ H k+2 0 sk+2 C1 H sk + C2 sL2 .

(A.3.4)

We have a variant of the basic elliptic estimate for the quadratic form (cf. [238, Chap.7, Th.6.1]). Theorem A.3.3 (G˚ arding’s inequality). Let K ⊂ X be compact. There exists C > 0 such that for any s ∈ H 10 (K, E), we have s21 C(Hs, s + s2L2 ).

(A.3.5)

Theorem A.3.4 (Regularity). Let s ∈ L2loc (X, E) with compact support, and let K = supp(s). Assume that Hs ∈ H k0 (K, E) (in the sense that the current Hs is defined by an element of H k0 (K, E)). Then s ∈ H k+2 (K, E). In particular, if 0 Hs ∈ C ∞ (X, E), then s ∈ C ∞ (X, E).

Appendix B

Elements of Analytic and Hermitian Geometry We collect in this appendix the necessary tools of complex analytic geometry that we used.

B.1 Analytic sets and complex spaces A pair (X, OX ) of a topological space X and a subsheaf of rings OX of the sheaf of continuous functions CX such that each stalk OX,x is a local C-algebra is called a C-ringed space. Let X, Y be topological spaces, f : X −→ Y be a continuous map and A be a sheaf of abelian groups on X. The presheaf on Y defined by (f∗ A )(U ) = A (f −1 (U )) for any open set U ⊂ Y is a sheaf, called a direct image and denoted f∗ A . Let f : X −→ Y be a continuous map. Then f induces a canonical comorphism f : CY −→ f∗ CX by lifting the continuous functions: CY (U ) t −→ t ◦ f ∈ CX (f −1 (U )) = (f∗ CX )(U ).

(B.1.1)

A morphism of C-ringed spaces f : (X, OX ) −→ (Y, OY ) is a continuous map f : X −→ Y such that the canonical morphism (B.1.1) induces a morphism f : OY −→ f∗ OX , i.e. pull-backs of sections of OY lie in OX . Let G be a sheaf of OY -modules over Y . The topological inverse image f −1 (G ) is a sheaf on X determined uniquely by the property f −1 (G )x = Gf (x) , for any x ∈ X. The analytic inverse image sheaf G is defined by Gx = f −1 (G )x ⊗OY,f (x) OX,x , x ∈ X. G is a sheaf of OX -modules.

352

Appendix B. Elements of Analytic and Hermitian Geometry

Definition B.1.1. Let X be a complex manifold and OX be the sheaf of holomorphic functions on X. We say A ⊂ X is an analytic subset of X if for any x ∈ A there exist an open neighbourhood Ux in X and holomorphic functions f1 , . . . , fk ∈ OX (Ux ) on Ux such that A ∩ Ux = {y ∈ Ux : f1 (y) = · · · = fk (y) = 0}. A point x ∈ A is called a regular point if there exists an open neighbourhood Ux of x such that A ∩ Ux is a complex submanifold of X. The set of regular points of A is denoted by Areg . A point x ∈ A is a singular point of A if it is not regular; we denote Asing = A Areg . A is said to be irreducible if A can not be written as the union of two analytic subsets A1 , A2 with A1 , A2 = A. A ⊂ X is called an analytic hypersurface if A is locally the zero locus of a single non-zero holomorphic function g. Lemma B.1.2 ([107, § 2.15], [79, II.4.31]). If A ⊂ X is an analytic subset of a complex manifold X, then Areg , Asing are analytic subsets of X, and dim Asing < dim Areg . For the following basic result we refer to [79, III.7.6]. Definition and Theorem B.1.3. Let A be an analytic set of dimension k in a complex manifold X with dim X = n. Let x ∈ A. For a generic choice of local coordinates z = (z1 , . . . , zk ) and z = (zk+1 , . . . , zn ) around x there exists a neighborhood U of x in X and C > 0 such that A ∩ U ⊂ {|z | C|z |}. Denote by B (0, r) the ball of center 0 and radius r in Ck and B (0, Cr) the ball of center 0 and radius Cr in Cn−k . For r > 0 sufficiently small the projection π : A∩(B (0, r)×B (0, Cr)) → B (0, r) is a ramified covering with finite sheet number m, i.e., there exists a nowhere dense analytic set S ⊂ B (0, r) (called ramification locus) such that π : A π −1 (S) → B (0, r) S is a topological covering with m sheets. The sheet number is independent on the choice of generic coordinates. It is called the multiplicity of A at x and is denoted by mx (A). Definition B.1.4. Let V ⊂ X be an irreducible analytic hypersurface of a complex manifold X, and g be a local defining function for V near y ∈ V . For any holomorphic function f defined near y, we define the order ordV,y (f ) of f along V at y to be the largest integer k such that there exists h ∈ OX,y such that f = g k · h in the local ring OX,y . As relatively prime elements of OX,y are still relatively prime in OX,x for x near y, ordV,y (f ) is independent of y and we denote it simply by ordV (f ), the order of f along V . If f is a meromorphic function on X, written locally as f = f1 /f2 with f1 , f2 holomorphic, then we define ordV (f ) = ordV (f1 )−ordV (f2 ). Certainly, there exist only a locally finite number of irreducible analytic hypersurfaces V of X such that ordV (f ) = 0. The divisor Div(f ) of a meromorphic function f on X is defined by Div(f ) =

V

ordV (f ) · V.

(B.1.2)

B.1. Analytic sets and complex spaces

353

Let U be an open set in Cn and let I ⊂ OU be a coherent ideal sheaf. The set U /I ) := {x ∈ U | I (x) = OU,x } is an analytic subset of U and

A := supp(O A, (OU /I )|A is a C-ringed space called a local model. Definition B.1.5 (complex spaces). A complex space (X, OX ) is a C-ringed space which is Hausdorff and has the property that each x ∈ X possesses an open neighbourhood W such that (W, OW ) is isomorphic to a local model (as C-ringed spaces). A complex space is called reduced if its local models are of the form

A, (OU /IA )|A , where A is an analytic subset of U and IA ⊂ OU is the sheaf of all holomorphic functions which vanish on A. A section of the sheaf OX is called a holomorphic function. A complex space (X, OX ) is called irreducible if X cannot be represented as the union of two analytic subsets X1 , X2 with X1 , X2 = X. When not otherwise stated, we consider only reduced complex spaces. We will usually denote a complex space (X, OX ) simply by X. The property from the definition of being locally isomorphic to a model, means that there exists 1 ⊂ CN , an analytic subset W ⊂ W 1 and a homeomorphism ı : an open set W W −→ W , such that the canonical homomorphism ı : OW −→ ı∗ OW induces an isomorphism of sheaves of rings, where ı∗ OW is the direct image sheaf. The map 1 , obtained by composing ı with the inclusion W −→ W 1 , is called a ι : W −→ W local chart on X. Note that we have induced isomorphisms ιx : OW , ι(x) −→ OX,x , ιx (tx ) = (t ◦ ι)x , tx ∈ OW , ι(x) . We denote by Xreg the set of regular points, i.e. points for which a local chart 1 can be found, such that ι is surjective. In other words, a local model ι : W −→ W is an open set in the Euclidean space. The other points of X are called singular, and the set of such points is denoted by Xsing . For any complex space the singular locus Xsing is a nowhere dense analytic subset (cf. [107, § 2.15–16]). A reduced complex space is irreducible if and only if Xreg is connected. We state now two useful extension theorems. Theorem B.1.6 (Riemann’s second extension theorem). Let X be a complex space and A an analytic subset of codimension at least two at every point. Then every holomorphic function on X A has a unique holomorphic extension to X. For the proof we refer to [107, p. 120], [79, I.6.4]. Theorem B.1.7 (Remmert-Stein extension theorem). Let X be a complex space, A an analytic subset of X, Y an analytic subset of X A. Assume that there exists k ∈ N such that dim A k and dimy Y > k for all y ∈ Y . Then the closure Y of Y in X is an analytic set. For the proof see [120, p. 395-400], [79, II.8.6]. Theorem B.1.8 (Oka [79, II.3.19]). Let E be a holomorphic vector bundle over a complex space X. The sheaf OX (E) of holomorphic sections of E is a coherent sheaf.

354


Definition B.1.9 (holomorphic map). If (X, OX ), (Y, OY ) are arbitrary complex spaces, a morphism ϕ : X → Y of C-ringed spaces is called holomorphic map. A holomorphic map is called biholomorphic if there exists a holomorphic map ψ : Y → X such that ϕ ◦ ψ = IdY , ψ ◦ ϕ = IdX . A holomorphic map ϕ : X → Y is called proper if the preimage of any compact set is a compact set. A holomorphic map ϕ : X → Y is called finite if it is proper and each point x ∈ X is isolated in ϕ−1 (ϕ(x)). The following result shows that the fibers of a holomorphic map can be considered as a complex space. It is useful to determine if the fibers are connected. Theorem B.1.10 (Stein factorization). Let X, Y be complex spaces and ϕ : X → Y a proper surjective holomorphic map. Then there exists a complex space Z and surjective holomorphic maps ψ : X → Z, τ : Z → Y and a commutative diagram /Y X@ O @@ @@ τ ψ @@ Z ϕ

(B.1.3)

such that (a) The fibers of ψ are connected. (b) The fibers of τ are finite, i.e., τ is a finite holomorphic map . (c) For any y ∈ Y there is a bijection between the points of τ −1 (y) and connected components of ϕ−1 (y). For the proof we refer to [107, Th. 1.24, p. 70] and the references therein. Theorem B.1.11 (Zariski’s main theorem). Let X, Y be irreducible complex spaces and ϕ : X → Y a finite holomorphic map which is a proper modification. Then ϕ is a biholomorphic map. For the proof we refer to [107, 4.9, p. 187 ]. Theorem B.1.12 (Remmert’s proper mapping theorem). Let f : X → Y be a proper holomorphic mapping between complex spaces. Then f (X) is an analytic subset of Y . For the proof we refer to [107, 1.18, p. 64 ], [79, II.8.8].

B.2 Currents on complex manifolds In this section, we collect the necessary facts about currents on complex manifolds. We refer the reader to the textbooks [120, 79] for more details. If not otherwise stated, X will be a paracompact complex manifold of complex dimension n. We denote by T X the real tangent bundle of the underlying real manifold X.

B.2. Currents on complex manifolds

355

Proposition B.2.1. There exists a unique complex structure J on Tx X which satisfies the intrinsic Cauchy-Riemann equations: √ dfx (Jv) = −1dfx (v), v ∈ Tx X for all germs of holomorphic function f ∈ OX,x at x.

√ If (U, z1 , . . . , zn ) is a holomorphic system of coordinates and zj := xj + −1yj , then (U, x1 , y1 , . . . , xn , yn ) is a smooth system of coordinates for the real differentiable structure. The complex structure on Tx X given by J(∂/∂xj ) = ∂/∂yj , J(∂/∂yj ) = −∂/∂xj . We get in this way a bundle morphism J : T X −→ T X, J 2 = − Id. Definition B.2.2. The complex vector bundle Th X = (T X, J) is called the holomorphic tangent bundle of X. There is a natural splitting of the bundle of R-linear maps HomR (T X, C) = HomC (Th X, C) ⊕ HomC (Th X, C) in C-linear and C-antilinear maps. We set T ∗(1,0) X = HomC (Th X, C) , T ∗(0,1) X = HomC (Th X, C) . (B.2.1) √ √ Then dzj := dxj + −1dyj (resp. dz j := dxj − −1dyj ) is a local frame of T ∗(1,0) X (0,1) (resp. T ∗(0,1) X). The duals, T (1,0) √ X and T√ X, are obtained similarly from the decomposition of T X ⊗R C in −1 and − −1 eigenspaces. It is easy to check that T (1,0) X is a holomorphic vector bundle √ (i.e. the transition functions are holomorphic). The map Y ∈ Th X → 12 (Y − −1JY ) ∈ T (1,0)X induces the natural identification of Th X and T (1,0) X as complex vector bundles. Hence Th X has also the structure of a holomorphic vector bundle. The bundle of (p, q)-forms is Λp,q (T ∗ X) := Λp (T ∗(1,0) X) ⊗ Λq (T ∗(0,1) X) .

(B.2.2)

For any open set U ⊂ X we denote by Ω (U ) (resp. Ω (U )) the space of smooth r-forms (resp. (p, q)-forms) on U . The correspondence U → Ωr (U ) (resp. U → Ωp,q (U )) defines the sheaf Ωr (resp. Ωp,q ) of smooth r-forms (resp. smooth (p, q)-forms). We have the direct sum decomposition Ωr = ⊕p+q=r Ωp,q and the differentials d : Ωr −→ Ωr+1 (resp. ∂ : Ωp,q −→ Ωp+1,q and ∂ : Ωp,q −→ Ωp,q+1 ) with d = ∂ + ∂. We let Ωr0 (X) (resp. Ωp,q 0 (X)) denote the space of smooth r-forms (resp. (p, q)forms) on X with compact support. In this context, forms with compact support are called test forms. For K ⊂ X compact, let ΩrK (X) be the set of r-forms with support in K, endowed with C ∞ -topology, which makes it a Fréchet space. We introduce the direct limit topology on Ωr0 (X) by declaring a set E ⊂ Ωr0 (X) open if and only if E ∩ ΩrK (X) is open in ΩrK (X) for any compact set K ⊂ X. It is easy to see that for ϕν , ϕ0 ∈ Ωr0 (X), ϕν → ϕ0 in this topology, if and only if there exists a compact set K such that ϕν , ϕ0 ∈ ΩrK (X) and ϕν → ϕ0 in ΩrK (X). In the same vein we introduce the direct limit topology on Ωp,q 0 (X). r

p,q

Definition B.2.3. Let M be a real manifold of real dimension m. The space of currents of degree m−r is the space of complex linear continuous forms on Ωr0 (M ).

356


This space is denoted by Ω m−r (M ). We denote by (T, ϕ) := T (ϕ) the pairing between a current T ∈ Ω m−r (M ) and a test form ϕ ∈ Ωr0 (M ). A current is said to be of order k if it is continuous in the C k -topology. The space Ω 0 (M ) dual to Ωm 0 (M ) is called the space of distributions on M ∼ 0 and is denoted by D (M ). If M is orientable we can identify Ωm 0 (M ) = Ω0 (M ) using a volume element on M , so we can think D (M ) = Ω 0 (M ) as the dual of Ω00 (M ). In particular, let U be an open subset in Rm . The space of distributions on U is denoted D (U ) and is the dual of the space C0∞ (U ). The Schwartz space of rapidly decreasing test functions S(Rm ) (cf. (A.1.11)) ∂β endowed with the seminorms |f |α,β := supx∈Rm |xα ∂x echet space. β f (x)| is a Fr´ m m The dual S (R ) of S(R ) is called the space of tempered distributions. Assume that E is a complex vector bundle on M . The dual of the space Ωr0 (M, E ∗ ), denoted by Ω m−r (M, E), is called the space of currents of degree m − r with values in E. Assume now that X is a complex manifold of complex dimension n. The space of currents of bidegree (n−p, n−q) is the space of complex linear continuous forms n−p,n−q (X). on Ωp,q 0 (X). This space is denoted by Ω The spaces of currents (in particular distributions) are endowed with the weak topology. For example, a sequence {Tk } converges in Ω m−r (X) to T if and only if for any ϕ ∈ Ωr0 (X) we have limk→∞ (Tk , ϕ) = (T, ϕ). Note that by the definition of the direct limit topology, a linear form T on r Ωr0 (X) (resp. Ωp,q 0 (X)) is a current if and only if T is continuous on ΩK (X) (resp. p,q ΩK (X)) for any compact K ⊂ X. Equivalently, for any compact set K ⊂ X, there exist k ∈ N and C > 0 such that |(T, ϕ)| C|ϕ|C k (X) for any ϕ ∈ ΩrK (X). We have the following direct sum decompositions: Ωr (X) = ⊕p+q=r Ωp,q (X) ,

Ω r (X) = ⊕p+q=r Ω p,q (X).

(B.2.3)

For two open sets U ⊂ V ⊂ X we define the restrictions Ω r (V ) → Ω r (U ) and Ω p,q (V ) → Ω p,q (U ), T −→ T |U simply by (T |U , ϕ) := (T, ϕ) for ϕ ∈ Ωn−r (U ) (resp. ϕ ∈ Ωn−p,n−q (U )). It is easy to see that this defines sheaves 0 0 Ω r and Ω p,q on X. We define the support of T as supp(T ) := X ∪{V ⊂ X : V open, T |V = 0}. Definition B.2.4 (operations with currents). 1. For T ∈ Ω r (X), α ∈ Ωs (X) the wedge product T ∧ α is defined by (T ∧ α , ϕ) = (T , α ∧ ϕ) ,

(X). ϕ ∈ Ω2n−r−s 0

(B.2.4)

2. The operator d : Ω r (X) −→ Ω r+1 (X) is defined by (dT, ϕ) = (−1)r+1 (T, dϕ) , A current T is called closed if dT = 0.

ϕ ∈ Ω2n−r−1 (X). 0

(B.2.5)


357

3. The operator ∂ : Ω p,q (X) −→ Ω p,q+1 (X) is defined by (∂T, ϕ) = (−1)p+q+1 (T, ∂ϕ) ,

ϕ ∈ Ω0n−p,n−q−1 (X).

(B.2.6)

The operator ∂ : Ω p,q −→ Ω p+1,q (X) is defined analogously and d = ∂ + ∂. 4. Let f : X −→ Y be a smooth map and T ∈ Ω r (X). If f |supp(T ) is proper then (f∗ T, ϕ) = (T, f ∗ ϕ) , ϕ ∈ Ω02n−r (Y ) (B.2.7) defines a current f∗ T called the push-forward of T by f . Example B.2.5. Let u ∈ L2loc (X, Λr (T ∗ X) ⊗R C) be alocally L2 integrable r-form. Then u defines a current Tu ∈ Ω r (X) by (Tu , ϕ) = X u ∧ ϕ, for ϕ ∈ Ω02n−r (X). The map L2loc (X, Λr (T ∗ X) ⊗R C) → Ω r (X), u → Tu is injective, and we can identify Tu with u. Definition B.2.6. A current T ∈ Ω r (X) is said to be smooth on an open set U if T |U = Tu for some u ∈ Ωr (U ). The singular support sing supp(T ) of a current T ∈ Ω r (X) is defined as the smallest subset Y of X such that T is a smooth form on X Y . Let us recall the classical Schwartz kernel theorem [237, p. 296]. Let M be a real orientable manifold of dimension m and E, F be two vector bundles on M . We denote by D0 (M, E) the space of E-valued distributions with compact support. Let us fix a volume form dµ on M . Then u ∈ L1loc (M, E) defines the distribu tion Tu ∈ D (M, E) by setting (Tu , ϕ) = M u(y) · ϕ(y) dµ(y), for ϕ ∈ C0∞ (M, E ∗ ), where u(y) · ϕ(y) is the pairing E × E ∗ → C. We denote by pr1 , pr2 the projections from M × M on the first and second factor M respectively. We introduce the vector bundle E F = pr∗1 E ⊗ pr∗2 F on M ×M . For u ∈ C0∞ (M, E), v ∈ C0∞ (M, F ∗ ) we define v⊗u ∈ C0∞ (M ×M, F ∗ E) by (v ⊗ u)(x, y) := v(x) ⊗ u(y). Let K ∈ D (M × M, F E ∗ ). For any fixed u ∈ C0∞ (M, E), the linear map ∞ C0 (M, F ∗ ) v → (K, v ⊗ u) ∈ C defines a distribution AK (u) ∈ D (M, F ) . The operator AK : C0∞ (M, E) → D (M, F ), u → AK (u), is linear and continuous. Theorem B.2.7 (Schwartz kernel theorem). (a) Let A : C0∞ (M, E) → D (M, F ) be a linear continuous operator. Then there exists a unique distribution K ∈ D (M × M, F E ∗ ), called the Schwartz kernel distribution such that A = AK , i.e., (Au, v) = (K, v ⊗ u)

(B.2.8)

for any u ∈ C0∞ (M, E), v ∈ C0∞ (M, F ∗ ). (b) Assume A : C0∞ (M, E) −→ C ∞ (M, F ) is linear continuous. Then for any volume form dµ, the Schwartz kernel of A is represented by a smooth kernel

358


K ∈ C ∞ (M × M, F E ∗ ) called the Schwartz kernel of A with respect to dµ such that K(x, y)u(y) dµ(y) , for any u ∈ C0∞ (M, E) . (B.2.9) (Au)(x) = M

Moreover, A can be extended as a linear continuous operator A : D0 (M, E) → C ∞ (M, F ) by setting

(Au)(x) = u(·), K(x, ·) , x ∈ M , (B.2.10) for any u ∈ D0 (M, E). Linear continuous operators A : D0 (M, E) → C ∞ (M, F ) are called smoothing operators. For I ⊂ {1, 2, . . . , n}, we set I = {1, 2, . . . , n} I. If I = {i1 < i2 < · · · < ip }, I = {j1 < j2 < · · · < jn−p }, we denote by (II) the permutation {i1 , i2 , . . . , ip , j1 , j2 , . . . , jn−p } of {1, 2, . . . , n}. Let (z1 , . . . , zn ) be local coordinates on an open set U . We can identify the space of currents Ω p,q (U ) with the space of (p, q)-forms with distribution coefficients. For T ∈ Ω p,q (X) we have TIJ ∧ dzI ∧ dz J , TIJ ∈ Ω 0,0 (U ), (B.2.11) T |U = |I|=p,|J|=q

where dzI := dzi1 ∧ · · · ∧ dzip and TIJ (dz1 ∧ · · · ∧ dzn ∧ dz 1 ∧ · · · ∧ dz n ) = (−1)(n−p)q sgn(I I) sgn(J J) T (dzI ∧ dz J ) . We can of course identify TIJ ∈ Ω 0,0 (U ) with a usual distribution TIJ ∈ Ω n,n (U ) due to the canonical isomorphism Ω0,0 (U ) ∼ = Ωn,n (U ). We discuss now positivity notions for currents and start with the definition for (1, 1)-forms. For a thorough discussion see [79, III.1]. Definition and Theorem B.2.8. Let ω : T X × T X → R be a real (1, 1)-form on the complex manifold X. Then the following assertions are equivalent: 1. For all v ∈ T X, v = 0, we have ω(v, Jv) > 0 (resp. 0). √ 2. If ω has the local form ω = 2−1 jk hjk (z)dzj ∧ dz k , the Hermitian matrix (hjk (z)) is positive definite (resp. semidefinite) for all z. √ 3. ω : T (1,0) X × T (0,1) X −→ C satisfies − −1ω(u, u) > 0 (resp. 0) for all u ∈ T (1,0) X, u = 0. If they are satisfied, ω is called positive (resp. semipositive). In the case of a positive form the conditions are also equivalent to 4. ω = − Im hTh X , where hTh X is a Hermitian metric on Th X.


359

Definition B.2.9. (i) Let U be an open set in C. A function f : U → [−∞, ∞[ is called subharmonic if it is upper semicontinuous and satisfies the mean value inequality on any closed ball B(a, r) ⊂ U with center a and radius r, i.e., 2π 1 u(a + reiθ ) dθ . (B.2.12) u(a) 2π 0 (ii) A function f : X −→ [−∞, ∞[ is called plurisubharmonic if it is upper semicontinuous and for any holomorphic function u : B(a, 1) ⊂ C → X, f ◦ u is subharmonic. Proposition B.2.10 ([79, I.Th. √ 5.8]). Let ϕ : X −→ R be smooth. Then ϕ is plurisubharmonic if and only if −1∂∂ϕ is a semipositive form. Definition B.2.11. (i) A (1, 1)-current T is called real if T = T in the sense that T (ϕ) = T (ϕ) for √ 2 (X) and a real current T is positive when ( −1)(n−1) T (ψ ∧ all ϕ ∈ Ωn−1,n−1 0 ψ) 0 for all ψ ∈ Ωn−1,0 (X). 0 (ii) A real (1, 1)-current T on X is called strictly positive if there exists a positive C ∞ -(1, 1)-form ω on X such that T − ω is a positive current on X. Note that the terminology for forms and currents is not symmetric. We kept the traditional terminology of Lelong, although one might call “semi-positive” a current as in Definition B.2.11 (i). If a current T is positive we write T 0. T1 T2 means T1 − T2 0. 1,1 (X) A current T ∈ Ω√ can be written in local coordinates (z1 , . . . , zn ) on an open set U as T |U = −1 jk Tjk dzj ∧dz k with distributions TIJ ∈ Ω n,n (U ) (cf. (B.2.11)). T is real if Tjk = T kj and positive if and only if the current jk Tjk λj λk is a positive measure for every (λ1 , . . . λn ) ∈ Cn . In particular Tjk are measures and T is of order zero. Definition B.2.12. For a measure µ on a manifold M we denote by µac and µsing the uniquely determined absolute continuous and singular measures (with respect to the Lebesgue measure on M ) such that µ = µac + µsing

(B.2.13)

which is called the Lebesgue decomposition of µ. √ If T is a (1, 1)-current of order 0 on X, written locally T = −1 Tij dzi ∧ dz j , we define its absolute continuous and singular components by √ Tac = −1 (Tjk )ac dzj ∧ dz k , (B.2.14) √ (Tjk )sing dzj ∧ dz k . Tsing = −1

360


The Lebesgue decomposition of T is then T = Tac + Tsing .

(B.2.15)

Remark B.2.13. 1) If T 0, it follows that Tac 0, Tsing 0. Moreover, if T α for a continuous (1, 1)-form α, then Tac α, Tsing 0. 2) The Radon–Nikodym theorem insures that Tac is (the current associated to) a (1, 1)-form with L1loc coefficients. The form Tac (x)n exists for almost all x ∈ X n and is denoted Tac . Proposition B.2.14 ([79, I,Th. 5.8]). Let ϕ : X −→ [−∞, ∞[ √ be a plurisubharmonic function such that ϕ ∈ L1loc (X). Then the (1,1)-current −1∂∂ϕ is positive. √ Conversely, if ϕ ∈ L1loc (X) and −1∂∂ϕ is positive, then there exists a plurisubharmonic function ψ : X −→ [−∞, ∞[ such that ϕ = ψ almost everywhere. Example B.2.15. If f is a holomorphic function on X, log |f | is plurisubharmonic. Definition B.2.16. A plurisubharmonic function √ ϕ : X −→ [−∞, ∞[ is called strictly plurisubharmonic if ϕ ∈ L1loc (X) and −1∂∂ϕ is strictly positive. A function ϕ : X −→ [−∞, ∞[ is called quasi-plurisubharmonic if ϕ can be written locally as the sum of a smooth function and a plurisubharmonic function. Proposition B.2.17. Let ϕ : X −→ [−∞, ∞[ be a quasi-plurisubharmonic function such √ that ϕ ∈ L1loc (X). Then there exists a continuous (1, 1) form α on X such that −1∂∂ϕ√ α in the sense of currents. Conversely, if ϕ ∈ L1loc (X) has the property that −1∂∂ϕ α for some continuous (1, 1)-form α on X, then ϕ equals a quasi-plurisubharmonic function almost everywhere. Example B.2.18 (cf. [79, III-1.20]). Let Y be an analytic subset of X of pure dimension m. The current of integration on the analytic set Y , denoted [Y ] ∈ Ω n−m,n−m (X), is defined by ϕ , ϕ ∈ Ωm,m (X) , (B.2.16) ( [Y ], ϕ) = 0 Yreg

where Yreg is endowed with the canonical orientation. P. Lelong showed that this is well defined (cf. [79, III-2.6], [120, p. 32]) which amounts to showing that Yreg has finite volume locally near Ysing . Moreover, a fundamental theorem of Lelong [79, III-2.7], [120, p. 33] states that [Y ] is closed. Another important result due to Lelong is the following (see [79, III-2.15]). If D = cj Dj with cj ∈ Z is a divisor on X, where Dj are irreducible hypersurfaces, we define the current of integration on D by (B.2.17) [D] = cj [Dj ] . Theorem B.2.19 (Poincaré-Lelong formula). Let f be a meromorphic function on X which is non-zero on any open set. Then √

−1 2π ∂∂

log |f |2 = [ Div(f ) ] .

(B.2.18)


361

Let T be a closed positive (1, 1)-current in Cn . We set 1 T ∧ ω n−1 , (B.2.19) ν(T, x, r) = (πr2 )n−1 B(x,r) √ where ω = 2−1 j dzj ∧ dz j is the Euclidean K¨ ahler form on Cn . Then ν(T, x, r) is an increasing function of r. We define the Lelong number of T at x by ν(T, x) = limr→0 ν(T, x, r). By a result of Siu, the Lelong number is invariant by holomorphic changes of coordinates, and can be therefore defined for currents on manifolds. We refer the reader to [79] for detailed information about Lelong numbers. We close this section with some remarks about cohomology of currents. It is 2 easy to see that d2 = 0 and ∂ = 0 also at currents level. Thus we get complexes of sheaves X: (Ω • , d) : Ω 0 −→ Ω 1 −→ Ω 2 −→ · · · , (B.2.20) (Ω r,• , ∂) : Ω r,0 −→ Ω r,1 −→ Ω r,2 −→ · · · . Note that the sheaves Ωr and Ω r , (r 0) are fine. Moreover, the Poincaré and Dolbeault-Grothendieck lemmas (∂-Lemma) are still valid at current level. Thus (Ω• , d) and (Ω • , d) are resolutions of C (or R) on X. There are natural homomorphisms of complexes of sheaves Z −→ R −→ C −→ (Ω• , d) −→ (Ω • , d) which induce homomorphisms of hypercohomology groups H ∗ (X, Z) −→ H ∗ (X, R) −→ H ∗ (X, Ω• ) −→ H ∗ (X, Ω • ).

(B.2.21)

By the fineness of Ωr and Ω r (r 0), the canonical edge homomorphisms H ∗ (Γ(X, Ω• )) −→ H ∗ (X, Ω• ) ,

H ∗ (Γ(X, Ω • )) −→ H ∗ (X, Ω • )

are isomorphisms. We denoted by Γ(X, Ω • ) the space of sections of the sheaf Ω • over X. Lemma B.2.20 (∂∂-Lemma for currents). Let T be a closed, real, (1, 1)-current of order 0. Then for every contractible open set U √there exists a real function ϕ ∈ L1loc (U ), called a local potential, such that T = −1∂∂ϕ. If T is a (strictly) positive (1, 1)-current, then ϕ is (strictly) plurisubharmonic. Indeed, since T is closed, as in [120, p. 387] and [79, III-1.18], √the Poincaré and Dolbeault-Grothendieck lemmas for currents imply that T = −1∂∂ϕ for some distribution ϕ. (Compare to the global version of the ∂∂-Lemma 1.5.1 given by Hodge Theory.) Then, locally ∆ϕ = −2 Tr(∂∂ϕ) is of order 0 and by the regularity theory for elliptic equations, we have ϕ ∈ L1loc . If T is (strictly) positive then ϕ is (strictly) plurisubharmonic. A real (1, 1)-current ω on X is said to be a K¨ ahler current if it is d-closed and strictly positive on X. A d-closed (1, 1)-current or a d-closed C ∞ -(1, 1)-form is said to be integral if its cohomology class is in the image of H 2 (X, Z) under the map in (B.2.21).

362


B.3 q-convex and q-concave manifolds We recall here briefly some notions linked to pseudoconvexity. Definition B.3.1. Let X be a complex manifold, ϕ ∈ C ∞ (X). The complex Hessian of ϕ is the (1, 1)-form ∂∂ϕ. Let us introduce an auxiliary Riemannian metric on X; as in (1.5.15), we identify ∂∂ϕ with a Hermitian matrix H(ϕ) ∈ C ∞ (X, End(T (1,0) X)) and we will speak about the eigenvalues of ∂∂ϕ at a point x ∈ X. Definition B.3.2 (Andreotti-Grauert [4]). Let X be a complex manifold of complex dimension n. Let 1 q n. (i) X is called q-convex if there exists a smooth function ϕ : X −→ [a, b[, a ∈ R, b ∈ R ∪ {+∞} such that the sublevel set Xc = {ϕ < c} X for all c ∈ [a, b[ and the complex Hessian ∂∂ϕ has at least n − q + 1 positive eigenvalues outside a compact set (exceptional set) K. Here Xc = {ϕ < c} X means that the closure Xc of Xc is a compact set in X. (ii) X is called q-complete if it is q-convex with K = ∅. (iii) X is called q-concave if there exists a smooth function ϕ : X −→]a, b], a ∈ R ∪ {−∞}, b ∈ R such that the superlevel set Xc = {ϕ > c} X for all c ∈]a, b] and ∂∂ϕ has at least n − q + 1 positive eigenvalues outside a compact set. In all these cases we call ϕ an exhaustion function . Immediate examples of 1-complete manifolds are Cn (resp. balls B(a, r) = {z ∈ Cn : |z − a| < r}) with exhaustion function ϕ = |z|2 (resp. ϕ = |z − a|2 ). Definition B.3.3. Let X be a complex space. We say that: (a) X is holomorphically separable if for any x, y ∈ X, x = y, there exists f ∈ OX (X) with f (x) = f (y). (b) X is holomorphically regular if for any x ∈ X the set of germs {fx : f ∈ OX (X)} generates the maximal ideal mx of OX,x . " = {x ∈ X : (c) X is holomorphically convex if for any compact set K ⊂ X, K |f (x)| supK |f | , for all f ∈ OX (X)} is compact. (d) X is called Stein space if X is holomorphically separable, regular and convex. By definition, a complex manifold is 1-complete if it admits a smooth strictly plurisubharmonic function ϕ such that {ϕ < c} X for any c ∈ R. It is easy to see that a Stein manifold is 1-complete. The problem whether the converse is true was known as the Levi problem on complex manifolds. It was solved affirmatively by Hans Grauert [115] (see also [132, 108, 78]). Theorem B.3.4 (Grauert). A complex manifold is Stein if and only if it admits a smooth strictly plurisubharmonic exhaustion function.

B.3. q-convex and q-concave manifolds

363

We need the solution of the Levi problem in a more general form. Theorem B.3.5 (Grauert). Let X be a 1-convex manifold. Then there exists (a) a compact analytic set A ⊂ X with dimx A > 0 for any x ∈ A, (b) a Stein space X with at worst isolated singularities, a finite set D ⊂ X and a proper holomorphic map π : X → X , such that π(A) = D, π : X A → X D is a biholomorphism, and π∗ OX = OX . The Stein space X is called the Remmert reduction of X and A the exceptional analytic set of X. We say that A can be blown down to a finite set. The following result shows when Stein spaces can be embedded in the Euclidean space. Theorem B.3.6 (Remmert-Bishop-Narasimhan [184]). Assume that X is a Stein space of dimension n and of finite type m > n, that is, X can be locally realized as an analytic set in Cm . Then the set of proper regular embeddings of X in Cn+m is dense in the set of all holomorphic mappings of X in Cn+m endowed with the topology of uniform convergence. The analytic convexity of a manifold is determined by the behavior of the complex Hessian of an exhaustion function on the analytic tangent space of sublevel sets. Let M be a relatively compact domain with smooth boundary in a complex manifold X. Let ρ ∈ C ∞ (U ) be defined on an open neighbourhood U of M such that M = {x ∈ X : ρ(x) < 0} and dρ = 0 on ∂M . We say that ρ is a defining function of M . The analytic tangent space to ∂M at x ∈ ∂M is given by Tx(1,0) ∂M = {v ∈ Tx(1,0) X : ∂ρ(v) = 0}.

(B.3.1)

The definition does not depend on the choice of ρ. Definition B.3.7. The Levi form of ρ is the 2-form Lρ ∈ C ∞ (∂M, T (1,0) ∂M ⊗ T (0,1) ∂M ), for U, V ∈

(1,0) Tx ∂M ,

Lρ (U, V ) = (∂∂ρ)(U, V ),

(B.3.2)

x ∈ ∂M .

Lemma B.3.8. The number of positive and negative eigenvalues of the Levi form is independent of the choice of the defining function ρ. Proof. Consider another defining function ρ1 ∈ C ∞ (U ). Let x ∈ ∂M . We can take ρ as part of a set of smooth coordinate functions (ρ, x ), x = (x2 , . . . , x2n ) centered at x = 0. By applying the Taylor formula with integral rest we find ρ1 (ρ, x ) = ρ1 (ρ, x ) − ρ1 (0, x ) = ρ · h(ρ, x ), where h is smooth and h(x) = 0. Since ρ1 < 0 and ρ < 0 on M we have h(x) > 0. Computing ∂∂(ρ1 ) = ∂∂(ρh) = ρ∂∂h + ∂ρ ∧ ∂h + ∂h ∧ ∂ρ + h∂∂ρ (1,0)

(B.3.3)

and taking into account that ρ(x) = 0 and ∂ρ, ∂ρ vanish on Tx ∂M we obtain (1,0) Lρ1 (U, V ) = h(x)Lρ (U, V ) for any U, V ∈ Tx ∂M . This implies immediately Lemma B.3.8.

364


Definition B.3.9. The domain M is called strongly pseudoconvex (resp. (weakly) pseudoconvex ) if the Levi form is positive definite (resp. semidefinite). Example B.3.10. Let X be a compact complex manifold and let (L, hL ) be a positive line bundle on X. We consider the Grauert tube M = {v ∈ L∗ : |v|hL∗ < 1} . If we set ρ = |v|2hL∗ − 1, then ∂∂ρ|T (1,0) ∂M = π ∗ (RL )|T (1,0) ∂M ,

(B.3.4)

where π : M −→ X is the projection. Thus Lρ is positive definite and M is a strongly pseudoconvex domain called the Grauert tube. Example B.3.11. Assume that M = {ρ < 0} ⊂ X is strongly pseudoconvex. By replacing ρ with eCρ − 1 for C 1, we can achieve that ∂∂ρ is positive definite (1,0) on the whole tangent space Tx X, x ∈ ∂M , and therefore we can assume that ρ is strictly plurisubharmonic in a neighbourhood of ∂M . It follows that M is a 1-convex manifold. Indeed, the function ϕ : M −→ R, ϕ = ρ12 is a strictly plurisubharmonic exhaustion function. Conversely, if X is a 1-convex manifold and Xc ⊃ K is smooth, then Xc is strongly pseudoconvex. Let X be a compact complex manifold and E be a holomorphic vector bundle of rank m over X. Let us consider the projectivized bundle π : P(E ∗ ) −→ X whose fiber π −1 (x), x ∈ X, is the projective space P(Ex∗ ), i.e., the set of complex lines through the origin in Ex∗ . The tautological line bundle OP(E ∗ ) (−1) ⊂ π ∗ E ∗ over P(E ∗ ) is the line bundle such that the fiber over (y, [ξ]) is Cξ ⊂ Ey∗ . Let OP(E ∗ ) (1) be its dual line bundle. Definition B.3.12. E is said to be ample if OP(E ∗ ) (1) is ample on P(E ∗ ) (cf. Definition 5.1.1). E is called Grauert positive if the zero section Z(E ∗ ) of the dual bundle E ∗ has a strongly pseudoconvex (equivalently, 1-convex) neighborhood. The previous example shows that a positive line bundle is Grauert positive. The converse is proved in Problem 5.1. Thus, positivity, ampleness and Grauert positivity are equivalent for holomorphic line bundles. Theorem B.3.13 (Grauert’s ampleness criterion [116]). A holomorphic vector bundle E on a compact complex manifold X is ample if and only if it is Grauert positive. In this case one can show that Z(E ∗ ) is the exceptional analytic set of its 1-convex neighborhood in E ∗ . Hence Z(E ∗ ) can be blown down to a finite set of points (cf. Theorem B.3.5). The notion of weakly 1-complete manifold was introduced by S. Nakano [182] in order to solve the problem of the inverse of the monoidal transformation. Definition B.3.14. A complex manifold X is called weakly 1-complete if there exists a smooth plurisubharmonic function ϕ : X → R such that {ϕ < c} X for any c ∈ R. ϕ is called an exhaustion function.

B.4. L2 estimates for ∂

365

Any 1-convex (and therefore any compact or Stein) manifold is weakly 1complete. A proper modification (cf. Section 2.1) of a weakly 1-complete manifold is again weakly 1-complete. Proposition B.3.15. Every weakly 1-complete K¨ ahler (in particular Stein) manifold carries a complete K¨ ahler metric. Proof. By adding a constant we can consider a smooth exhaustion function ϕ > 0. Let χ : [0, +∞[→ R be a convex increasing function satisfying χ (t) > 0 and ∞ χ (t)dt = ∞. Let ω be a K¨ ahler form on X. Then 0 ω0 = ω +

√ √ √ −1∂∂χ(ϕ) = ω + −1χ (ϕ)∂ϕ ∧ ∂ϕ + −1χ (ϕ)∂∂ϕ

(B.3.5)

is the K¨ ahlerform of a complete metric on X. To prove this, let’s observe that |∂ϕ|ω0 1/ χ (ϕ) by (B.3.5). Introduce the function λ : X → R, λ(x) = ϕ(x) χ (t)dt. Since ϕ is an exhaustion function, λ is proper. Moreover, ∂λ = 0 χ (ϕ)∂ϕ, hence |∂λ|ω0 1, so |dλ|ω0 2. By the same argument after (3.4.9) we conclude √ that ω0 is complete. If X is 1-complete the above argument works with ω0 = −1∂∂χ(ϕ).

B.4 L2 estimates for ∂ We recall here the standard L2 existence theorem of Hörmander–Andreotti–Vesentini [7, 131]. For excellent introductory books see [132, 79, 189]. Let (X, J, Θ) be a complex Hermitian manifold of dimension n as in Def. 1.2.7 and (E, hE ) be a holomorphic Hermitian vector bundle over X. We denote by dvX the Riemannian volume form on (X, Θ). √ We use the notation from Section 1.4.3 for [ −1RE , i(Θ)]. On the space of 2 smooth forms with compact support Ω•,• 0 (X, E), we introduce the L scalar product (1.3.14). The corresponding Hilbert space completion is denoted by L2•,• (X, E). Theorem B.4.1 (H¨ ormander-Andreotti-Vesentini). Let (X, Θ) be a complete K¨ ahler manifold of dimension n. Assume that for some (r, q), q 1, there exists a continuous function ψ : X → [0, ∞[ such that pointwise √ [ −1RE , i(Θ)]s, sΛr,q ⊗E ψ |s|2 , for all s ∈ Ωr,q (B.4.1) 0 (X, E). E

Then for any form f ∈ L2r,q (X, E) satisfying ∂ f = 0 and E

there exists g ∈ L2r,q−1 (X, E) such that ∂ g = f and |g|2 dvX ψ −1 |f |2 dvX . X

X

Moreover, if f is smooth, g can be chosen smooth.

X

ψ −1 |f |2 dvX < ∞,

(B.4.2)

366


Proof. Consider the complex of closed, densely defined operators analogue to (3.1.3) T =∂

E

S=∂

E

L2r,q−1 (X, E) −→ L2r,q (X, E) −→ L2r,q+1 (X, E) ,

(B.4.3)

E

where T and S are the maximal extensions of ∂ . We apply the Nakano inequality (Theorem 1.4.14) and obtain for all s ∈ Ωr,q 0 (X, E) √

E E,∗ 2 2 E ψ |s|2 dvX . (B.4.4) ∂ sL2 + ∂ sL2 [ −1R , i(Θ)]s, s X

Since the metric associated to Θ is complete, Lemma 3.3.1 shows that (B.4.4) holds for any s ∈ Dom(S) ∩ Dom(T ∗ ). By the Cauchy–Schwarz inequality and (B.4.4) we have for any s ∈ Dom(S) ∩ Dom(T ∗ )

| f, s |2 = | ψ −1/2 f, ψ 1/2 s |2 ψ −1 |f |2 dvX · ψ|s|2 dvX X X (B.4.5)

E E,∗ ψ −1 |f |2 dvX · ∂ s2L2 + ∂ s2L2 . X

Consider now s ∈ Dom(T ∗ ) and write the decomposition s = s1 + s2 , s1 ∈ Ker(S), s2 ∈ Ker(S)⊥ . Since f ∈ Ker(S) by hypothesis, | f, s | = | f, s1 |. Applying (B.4.5) for s1 ∈ Ker(S) ∩ Dom(T ∗ ) we obtain

| f, s |2 = | f, s1 |2 (B.4.6) ψ −1 |f |2 dvX · T ∗ s1 2L2 = ψ −1 |f |2 dvX · T ∗ s2L2 , X

X ⊥

∗

where we used that s2 ∈ Ker(S) = [Im(S )] ⊂ Ker(T ∗ ) (cf. (C.1.2)).

We consider the antilinear functional Im(T ∗ ) T ∗ s → f, s ∈ C. By (B.4.6) 1/2

it is well defined and bounded, with norm X ψ −1 |f |2 dvX . We can thus extend it to a bounded functional on [Im(T ∗ )] and then by zero on the complement of

1/2 [Im(T ∗ )] to a bounded functional on L2r,q−1 (X, E), of norm X ψ −1 |f |2 dvX . By the Riesz representation theorem, there exists g ∈ L2r,q−1 (X, E) such that f, s = g, T ∗ s for all s ∈ Dom(T ∗ ) and g2L2 X ψ −1 |f |2 dvX . But this means that g ∈ Dom(T ), T g = f and (B.4.2) is satisfied. We wish to prove the regularity statement now. By replacing g with its projection on Ker(T )⊥ (which is still a solution of T g = f ) we can consider g ∈ Ker(T )⊥ . This solution is sometimes called the canonical solution of T g = f , since U=∂

E

it is the unique solution of minimal L2 norm. Let L2r,q−2 (X, E) −→ L2r,q−1 (X, E) E

be the maximal extension of ∂ in bidegree (r, q − 2). Since g ∈ Ker(T )⊥ = [Im(T ∗ )] ⊂ Ker(U ∗ ) (cf. (C.1.2)), we have U U ∗g = 0 ,

(U U ∗ + T ∗ T )g = T ∗ f ∈ C ∞ (X, Λr,q−1 (T ∗ X) ⊗ E).

(B.4.7)

B.4. L2 estimates for ∂

367

Applying the regularity theorem A.3.4 for the elliptic operator U U ∗ + T ∗T = E , we obtain that g is also smooth. The proof is complete. Using a weight function χ(ϕ) as in (3.5.8) and (B.3.5), with a convex rapidly increasing function χ : [0, ∞[→ R and a positive exhaustion function ϕ, in order to temper the growth of smooth forms at infinity, we get: Corollary B.4.2. If X is a Stein manifold, then for any (r, q) with q 1 and f ∈ E E Ωr,q (X, E) satisfying ∂ f = 0, there exists g ∈ Ωr,q−1 (X, E) such that ∂ g = f . In particular, if X = Bn , the unit ball in Cn , we obtain: Corollary B.4.3 (Dolbeault-Grothendieck Lemma or ∂-Lemma). If q 1 and f ∈ Ωr,q (Bn ) satisfying ∂f = 0, then there exists g ∈ Ωr,q−1 (Bn ) such that ∂g = f . We apply the results above to cohomology theory. For a sheaf F over X, we define the qth cohomology group of F by H q (X, F ) := H q (Γ(X, S • )), where 0 → F → S • is the canonical (Godement) flasque resolution of F where we denote by Γ(X, G ) the space of sections of the sheaf G over X. •,• (U, E), Consider the sheaves Ω•,• X (E) of smooth forms, defined by U → Ω for any open set U ⊂ X. We are interested in computing the cohomology of the r (E) of holomorphic r-forms with values in E, i.e. holomorphic sections sheaf OX r,0 0 of ΩX (E). For r = 0 we set OX (E) := OX (E), the sheaf of holomorphic sections • • of E. We will denote H (X, E) := H (X, OX (E)). By the Dolbeault–Grothendieck lemma B.4.3 applied to small coordinate r (E)): balls, the following sequence of sheaves is exact (forms a resolution of OX ∂

E

∂

E

∂

E

r,1 r,n r 0 −→ OX (E)−→Ωr,0 X (E) −→ ΩX (E) −→ · · · −→ ΩX (E) −→ 0 .

(B.4.8)

The complex of global sections ∂

E

∂

E

∂

E

0 −→ Ωr,0 (X, E) −→ Ωr,1 (X, E) −→ · · · −→ Ωr,n (X, E) −→ 0 ,

(B.4.9)

is called the Dolbeault complex . The cohomology of (B.4.9) is denoted by H •,• (X, E) and called the Dolbeault cohomology of X with values in E. Since Ω•,• X (E) are fine sheaves, hence acyclic, the abstract de Rham theorem entails: Theorem B.4.4 (Dolbeault isomorphism). The canonical morphism r H r,q (X, E) → H q (X, OX (E))

(B.4.10)

is an isomorphism. For a general complete K¨ ahler manifold endowed with a semi-positive line bundle, the L2 method applies to solve the ∂-equation for (n, q)-forms.

368


Corollary B.4.5. Let (L, hL ) be a semi-positive line bundle over a complete K¨ ahler manifold (X, Θ) of dimension n. Let a1 · · · an be the eigenvalues of R˙ L ∈ End(T (1,0) X) (cf. (1.5.15)) with respect to Θ. Then for any q 1 and any form L f ∈ L2n,q (X, L) satisfying ∂ f = 0 and X (a1 + · · · + aq )−1 |f |2 dvX < ∞ there L

exists g ∈ L2n,q−1 (X, L) such that ∂ g = f and

|g|2 dvX X

(a1 + · · · + aq )−1 |f |2 dvX .

(B.4.11)

X

Proof. The Bochner–Kodaira–Nakano inequality (1.4.51) together with (1.4.61) (cf. also Problem 1.10) for r = n delivers (B.4.1) with ψ = a1 + · · · + aq . In the seminal works of Bombieri [39] and Skoda [229] it has been already observed that Corollary B.4.5 still applies if hL is singular; the following result was proved in full generality by Demailly [71, Th. 5.1] (cf. also [78, Cor. 5.3]). The idea is to reduce the problem to the case of smooth metrics by regularization (convolution of plurisubharmonic functions with smooth kernels). The solutions obtained by Theorem B.4.1 turn out to converge weakly to a solution for the singular metric, due to the uniformity of the constants in the L2 estimates. Let L2n,q (X, L, loc) be the usual space of forms with locally L2 coefficients with respect to smooth metrics on X and L. In the case of a singular Hermitian metric hL in the sense of Definition 2.3.1, we set |s|2hL dvX < ∞} . (B.4.12) L2n,q (X, L) := {s ∈ L2n,q (X, L, loc) : X

Theorem B.4.6 (Demailly). Let (X, Θ) be a complete K¨ ahler manifold, dim X = n, L ) be a holomorphic singular Hermitian line bundle on X such that and let (L, h √ −1RL εΘ in the sense of currents, for some constant ε > 0. Then for any q L 1 and any form f ∈ L2n,q (X, L) satisfying ∂ f = 0, there exists g ∈ L2n,q−1 (X, L) L

such that ∂ g = f and |g|2hL dvX X

1 qε

|f |2hL dvX .

(B.4.13)

X

Let us consider the fine sheaf L n,q given by L

U −→ L n,q (U, L) = {s ∈ L2n,q (U, L, loc) : |s|hL , |∂ s|hL ∈ L2loc } .

(B.4.14)

L The sequence 0 → OX (L ⊗ KX ) ⊗ I (hL ) → L n,• , ∂ is exact, where I (hL ) L is the Nadel ideal multiplier sheaf associated to h (cf. Definition 2.3.13). This L −χ(ϕ) follows by applying Theorem B.4.6 on√a ball B(0, r); we consider hL 1 = h e and the complete K¨ ahler form Θ = −1∂∂χ(ϕ) on B(0, r), constructed as in

B.5. Chern-Weil theory

369

(3.5.8) or Proposition B.3.15, where ϕ(z) = − log(r2 − |z|2 ) is the exhaustion function of B(0, r). Therefore H q (X, L ⊗ KX ⊗ I (hL )) = H q (Γ(X, L n,• )) .

(B.4.15)

As a consequence we have the singular variant of the Kodaira and Nakano vanishing theorem 1.5.4, cf. [182, 180, 77]. Theorem B.4.7 (Nadel). Let (X, Θ) be a weakly 1-complete manifold endowed with L a complete K¨ ahler metric with associated (1, 1)-form Θ. Let √ (L, hL ) be a holomorphic singular Hermitian line bundle on X. Assume that −1R ε Θ for some ε > 0. Then H q (X, L ⊗ KX ⊗ I (hL )) = 0 ,

for all

q 1.

(B.4.16)

Indeed, applying again Theorem B.4.6 globally we obtain H q (Γ(X, L n,• )) = 0 for q 1 and (B.4.15) entails (B.4.16). Note that if hL is smooth, we obtain as a special case the Kodaira vanishing theorem. Theorem B.4.7 is actually very strong: it contains also the KawamataViehweg vanishing theorem. The L2 estimates for ∂ imply also the following version of the finiteness theorem of Andreotti-Grauert. Theorem B.4.8 (Andreotti-Grauert). Let X be an n-dimensional complex manifold and E be a holomorphic vector bundle on X. If X is q-convex (resp. q-concave), then dim H j (X, E) < ∞ for j q (resp. j n − q − 1) . The proof of Andreotti and Grauert is sheaf-theoretic and makes use of the “bumping lemma” and it works on complex spaces too. On the analytic side, there are proofs based on L2 estimates on complete manifolds (Andreotti-Vesentini [7], H¨ ormander [132], Ohsawa [187]), ∂-Neumann problem (Kohn [108]), or integral representations (Henkin-Leiterer [126]).

B.5 Chern-Weil theory Some good references for this section are [263], [31] and [15]. In [55], one can also find the relative Euler class for manifold with boundary. We will use the notation in Section 1.1.1. Let X be a smooth manifold. For R = C or R, let Ω• (X, R) be the space of smooth R-valued forms on X. Let d : Ω• (X, R) → Ω•+1 (X, R) be the exterior differential. Then d2 = 0. The elements of the kernel Ker(d) (resp. image Im(d)) of d are called closed (resp. exact) forms. We will denote d also by dX when we wish to emphasize the manifold X. The de Rham cohomology of X is defined by H j (X, R) :=

Ker(d) ∩ Ωj (X, R) , Im(d) ∩ Ωj (X, R)

H • (X, R) :=

) j

H j (X, R).

(B.5.1)

370


For a closed differential form η, we denote by [η] ∈ H • (X, R) its cohomology class. In the same way we introduce the complex de Rham cohomology group H • (X, C), by replacing R with C in the definition above. Let E be a complex vector bundle on X. Let ∇E be a connection on E, and E ∇ : Ω• (X, E) → Ω•+1 (X, E) be its unique extension verifying the Leibniz rule (1.1.4). Its curvature is RE = (∇E )2 ∈ Ω2 (X, End(E)) and (1.1.5) holds. The fiberwise trace Tr : C ∞ (X, End(E)) → C ∞ (X), A → Tr[A], extends to a map Tr : Ω• (X, End(E)) → Ω• (X, C), such that for η ∈ Ω• (X, C), A ∈ C ∞ (X, End(E)) we have Tr[η ⊗ A] = η Tr[A]. For A, B ∈ C ∞ (X, End(E)) and η, γ ∈ Ω• (X, C) we have (η⊗A)(γ⊗B) = (η∧γ)⊗(AB). Let us extend the fiberwise commutator [A, B] of two sections A, B ∈ C ∞ (X, End(E)) to a supercommutator [· , ·] on Ω• (X, End(E)) by setting [η ⊗ A, γ ⊗ B] := (η ⊗ A)(γ ⊗ B) − (−1)deg η·deg γ (γ ⊗ B)(η ⊗ A) = (η ∧ γ) ⊗ [A, B]. (B.5.2) We have thus Tr

A, B

= 0,

for any A, B ∈ C ∞ (X, End(E)) .

(B.5.3)

Let f ∈ R[z] be a real polynomial on z. We set √

−1 E ∈ Ω• (X, C) . R F RE = Tr f 2π

(B.5.4)

Theorem B.5.1. F RE is a closed differential form and its de Rham cohomology 2• (X, C) does not depend on the choice of ∇E . If ∇E is a class [F RE ] ∈ HdR Hermitian connection with respect to aHermitian metric hE on E, then F RE is a real differential form. Thus [F RE ] ∈ H 2• (X, R). Proof. Let us start by proving that for any A ∈ C ∞ (X, End(E)) we have d Tr[A] = Tr [∇E , A] .

(B.5.5)

E We first observe that (B.5.5) does not depend on the choice of ∇E . Indeed, if ∇ E − ∇E ∈ is another connection on E, the definition of a connection implies that ∇ 1 E E Ω (X, End(E)), so by (B.5.3) we have Tr [∇ − ∇ , A] = 0. It is also clear that (B.5.5) has local character, i.e., it suffices to prove it in a small neighborhood of each point. Moreover, in a neighborhood U such that E|U ∼ = U × Cm and for E the trivial connection ∇ (ϕ1 , . . . , ϕm ) = (dϕ1 , . . . , dϕm ), formula (B.5.5) clearly holds. Thus, the local character and the independence on the connection yield (B.5.5) in general.


371

Under the notation (1.1.4), the Bianchi identity holds [∇E , RE ] = [∇E , (∇E )2 ] = 0,

(B.5.6)

which together with (B.5.5) yield dF (RE ) = Tr

√−1 E = 0. R ∇E , f 2π

(B.5.7)

Thus F (RE ) is closed. E two connections on E. Let π : X × R → X be the natural If ∇E 0 , ∇1 are ∗ ∗ projection. Let ∇π E be a connection on π ∗ E over X ×R with curvature Rπ E such ∗ ∗ π E π E |X×{i} for i = 0, 1. We denote ∇E |X×{t} the connection that ∇E t = ∇ i =∇ E π∗ E on E with curvature Rt . Then we can write F (R ) as follows: F (Rπ

∗

E

) = F (RtE ) + dt ∧ Qt ,

(B.5.8)

∗

with Qt a differential form on X. Since F (Rπ E ) is closed on X × R by (B.5.7), ∗ after considering the coefficient of dt in the relation dX×R F (Rπ E ) = 0, we get ∂ F (RtE ) = dQt . ∂t Thus

(B.5.9)

1

F (R1E ) − F (R0E ) = d

Qt dt.

(B.5.10)

0

Hence [F (R1E )] = [F (R0E )] ∈ H 2• (X, C). we know that for If ∇E is a Hermitian connection on (E, hE ), then by (1.1.6), √ any U, V ∈ T X, RE (U, V ) ∈ End(E) is skew-adjoint, thus −1RE is a Hermitian matrix, and F (RE ) is a real form. But there always exist Hermitian metric hE and Hermitian connections on (E, hE ); thus [F (RE )] ∈ H 2• (X, R). E E Remark B.5.2. Let ∇E t be any path connecting ∇0 and ∇1 , a natural choose of π∗ E ∇ is given by

∇π Then Rπ

∗

E

= RtE + dt ∧

∂∇E t ∂t ,

∗

E

= ∇E t + dt ∧

∂ . ∂t

(B.5.11)

and Qt in (B.5.8) is

√ √ −1 ∂∇E −1 E t Qt = Tr f( R ) ∈ Ω2•−1 (X, C) . 2π ∂t 2π t

(B.5.12)

372

Appendix B. Elements of Analytic and Hermitian Geometry ∗

E π E Definition B.5.3. Let ∇E 0 , ∇1 be two connections on E. For a connection ∇ E E on X × R connecting ∇0 and ∇1 , with Qt in (B.5.8), we define the Chern-Simons form of F by 1 π∗ E F (∇ )= Qt dt. (B.5.13) 0 E The Chern-Simons class of F associated to ∇E 0 and ∇1 is defined by E π∗ E )] ∈ Ω• (X, C)/dΩ• (X, C). F (∇E 0 , ∇1 ) := [F (∇

(B.5.14)

The following result is a special case of [31, Theorem 2.10]. ∗ Theorem B.5.4. Modulo exact forms, F (∇π E ) does not depend on the choice of ∗ E • • the connection ∇π E . Thus F(∇E 0 , ∇1 ) ∈ Ω (X, C)/dΩ (X, C) is well defined, and E E E dF(∇E (B.5.15) 0 , ∇1 ) = F (R1 ) − F (R0 ).

E If ∇E i , (i = 0, 1) are Hermitian connections on (E, hi ), then the imaginary E • • E E part of F (∇E 0 , ∇1 ) is exact, thus F (∇0 , ∇1 ) ∈ Ω (X, R)/dΩ (X, R). ∗

∗

Proof. By (B.5.10), we get (B.5.15). Let ∇π0 E , ∇π1 E be two connections on X ×R E connecting ∇E 0 , ∇1 . Let us consider the trivial fibration ρ : X × R × R → X. Let ∗ ∇ρ E be a connection on ρ∗ E over X × R × R defined by ∇πs

∗

E

= (1 − s)∇π0 ∗

ρ Set ∇E t,s = ∇ ∗

F (Rρ

E

∗

E

+ s∇π1

∗

E

∗

∇ρ

,

E

= ∇πs

∗

+ ds ∧

E

∗

|X×{t}×{s} . Then we can decompose F (Rρ

E

∂ . ∂s

) as

2 + dt ∧ Qt,s + ds ∧ Tt,s + dt ∧ ds ∧ γ, ) = F (∇E t,s )

E

(B.5.16)

(B.5.17)

where Qt,s , Tt,s , γ are forms on X. Again by considering the coefficient of dt ∧ ds ∗ in dX×R×R F (Rρ E ) = 0, we get −

∂ ∂ Qt,s + Tt,s + dX γ = 0. ∂s ∂t

(B.5.18)

Integrating on [0, 1]2 we obtain

1

Qt,0 dt− 0

1

1

Qt,1 dt+ 0

0 ∗

Now we observe that ∇ρ (B.5.12). Therefore

E

1

1

T0,s ds+dX

T1,s ds−

1

γ dt ds = 0. (B.5.19)

0

0

0

∂ |X×{i}×R = ∇E i,0 + ds ∂s , thus T1,s = T0,s = 0 by

∗ ∗ F (∇π1 E ) − F(∇0π E ) = dX

1

1

γ dt ds. 0

0

(B.5.20)


373

E Assume that ∇E i , (i = 0, 1) are Hermitian connections on (E, hi ). We de∞ E E fine G ∈ C (X, End(E)) by h1 (ξ1 , ξ2 ) = h0 (Gξ1 , ξ2 ). Then G is positive and E = G1/2 ∇E G−1/2 is a Hermitian conself-adjoint, so G1/2 is well defined. Now ∇ 1 1 E E E nection on (E, hE 0 ) and ∇t = (1 − t)∇0 + t ∇1 are Hermitian connections on −1 E 1/2 E E , hE (E, hE t ∇t gt 0 ). If we take gt = (1 − t) Id +tG t = h0 (gt ·, gt ·), then ∇t = g∗ E π E are Hermitian connections on (E, ht ). Now we choose the connection ∇ by ∂ ∗ 1 −1 ∂ E (B.5.21) + (hE ht . ∇π E = ∇E t + dt ∧ t ) ∂t 2 ∂t ∗

∗

Then ∇π E is a Hermitian connection on π ∗ E with the metric hπ E induced by hE t . Now by the second part of Theorem B.5.1 and (B.5.8), the second part of Theorem B.5.4 follows. Example B.5.5. Let ∇E be a Hermitian connection on a Hermitian vector bundle (E, hE ). Taking f (z) = z, e−z , log( 1−ez −z ), we get from Theorem B.5.1 and (1.3.45) that c1 (E, ∇E ), ch(E, ∇E ), Td(E, ∇E ) are closed real differential forms on X and their cohomology classes in H 2• (X, R) do not depend on the choice of hE and ∇E . We define now √ rk(E) −1 E = 1+ R det 1 + ci (E, ∇E ), with ci (E, ∇E ) ∈ Ω2i (X, R) . (B.5.22) 2π i=1 Since for any positive matrix A, we have det A = exp(Tr[log A]), by taking f (z) = log(1 + z), we see that ci (E, ∇E ) are closed real differential forms on X. The 2• corresponding cohomology classes ci (E) ∈ H (X, R) are called ith Chern classes of E, and c(E) = 1 + i ci (E) is called the total Chern class of E. Remark B.5.6. If E is a real vector bundle and ∇E is a connection on E, then by applying the above construction to E ⊗R C, we obtain the corresponding theory for E. Especially, if ∇E preserves an Euclidean metric hE , then RE (U, V ) is antisymmetric, thus from (B.5.22), we get c2i+1 (E, ∇E ) = 0 and pi (E) := (−1)i c2i (E) is the ith Pontryagin class of E. Example B.5.7 (Chern-Simons functional [66]). We suppose that X is an oriented manifold of dimension 3. By a classical theorem of Stiefel (cf. N. Steenord, The topology of fiber bundles, Princeton Math. Series, Vol 14, 1951), the tangent bundle T X is trivial, and we fix a trivialization of T X. If ∇T X is a connection on T X, there exists A ∈ C ∞ (X, T ∗ X ⊗ End(T X)) such that ∇T X = d + A.

(B.5.23)

Now we choose the special path ∇Tt X = d + tA of connections from d to ∇T X . Let π : X × R → X be the natural projection, and E = T X. We define a connection ∗ ∗ ∂ . Then ∇π E on π ∗ E on X × R as before: ∇π E = ∇Tt X + dt ∂t Rπ

∗

E

∗ 2 := ∇π E = t dA + t2 A ∧ A + dt ∧ A.

(B.5.24)

374


On X × R we have

∗ 2 Tr[ Rπ E ] = dt ∧ Tr[A ∧ (tdA + t2 A ∧ A)].

(B.5.25)

Thus for the polynomial f (z) = z 2 , from (B.5.13), we find ∗ −1 t F (∇π E ) = Tr[A ∧ (tdA + t2 A ∧ A)] dt 4π 2 0 2 −1 Tr[A ∧ dA + A ∧ A ∧ A)]. = 8π 2 3 Tr[A ∧ dA + 23 A ∧ A ∧ A)] is the classical Chern-Simons functional .

(B.5.26)

Appendix C

Spectral Analysis of Self-adjoint Operators We collect in this appendix some of the properties related to self-adjoint extensions and their spectrum that we used. A short and general introduction on functional analysis for applications in partial differential equations is [237, Appendix A] which is very helpful to understand our book.

C.1 Quadratic forms and Friedrichs extension Let H be a complex Hilbert space with inner product , and norm . Let A : Dom(A) ⊂ H −→ H be a linear operator, where Dom(A) is a dense linear subspace of H. A is called closed if Graph(A) = {(u, Au); u ∈ Dom(A)} is closed and preclosed if the closure [Graph(A)] is again the graph of a linear operator denoted A (the closure of A). An operator B : Dom(B) ⊂ H −→ H is called an extension of A if Graph(A) ⊂ Graph(B), and we write A ⊂ B. We define the adjoint operator of A by Dom(A∗ ) = {u ∈ H; there exists C > 0 such that |Av, u| Cv for any v ∈ Dom(A)}, and for u ∈ Dom(A∗ ) we set A∗ u to be the unique w ∈ H such that Av, u = v, w for any v ∈ Dom(A), by using the Riesz representation theorem. Clearly, A ⊂ (A∗ )∗ and A∗ is closed. If we denote Φ : H × H → H × H, Φ(u, v) = (v, −u), we have Graph(A)⊥ = Φ(Graph(A∗ )).

(C.1.1)

We can also verify the important relations: [Im(A∗ )] = Ker(A)⊥ ,

[Im(A)] = Ker(A∗ )⊥ ,

(C.1.2)

where [V ] denotes the closure of a linear space V , and V ⊥ denotes the orthogonal complement of V . We need a criterion for an operator to have closed range.

376

Appendix C. Spectral Analysis of Self-adjoint Operators

Lemma C.1.1. Let T : Dom(T ) ⊂ H −→ H be a linear, closed, densely defined operator. The following conditions on T are equivalent: 1. Im(T ) is closed. 2. There is a constant C such that f CT f

for all

f ∈ Dom(T ) ∩ [Im(T ∗ )] .

(C.1.3)

f ∈ Dom(T ∗ ) ∩ [Im(T )] .

(C.1.4)

3. Im(T ∗ ) is closed. 4. There is a constant C such that f CT ∗ f

for all

The best constants in (C.1.3) and (C.1.4) are the same. Proof. We assume that (1) holds, hence Im(T ) is a Hilbert space. From (C.1.2), T : Dom(T ) ∩ [Im(T ∗ )] −→ Im(T ) is one-to-one, and its inverse T −1 : Im(T ) −→ Dom(T ) ∩ [Im(T ∗ )] is a well-defined closed operator. Thus from the closed graph theorem, T −1 is continuous and this proves (2). It is obvious that (2) implies (1). Similarly, (3) and (4) are equivalent. To prove that (2) implies (4), notice that |g, T f | = |T ∗ g, f | CT ∗ gT f , for g ∈ Dom(T ∗ ) and f ∈ Dom(T ) ∩ [Im(T ∗ )]. Thus |g, h| CT ∗ gh ,

for g ∈ Dom(T ∗ )

which implies (4). Similarly, (4) implies (2).

and h ∈ Im(T ) ,

We say that A is self-adjoint if A = A∗ . We say A is symmetric if A ⊂ A∗ . If A is symmetric, then A is preclosed and A is said to be essentially self-adjoint if A = A∗ . We say A is positive if Au, u 0 for any u ∈ Dom(A). Lemma C.1.2. Let A be an injective self-adjoint operator. Then Im(A) is dense and the operator A−1 with Dom(A−1 ) = Im(A) is self-adjoint. Proof. Since Im(A)⊥ = Ker(A∗ ), A = A∗ and Ker(A) = 0, we see that Im(A) is dense. Moreover, using (C.1.1) and Graph(A−1 ) = Φ(Graph(−A)), we deduce that (A−1 )∗ = (A∗ )−1 , hence (A−1 )∗ = A−1 . Lemma C.1.3 (von Neumann). Let A be a closed operator. Then A∗ A is self-adjoint and 1 + A∗ A has a bounded inverse. Proof. Let w ∈ H. We write (w, 0) = (u, v) + (u0 , v0 ) ∈ Graph(A) ⊕ Graph(A)⊥ . But Graph(A)⊥ = Φ(Graph(A∗ )) so u ∈ Dom(A), Au = v, v0 ∈ Dom(A∗ ), A∗ v0 = −u0 . Hence w = u − A∗ v0 , 0 = Au + v0 , thus Au = −v0 ∈ Dom(A∗ ), w = u + A∗ Au = (1 + A∗ A)u. Thus (1 + A∗ A) Dom(A∗ A) = H. Consequently

C.1. Quadratic forms and Friedrichs extension

377

1 + A∗ A is bijective, with inverse (1 + A∗ A)−1 having range Dom(A∗ A). Set f = (1 + A∗ A)−1 u, g = (1 + A∗ A)−1 v. Then u, (1 + A∗ A)−1 v = (1 + A∗ A)f, g = f, g + Af, Ag = (1 + A∗ A−1 )u, v , so (1 + A∗ A)−1 is symmetric. Since its domain is H, (1 + A∗ A)−1 is bounded and self-adjoint. Lemma C.1.2 shows that 1+A∗ A and hence A∗ A are self-adjoint. We shall work in general with the quadratic form associated to an operator rather than with the operator directly. A quadratic form is a sesquilinear map Q : Dom(Q) × Dom(Q) −→ C where Dom(Q) is a dense linear subspace of H. Q is called positive if Q(u, u) 0 for any u ∈ Dom(Q). A positive quadratic form Q is called closed if (Dom(Q), ·Q ) is complete, where uQ := (Q(u, u) + u2 )1/2 ,

for u ∈ Dom(Q).

(C.1.5)

There exists a basic correspondence between positive closed quadratic forms and positive self-adjoint operators. Proposition C.1.4. Let A be a positive self-adjoint operator. Then there exists a closed quadratic form QA such that . Dom(A) = u ∈ Dom(QA ) : there exists v ∈ H / (C.1.6) with QA (u, w) = v, w for any w ∈ Dom(QA ) , Au = v , for u ∈ Dom(A) . QA is called the associated quadratic form to A. Proof. On Dom(A) we consider the scalar product u, vA := (1 + A)u, v with 1/2 norm uA = u, uA and we consider the Hilbert space completion (D, ·, ·A ) of (Dom(A), ·, ·A ). This is the space of equivalence classes of Cauchy sequences {uν }, where two Cauchy sequences {uν }, {vν } are said to be equivalent if uν − vν → 0 as ν → ∞. Since uA u for u ∈ Dom(A), the canonical injection Dom(A) −→ H extends to an injective bounded linear map D −→ H. We will identify D with its image in H: . D = u ∈ H : there exists {uν } ⊂ Dom(A) , such that uν → u , and / uν − uµ A → 0, as µ, ν → +∞ . (C.1.7) Moreover uA u ,

u∈D,

u, vA = (1 + A)u, v ,

u ∈ Dom(A) , v ∈ D .

(C.1.8)

We define Dom(QA ) := D, QA (u, v) := u, vA − u, v, for u, v ∈ D. By construction, QA is a closed quadratic form. We prove now (C.1.6). Let us denote by

378


DA the right-hand side of (C.1.6). From (C.1.8), Dom(A) ⊂ DA . Conversely, let u ∈ DA . It follows from (C.1.6), (C.1.8) that Aw, u = w, v for all w ∈ Dom(A). Thus u ∈ Dom(A∗ ) and A∗ u = v. Since A = A∗ , DA ⊂ Dom(A). Proposition C.1.5. Let Q be a closed positive quadratic form. Then there exists a positive self-adjoint operator A with Q = QA . Proof. Consider the scalar product u, vQ := Q(u, v) + u, v with norm ·Q . By assumption (Dom(Q), ·, ·Q ) is a Hilbert space. For each w ∈ H the mapping u → w, u is a bounded form on (Dom(Q), ·, ·Q ), since w, u w · u w · uQ . By the Riesz representation theorem, there exists a unique element T w ∈ Dom(Q) such that w, u = T w, uQ

for all u ∈ Dom(Q).

(C.1.9)

It is easily seen that T : H → H is a bounded, injective, self-adjoint linear operator. Indeed, T w2 T w, T wQ = w, T w w·T w, so T is bounded. Moreover, T w = 0 implies T w, vQ = w, v = 0 for all v ∈ Dom(Q); since Dom(Q) is dense, w = 0, and T is injective. Next, T w, v = v, T w = T v, T wQ = T w, T vQ = w, T v, so T is self-adjoint. By Lemma C.1.2, T −1 is self-adjoint. We define the operator A by (C.1.10) Dom(A) := Dom(T −1 ), A := T −1 − 1. Then A is self-adjoint, positive (since Au, u = u2Q − u2 0 for u ∈ Dom(A)) and by construction of T , we have QA = Q. Let A : Dom(A) ⊂ H −→ H be a positive operator. We define QA by Dom(QA ) := Dom(A),

QA (u, v) := Au, v,

for u, v ∈ Dom(A).

(C.1.11)

Proposition C.1.6. Let A be a positive operator. Then QA is closable, i.e., there exists the smallest positive closed form QA extending QA . Proof. We repeat the construction given in Proposition C.1.4 word for word, and find a closed quadratic form QA (denoted there QA ) such that Dom(QA ) is dense in Dom(QA ). Definition C.1.7. Let A be a positive operator. The Friedrichs extension of A is the self-adjoint operator AF such that QAF = QA . The existence of AF follows from Propositions C.1.4-C.1.6. Moreover, we deduce . Dom(AF ) = u ∈ Dom(A∗ ) : there exists {uν } ⊂ Dom(A) ,

/ such that uν → u , and A(uν − uµ ), uν − uµ → 0 , (C.1.12)

and AF u = A∗ u for u ∈ Dom(AF ).

C.2. Spectral theorem

379

Definition C.1.8. A one-parameter semi-group of operators on a Hilbert space H is a set of bounded operators Pt : H −→ H, t ∈ R+ , satisfying for t1 , t2 ∈ R+ , Pt1 +t2 = Pt1 Pt2

and

P0 = Id,

(C.1.13)

and for any v ∈ H, Ptj v → Pt v when tj → t. We can find the following result in [237, Appendix A, §9]. Theorem C.1.9. If A : Dom(A) ⊂ H −→ H is positive self-adjoint, then −A generates a semi-group Pt = e−tA consisting of positive self-adjoint operators of norm ≤ 1. Pt is characterized by : for any v ∈ Dom(A), t ∈ R+ , Av = lim t−1 (Pt v − v), t→0

Pt Dom(A) ⊂ Dom(A), d − APt v = −Pt A v = Pt v. dt

(C.1.14)

C.2 Spectral theorem The point of extending an operator to a self-adjoint one is to study its spectral properties. Let A be a closed operator on a Hilbert space H. We say that a complex number λ lies in the resolvent set of A if λ−A is a bijection of Dom(A) onto H with a bounded inverse. Note that by the closed-graph theorem if λ−A : Dom(A) −→ H is a bijection, the inverse is automatically bounded. The spectrum of A, denoted σ(A), is the complement in C of the resolvent set. Let Bor(R) be the family of Borel sets in R. Let L (H) be the space of bounded operators of H. A map E : Bor(R) −→ L (H) is called a spectral measure if a). E(Ω) is an orthogonal projection for any Ω ∈ Bor(R); b). E(∅) = 0, E(R) = Id; c). If Ω = ∪j∈I Ωj with I ⊂ N, and Ωi ∩ Ωj = ∅ if i = j, then E(Ω) = s-limk→∞ j∈I∩[0,k] E(Ωj ); d). E(Ω1 ∩Ω2 ) = E(Ω1 )E(Ω2 ). Here s-lim is the strong limit of operator. Let E : Bor(R) −→ L (H) be a spectral measure. If f : R −→ C is a bounded Borel function we can define the integral f (t) dE(t) ∈ L (H) (C.2.1) R

using the usual pattern of defining the integral for step-functions first and then writing the integral of a general bounded Borel function f as the limit of the integrals of a sequence of step-functions, which converge uniformly to f . If E : Bor(R) −→ L (H) is a spectral measure we can define the associated scalar measures on R, by setting Bor(R) Ω −→ E(Ω)u, v, for each (u, v) ∈ H × H. For

380


any bounded Borel function f : R −→ C we have then f (t) dE(t) u, v = f (t) dE(t)u, v. R

(C.2.2)

R

We have the following fundamental result [202, Theorem VIII.6], [70, Theorem 2.5.5]. Theorem C.2.1 (Spectral theorem). Each self-adjoint operator A has a unique spectral measure E such that t2 dE(t)u, u < ∞ (C.2.3) Dom(A) = u ∈ H : R

and for u ∈ Dom(A) Au = lim

k→∞

R

1[−k,k] (t)t dE(t) u =: t dE(t) u.

(C.2.4)

R

From Theorem C.2.1 and (C.2.1), for any bounded Borel function f : R −→ C, we can define f (A) = R

f (t) dE(t) ∈ L (H).

(C.2.5)

Especially, we see that the operator e−tA defined in Theorem C.1.9 coincides with the definition in (C.2.5) for a positive self-adjoint operator A. More generally, consider a Borel measurable function f : [0, ∞[→ R. We define the self-adjoint operator f (A) by setting Dom(f (A)) = u ∈ H : |f (t)|2 dE(t)u, u < ∞ , (C.2.6) R

and for u ∈ Dom(f (A)), f (A)u = lim k→∞

R

1[−k,k] (t)f (t)t dE(t) u =: f (t) dE(t) u .

(C.2.7)

R

f (A) is√positive if f is positive. Let A be √a positive self-adjoint operator. For f (t) = t we obtain the positive operator A = A1/2 ; it is the unique positive self-adjoint operator B satisfying B 2 = A. Moreover Dom(QA ) = Dom(A1/2 ) and QA (u, v) = (A1/2 u, A1/2 v) for any u, v ∈ Dom(QA ). Therefore, Dom(QA ) = u ∈ H : |t| dE(t)u, u < ∞ , R (C.2.8) QA (u, v) = t dE(t)u, v , u, v ∈ Dom(QA ). R

C.3. Variational principle

381

Let us define the spectral resolution associated to A by (Eλ )λ∈R where Eλ = E(] − ∞, λ]). When we want to stress the dependence on A we note Eλ (A). We set E (λ) = E (λ, A) := Im(Eλ (A)). The spectrum counting function of A is defined as N (λ) := dim Im(Eλ ) = dim E (λ).

(C.2.9)

The discrete spectrum σd (A) is the set of all eigenvalues λ of finite multiplicity which are isolated, in the sense that ]λ − ε, λ[ and ]λ, λ + ε[ are disjoint from the spectrum for some ε > 0. The essential spectrum is the complement of the discrete spectrum, σess (A) = σ(A) σd (A). Definition C.2.2. B ∈ L (H) is compact, if for any U ⊂ H a bounded subset, the closure of B(U ) is compact in H. Theorem C.2.3. If B ∈ L (H) is compact and self-adjoint, then H has a complete orthonormal basis of eigenvectors {ϕj }∞ j=1 of B with corresponding eigenvalues λj which converge to 0 as j → ∞. We have the following criterion for the non-existence of the essential spectrum [70, Cor. 4.2.3]. Theorem C.2.4. Let A be a positive self-adjoint operator. Then the following conditions are equivalent: 1. The resolvent operator (1 + A)−1 is compact. 2. The operator A has empty essential spectrum. 3. There exists a complete orthonormal set of eigenvectors {ϕj }∞ j=1 of A with corresponding eigenvalues λj 0 which converge to +∞ as j → ∞. It follows from the proof of Proposition C.1.5 that A has compact resolvent T = (1+A)−1 if the injection (Dom(QA ), ·QA ) → (H, ·) is a compact operator. If the spectrum is discrete, we arrange the eigenvalues in increasing order and repeated according to multiplicity: λ1 λ2 · · · . Then N (λ) = #{j : λj λ}.

C.3 Variational principle The most important tool for estimating and for comparing the eigenvalues of different operators is the variational or minimax principle. The simplest but very useful form is the following. Lemma C.3.1 (Glazman lemma). The spectrum counting function of a positive self-adjoint operator A satisfies the variational formula N (λ) = sup{dim F : F closed ⊂ Dom(QA ) , QA (u, u) λu2 , for all u ∈ F } . (C.3.1)

382


Proof. Let u ∈ Im(Eλ ). Then E(Ω)u, u = E(Ω) E(] − ∞, λ])u, u = E(Ω∩] − ∞, λ])u, u for any Ω ∈ Bor(R), so formula (C.2.8) entails QA (u, u) =

λ −∞

t dE(t)u, u λE(] − ∞, λ])u, u = λu2 .

(C.3.2)

Hence QA (u, u) λu2 for all u ∈ Im(Eλ ) and N (λ) does not exceed the righthand side of (C.3.1). Consider a closed linear space V ⊂ Dom(QA ) such that QA (u, u) λu2 for all u ∈ V . We show that Eλ : V −→ Im(Eλ ) is injective. If u ∈ Ker(Eλ ) = Im(E(]λ, +∞[)) we have QA (u, u) = t dE(t)u, u > λE(]λ, +∞[)u, u = λu2 ]λ,+∞[

if u = 0. Thus any u ∈ V ∩ Ker(Eλ ) must vanish. We infer that dim V dim Im(Eλ ) = N (λ). Formula (C.3.1) is established. Let A be a positive self-adjoint operator and let QA be the associated closed quadratic form. We consider the sequence λ1 λ2 · · · λj · · · according to the formula inf sup QA (f, f ) (C.3.3) λj = F ⊂Dom(QA ) f ∈F, f =1

where F runs through the j-dimensional subspaces of Dom(QA ). We refer to [202, Vol IV, p.76-78], [70, Ch. 4.5] for the following result. Theorem C.3.2 (Variational principle). Let us define the bottom of the essential spectrum as inf σess (A) if σess (A) = ∅ and +∞ if σess (A) = ∅. Then for each fixed j, either following a) or b) should be true: (a) there are at least j eigenvalues (counted according to multiplicity) below the bottom of the essential spectrum and λj is the jth eigenvalue, (b) λj is the bottom of the essential spectrum, in which case λj = λj+1 = λj+2 = · · · and there are at most j − 1 eigenvalues (counting multiplicity) below λj . Theorem C.3.3 ([70, Lemma 8.4.1]). If A is a self–adjoint operator, then the following conditions are equivalent: (a) λ ∈ σess (A), (b) there exists a noncompact sequence {fk }k∈N in Dom(A) with fk = 1

for each

k∈N

and

lim (A − λ Id)fk = 0,

k→∞

(C.3.4)

(c) there exists a sequence {fk }k∈N in Dom(A) with fk , fl = δkl and which satisfies (C.3.4).

Appendix D

Heat Kernel and Finite Propagation Speed The heat kernel and the finite propagation speed of solutions of hyperbolic equations play important roles in our book. Basically, we only need to know the existence of the heat kernel, then we always use the finite propagation speed to localize the problem at hand. To help the reader understand the finite propagation speed, we explain it here in detail. This appendix is organized as follows. In Section D.1, we summarize the basic facts on heat kernels (cf. [148, Chap. 3], [15, Chap. 2]). In Section D.2, we explain the finite propagation speed of solutions of wave equations.

D.1 Heat kernel Let (X, g T X ) be a smooth complete Riemannian manifold without boundary, endowed with the Riemannian metric g T X on T X. Set dimR X = m. Let d(x, y) be the Riemannian distance on (X, g T X ). Let (E, hE ) be a Hermitian vector bundle on X with Hermitian connection ∇E . Since (X, g T X ) is complete, we know that the Bochner Laplacian ∆E is essentially self-adjoint, i.e., the closure of ∆E in L2 (X, E), the L2 -sections of E on X with norm · L2 as in (1.3.14), is self-adjoint (cf. Corollary 3.3.4, [9, Prop. 3.1], [148, Th. 2.5.7]). Let H = ∆E + Q be a generalized Laplacian on E defined in (A.3.3). We suppose that the Hermitian section Q ∈ C ∞ (X, End(E)) is bounded from below, i.e., there exists C > 0 such that hE (Qξ, ξ) −ChE (ξ, ξ) for any x ∈ X, ξ ∈ Ex . Then H is again essentially self-adjoint (cf. [238, §8.2]). Thus Lemma D.1.1. The operator ∆E + Q is essentially self-adjoint if the Hermitian section Q is bounded from below.

384

Appendix D. Heat Kernel and Finite Propagation Speed

The heat operator e−tH is defined in general by using operator theory (cf. Theorem C.1.9). For t > 0, the heat operator e−tH is an operator from L2 (X, E) to L2 (X, E) which is C 1 in t and which verifies the following properties: for s ∈ L2 (X, E), ∂ + H e−tH s = 0, (D.1.1a) ∂t (D.1.1b) lim e−tH s = s in L2 (X, E). t→0

The heat kernel e−tH (x, x ) ∈ Ex ⊗ Ex∗ , (x, x ∈ X), is the Schwartz kernel of the heat operator e−tH with respect to the Riemannian volume form dvX (x ), i.e., for s ∈ L2 (X, E), we have e−tH (x, x )s(x )dvX (x ). (D.1.2) (e−tH s)(x) = X

Theorem D.1.2. The heat kernel e−tH (x, x ) is smooth on x, x ∈ X, t ∈]0, ∞[, and e−tH (x, x ) = (e−tH (x , x))∗ ,

(D.1.3)

where for w ∈ Ex , we denote w∗ ∈ Ex∗ , defined by hE (w1 , w) = (w∗ , w1 ) for w1 ∈ Ex . Proof. By (C.2.5), for any t > 0, m1 , m2 ∈ N, there exists Cm1 ,m2 ,t > 0 such that for any s ∈ C0∞ (X, E), H m1 e−tH H m2 sL2 Cm1 ,m2 ,t sL2 .

(D.1.4)

(D.1.4) is equivalent to Hxm1 Hym2 e−tH (x, y)L2 Cm1 ,m2 ,t .

(D.1.5)

Here Hy acts on E ∗ by identifying E ∗ to E by hE . Now applying Theorems A.1.7 and A.3.2 to a compact subset K of X × X, we know that e−tH (x, y) is C ∞ on K. From (D.1.1a), we get that e−tH (x, y) is also C ∞ on t ∈]0, ∞[. Finally, for any s1 , s2 ∈ C0∞ (X, E), we have ∂t∂1 e−(t−t1 )H s1 , e−t1 H s2 = 0. Thus after taking the integral in t1 , we have e−tH s1 , s2 = s1 , e−tH s2 . This means (D.1.3) holds.

(D.1.6)

In the rest of this appendix, we suppose that X is compact. Theorem D.1.3. If X is compact, the spectrum Spec(H) of H is discrete and the eigensections of H form a complete basis of L2 (X, E). Let {ϕi }∞ i=1 be a complete orthogonal basis of L2 (X, E) consisting of eigensections of H with Hϕi = λi ϕi

D.1. Heat kernel

385

and −∞ < λ1 λ2 · · · → ∞; then ∞ e−tH (x, x ) = e−λi t ϕi (x) ⊗ ϕi (x )∗ ∈ Ex ⊗ Ex∗ .

(D.1.7)

i=1

Proof. By adding a constant on H, we can assume that there exists C > 0 such that for any s ∈ C0∞ (X, E), Hs, s Cs2L2 .

(D.1.8)

H : H k+2 (X, E) → H k (X, E)

(D.1.9)

We claim that for any k ∈ N, is one-to-one and onto. In fact, from (D.1.8), HsL2 CsL2 for any s ∈ C0∞ (X, E). Thus for any k ∈ N, H in (D.1.9) is injective and has closed rank in H k (X, E). If it were not onto, then there would exist 0 = s0 ∈ H k (X, E) that is orthogonal to Im(H), i.e., Hs, s0 = 0

for all s ∈ H k+2 (X, E).

(D.1.10)

Note that H k+2 (X, E) (and C ∞ (X, E)) is dense in H k (X, E). Thus for k 2, we can take s = s0 in (D.1.10), which contradicts (D.1.8). Thus H in (D.1.9) is onto for k 2. Since Im(H) is closed in H k (X, E) for any k, we know that the closure of H k+2 (X, E) in H k (X, E) (which is H k (X, E) itself) is contained in Im(H). Thus H −1 : L2 (X, E) → H 2 (X, E) is a bounded operator, by the open mapping theorem. Theorem A.3.1 implies that H −1 : L2 (X, E) → L2 (X, E) is a compact operator. From Theorem C.2.3, we get the first part of Theorem D.1.3. Now for any t > 0, we have ∞

e−λi t =

i=1

∞

e−tH ϕi , ϕi =

i=1

e−tH/2 ϕi 2L2

i=1

|e

=

∞

−tH/2

(D.1.11)

2

(x, y)| dvX (x)dvX (y) < +∞.

X×X

l For 1 k l, set Qk,l (x, y, t) = i=k e−λi t ϕi (x) ⊗ ϕi (x )∗ . Then from (D.1.11), we obtain that for any m1 , m2 ∈ N, there exists C > 0 such that for all k, l, Hxm1 Hym2 Qk,l (x, y, t)L2

l

1 +m2 e−λi t λm i

(D.1.12)

i=k ∞ −m1 −m2 −λk t/2 −λi t/4

Ct

e

e

< ∞.

i=1

Here we use the inequality e−λt/4 λm1 +m2 Ct−m1 −m2 for any λ > 0. By using again the Sobolev embedding theorem we get that for any r ∈ N , the C r -norms of Qk,l (x, y) are uniformly bounded, and tend to 0 as k → ∞. Thus Q1,l (x, y, t) converges uniformly to Q1,∞ (x, y, t). Certainly, Q1,∞ (x, y, t) verifies (D.1.1a) and (D.1.1b), so by the uniqueness of the heat kernel, we get (D.1.7).

386


We now recall some basic facts on trace class. Definition D.1.4. Let (H, ) be a Hilbert space. An operator A ∈ L (H) is a Hilbert-Schmidt operator if Ahj 2 = |Ahi , hj |2 < ∞. (D.1.13) A2HS = j

ij

Here {hj } is an orthonormal basis of H. The number AHS is called the HilbertSchmidt norm of A. An operator R ∈ L (H) is called a trace class operator if there exist HilbertSchmidt operators A, B such that R = AB. In this case, the trace of R is defined by (D.1.14) Tr[R] = j Rhj , hj . If A, B ∈ L (H) are Hilbert-Schmidt operators, then (D.1.13) implies that their adjoints A∗ , B ∗ are also Hilbert-Schmidt and A∗ HS = AHS . Thus i |ABhi , hi | AHS BHS , which means that (D.1.14) is well defined and Tr[AB] = Tr[BA] = Ahi , hj Bhj , hi . (D.1.15) ij

Recall that we assume that X is compact. Let K(x, y) be the Schwartz kernel of an operator K : L2 (X, E) → L2 (X, E) with respect to dvX (y). If K(x, y) is square integrable on X × X, then by (D.1.13), K is Hilbert-Schmidt, and 2 Tr[K(x, y)∗ K(x, y)]dvX (x) dvX (y). (D.1.16) KHS = X×X

Lemma D.1.5. If the kernel K(x, y) of K is smooth, then K is a trace class, and Tr[K] = Tr[K(x, x)]dvX (x). (D.1.17) X

Proof. As in the proof of Theorem D.1.3, we assume that H verifies (D.1.8). Then by using the Sobolev embedding theorem as before, we see that the operator H −(m+3) has continuous kernel H −(m+3) (x, y). Then K = H −(m+3) · H (m+3) K and H −(m+3) , H (m+3) K are Hilbert-Schmidt, thus K is a trace class. Let {si } be an orthonormal basis of L2 (X, E). Then from (D.1.15), we have Tr[K] = H (m+3) Ksi , sj sj , H −(m+3) si i,j

=

(H (m+3) K)(x, y), H −(m+3) (x, y)dvX (x)dvX (y)

X×X

(D.1.18) (m+3)

=

((H K)(x, y), (H X×X Tr[K(y, y)]dvX (y). =

−(m+3)

∗

(x, y)) )dvX (x)dvX (y)

X

Here we use the analogue of (D.1.6) and H −(m+3) (y, x) = (H −(m+3) (x, y))∗ .

D.1. Heat kernel

387

Theorem D.1.6. If Hv is a smooth family of generalized Laplacians associated to E −tHv is smooth in v and a family of gvT X , hE v , ∇v , Qv , then the heat operator e ∂ −tHv e =− ∂v

t

e−(t−t1 )Hv

0

∂Hv −t1 Hv e dt1 . ∂v

(D.1.19)

Proof. We will add a subscript v for the objects depending on v and work on E v ∈ [−1, 1]. Then the Sobolev norms induced by gvT X , hE v , ∇v are equivalent to TX E the corresponding ones induced by g0 , h0 . Thus from the proof of Theorem D.1.2, the heat kernel e−tHv (x, x ) of e−tHv with respect to dvX,v (x ), is smooth in x, x , uniformly for v ∈ [−1, 1]. We only need to prove (D.1.19) for v = 0. Set dvX,v (x) = κv (x)dvX,0 (x).

(D.1.20)

For s ∈ C ∞ (X, E ∗ ), by (D.1.3) and (D.1.6), we get

lim

t→0

(s(z), e−tHv (z, x))dvX,0 (z) ∗ −1 (e−tHv (x, z), (s(z))∗v ) κ−1 = lim v (z)dvX,v (z) = s(x)κv (x). X

u→0

X

(D.1.21)

v

By (D.1.1a), (D.1.1b), (D.1.21) and H0 e−tH0 = e−tH0 H0 , we infer −tH0 (x, w) e−tHv (x, w) − κ−1 v (w)e t ∂ e−(t−t1 )H0 (x, z), e−t1 Hv (z, w) dvX,0 (z)dt1 = 0 ∂t1 X t e−(t−t1 )H0 (x, z), (H0 (z) − Hv (z))e−t1 Hv (z, w) dvX,0 (z). (D.1.22) = dt1 0

X

From (D.1.22), we know that e−tHv (x, w) is continuous in v, and we can take the differential in v, so we get at v = 0 (as κ0 (x) = 1), ∂ (κv (w)e−tHv (x, w)) ∂v t ∂Hv e−(t−t1 )H0 (x, z), (z)e−t1 H0 (z, w) dt1 dvX,0 (z). (D.1.23) =− dt1 ∂v 0 X But (D.1.20) and (D.1.23) means exactly (D.1.19). By using (D.1.19), we conclude Theorem D.1.6.

As we suppose X is compact, we can also use the formal solution introduced by Minakshisundaram and Pleijel to construct the heat kernel (cf. [15, Chap. 2] for a detailed exposition). Especially, the following very useful asymptotic expansion

388


for the heat kernel holds: There exist hj ∈ C ∞ (X, End(E)) such that for any k, l ∈ N, there exists Ck,l > 0 so that k m −tH (x, x) − hj (x)tj− 2 e j=0

m

C l (X)

Ck,l tk+1− 2

(D.1.24)

as t → 0. The coefficients hj depend smoothly on the geometric data and their derivatives at x ∈ X. Moreover, if we have a family Hv as in Theorem D.1.6, then in the expansion (D.1.24), the C l -norm includes also the parameter v (cf. also Section 5.5.6 for a proof in the spirit of the present book). Assume now that the boundary ∂X of X is not empty. Then we need to fix boundary conditions in order to define a self-adjoint extension of H and the corresponding heat kernel. Usually, we impose Dirichlet (or Neumann) boundary conditions. Let en be the inward pointing unit normal at any boundary point of X. Then s ∈ C ∞ (X, E) satisfies the Dirichlet boundary conditions if s = 0 on ∂X. Likewise s ∈ C

∞

(D.1.25)

(X, E) satisfies the Neumann boundary condition if ∇E en s = 0

on ∂X.

(D.1.26)

D.2 Wave equation Now we explain the finite propagation speed property for the wave equation. We suppose X is a compact manifold with boundary ∂X. Without loss of generality, we assume that H = ∆E + Q in (A.3.3) is positive. If we consider Dom(H) = {s ∈ C ∞ (X, E) : s = 0 or ∇E en s = 0 on ∂X}, and take the Friedrichs extension, the resulting operator, denoted still by H, is positive. The basic references for this part are [65, §7.8] and [237, §2.8]. Theorem D.2.1. For ω = ω(t, x), t ∈ R, x ∈ X, we consider the wave equation ∂2 + H ω=0 (D.2.1) ∂t2 with the Dirichlet (or Neumann) boundary condition. Then for any f0 , f1 ∈ C ∞ (X, E), verifying the corresponding boundary condition, there exists a unique solution ω for (D.2.1) with initial conditions ω(0, x) = ∂ ω(0, x) = f1 . The solution ω is given by the operator expression f0 , ∂t √ √ sin(t H) √ ω(t, x) = cos(t H)f0 + f1 , (D.2.2) H and satisfies supp(ω(t, ·)) ⊂ Kt = {x ∈ X; d(x, y) t, for some y ∈ supp(f0 ) ∪ supp(f1 )}. (D.2.3)

D.2. Wave equation

389

Proof. In fact, to get the finite propagation speed (D.2.3) and the uniqueness of the solution, we only need to obtain an energy estimate. A vector v = (vt , vx ) ∈ T (R × X), is called timelike if |vx | < |vt |, and a hypersurface Σ ⊂ [0, T ] × (X \ ∂X) is called spacelike, if the unit normal vector (Nt , Nx ) is timelike. Suppose that Ω ⊂ [0, T ] × X is bounded by two spacelike surfaces Σ1 , Σ2 , and Σ3 := Ω ∩ [0, T ] × ∂X. We suppose that Ω is swept out by spacelike surfaces. Specifically, we assume that there is a smooth function h on a neighborhood of Ω such that grad(h) (defined by grad(h), Y = Y h for Y ∈ T X) is timelike, and we set Ω(u) := Ω ∩ {h u}, Σ2 (u) := Ω ∩ {h = u}. Suppose Ω is swept out by Σ2 (u), 0 u u1 with Σ2 = Σ2 (u1 ) and Ω(u) = ∅ if u < 0. Also set Σb1 (u) = Σ1 ∩ {h u}, Σb3 (u) = Σ3 ∩ {h u} (cf. Fig. D.1).

Σ2 (u)

Σ2

Σ3

t Ω (u)

Σb3 (u) Σb1 (u)

X Figure D.1.

Denote by ν2 = gradh/|gradh|, then ν2 is a timelike vector field. Set ∂

∂ ∂ ω, ∇E ω) = (| ∂t ω|2 + |∇E ω|2 )Nt − 2 ∂t ω, ∇E E( ∂t ν2 ω |Nx |,

(D.2.4)

on ∂Ω(u), then sign(Nt )E(ωt , ∇E ω) is positive as ∂Ω(u) is spacelike. For s1 , s2 ∈ C ∞ (X, E), let α be the one form on X defined by α(Y )(x) = E ∇Y s1 , s2 (x) for any Y ∈ Tx X. By (1.2.6), (1.2.9), (1.2.10) and Stokes’ theorem (cf. (1.4.67)), we have Tr(∇α)dvX = d(iW dvX ) = − α(en )dv∂X . (D.2.5) X

X

∂X

Thus by (1.3.20), (D.2.5), we get E E E ∆ s1 , s2 dvX = ∇ s1 , ∇ s2 dvX − Tr(∇α)dvX X X X = ∇E s1 , ∇E s2 dvX + ∇E en s1 , s2 dv∂X . X

∂X

(D.2.6)

390

Appendix D. Heat Kernel and Finite Propagation Speed 2

∂ Now from ( ∂t 2 + H)ω = f2 on [0, T ] × X with Dirichlet (or Neumann) ∂ ω(0, x) = f1 , by boundary conditions and the initial conditions ω(0, x) = f0 , ∂t applying (D.2.6) for Ω(u), we get ∂

E ∂ ∂2 ω, f − Qω dv dt = ∂t ω, ( ∂t 2 X 2 + ∆ )ωdvX dt ∂t Ω(u) Ω(u) 1 2 E 2 ∂ ∂ ∂ (| ω| + |∇ ω| )dv dt + ω, ∇E = X en,u ω dv∂Ω(u) (D.2.7) ∂t ∂t ∂t 2 Ω(u) ∂Ω(u) 1 ∂ ∂ = E( ∂t ω, ∇E ω)dS + ω, ∇E en,u ω |Nx |dS, ∂t 2 Σb1 (u)∪Σ2 (u) Σb3 (u)

where dS is the induced surface measure on ∂Ω(u) = Σb1 (u) ∪ Σ2 (u) ∪ Σb3 (u), and en,u is the inward pointing unit normal at ∂Ω(u). On Σb3 (u), the last term of the above equation is zero under the relative or absolute boundary condition (it can also be controlled under other suitable boundary conditions). Thus by using the Cauchy inequality and (D.2.7), we have ∂ ∂ E( ∂t ω, ∇E ω)dS E( ∂t ω, ∇E ω)dS Σ2 (u)

Σb1 (u)

+C Ω(u)

∂ 2E( ∂t ω, ∇E ω) + |ω|2 + |f2 |2 dvX dt.

The following argument is quite standard. First, we have ∂ |ω|2 dvX dt C |f0 |2 dS + C E( ∂t ω, ∇E ω)dvX dt. Ω(u)

Σ1 (u)

(D.2.8)

(D.2.9)

Ω(u)

Now set E(u) =

Ω(u)

Then by (D.2.4), dE C du

∂ E( ∂t ω, ∇E ω)dvX dt.

(D.2.10)

Σ2 (u)

∂ E( ∂t ω, ∇E ω)dS.

(D.2.11)

By (D.2.8), (D.2.9) and (D.2.11), we have the estimate dE CE(u) + F (u), du where

F (u) = C Σ1

(D.2.12)

∂ (E( ∂t ω, ∇E ω) + |f0 |2 )dS + C

(D.2.12) is equivalent to e

d −Cu E(u)) du (e −Cu

E(u) 0

|f2 |2 dvX dt.

(D.2.13)

Ω(u)

e−Cu F (u). Since E(0) = 0, we get u

e−Cu2 F (u2 )du2 .

(D.2.14)

D.2. Wave equation

391

From (D.2.14), we get the following energy estimate, for u ∈ [0, u1 ], E ∂ E( ∂t ω, ∇ ω)dvX dt C u (|f0 |2 + |∇E f0 |2 + |f1 |2 )dS Ω(u) Σ1 +C |f2 |2 dvX dt.

(D.2.15)

Ω(u)

From (D.2.15), we get the uniqueness of the solution of (D.2.1) and the finite propagation speed property (D.2.3). We turn now to the existence of the solution ω. One way to do it is as follows: Since X is compact, the heat kernel of H with boundary conditions exists, since H with corresponding boundary condition is essentially self-adjoint, thus the eigenfunctions of H span L2 (X, E). Let ϕj (x) be the eigenfunctions of H with eigenvalues λj . Then a solution of the Cauchy problem in the theorem is 1/2 −1/2 1/2 ω(t, x) = f0 , ϕj cos(λj t) + f1 , ϕj λj sin(λj t) ϕj (x), (D.2.16) j

√ which is equivalent to ω(t, x) = cos(t H)f0 + orem D.2.1.

√ sin(t H) √ f1 . H

Thus we conclude The

Note that from the finite propagation speed and the uniqueness of the problem, it is clear now that the solution ω can be defined on a complete manifold M with boundary. Namely, let Ωj ⊂ M be compact subsets with smooth boundary such that Ω1 ⊂ · · · ⊂ Ωj ⊂! M . Then we can work on each Ωj to get the result. From (D.2.2), we get the following important relation between the heat kernel and the wave equation for a non-negative operator H, √ 2 dv −t2 H/2 (D.2.17) e = cos(vt H)e−v /2 √ . 2π R

Problems Problem D.1. Let Q0 be a formally self-adjoint first order differential operator on E, and H be a generalized Laplacian on E associated to ∇E . Verify that there E such that H + Q0 is a generalized Laplacian exists a Hermitian connection ∇ E . operator associated to ∇ Problem D.2. Use the method of the proof of (D.1.6) to show that the heat operator e−tH is unique.

Appendix E

Harmonic Oscillator This appendix is organized as follows. In Section E.1, we explain in detail the spectrum and heat kernel of the harmonic oscillator on R. In Section E.2, we explain Mehler’s formula in the general case. Some good references for this appendix are [112, §1.5], [238, §8.6], [27, §6].

E.1 Harmonic oscillator on R We consider the harmonic oscillator H = ∆ + x2 − 1 on L2 (R),

with ∆ = −

∂2 . ∂x2

(E.1.1)

The inner product on L2 (R) is defined by f, g = R f · g¯dx with corresponding norm · L2 . We will establish the spectrum decomposition of H and Mehler’s formula for the heat kernel e−tH . We define the creation and annihilation operators a and a+ by ∂ a = − ∂x + x,

a+ =

∂ ∂x

+ x.

(E.1.2) / Theorem E.1.1. The spectrum of H is given by Spec(H) = 2k : k ∈ N and a 2 basis of the eigenspace of 2k is given by ak e−x /2 . .

Proof. From (E.1.2), a+ is the adjoint of a, and we have [a, a+ ] = −2.

(E.1.3)

From (E.1.2) and (E.1.3), we get H = a+ a − 2 = aa+ ,

[H, a] = 2a,

[H, a+ ] = −2a+ .

(E.1.4)

394

Appendix E. Harmonic Oscillator

Thus if u0 ∈ L2 (R) verifies a+ u0 = 0 and u0 L2 = 1, then u0 is an eigenfunction ∂ u0 = −xu0 , and we get of H with eigenvalue 0. But this means that ∂x u0 = π −1/4 e−x

2

/2

.

(E.1.5)

Certainly, e−x /2 belongs to C ∞ (R) ∩ L2 (R). Since aa+ is a non-negative operator, we see that the smallest element of Spec(H) is 0, and the corresponding 2 eigenfunction is Ce−x /2 with C ∈ C. Suppose that uj ∈ C ∞ (R) ∩ L2 (R) is an eigenfunction of H with eigenvalue λj , i.e., Huj = λj uj . (E.1.6) 2

By (E.1.4) and (E.1.6), we have H(a+ uj ) = (λj − 2)a+ uj .

H(auj ) = (λj + 2)auj ,

(E.1.7)

Thus auj , a+ uj are eigenfunctions of H, provided auj , a+ uj ∈ L2 (R). Especially ak u0 is an eigenfunction of H with eigenvalue 2k. To complete the proof, we need to know that the eigenfunctions ak u0 form a complete set in L2 (R). This will be proved in Lemma E.1.3. We define the Hermite polynomials Hk (x) by Hk (x)e−x

2

/2

= ak e−x

2

/2

.

(E.1.8)

Thus H0 = 1, H1 (x) = 2x. From (E.1.2), (E.1.8), we obtain Hk+1 (x) = (2x − Set ck,j =

∂ ∂x )Hk (x).

k! . j! 2j (k − 2j)!

(E.1.9)

(E.1.10)

Then we show by recurrence that k/2 ∂ k −x Hk (x) = (−1)k ex ( ∂x ) e = 2

2

(−1)j ck,j 2k−j xk−2j ,

(E.1.11)

j=0

where k/2 is the integer part of k/2. Set 2 ϕk = [π 1/2 2k (k!)]−1/2 Hk (x)e−x /2 .

(E.1.12)

Then ϕk is an eigenfuction of H with eigenvalue 2k and ϕk L2 = 1. In fact, by (E.1.4), (E.1.8), if ϕk L2 = 1, then ϕk+1 2L2 = (2k + 2)−1 aϕk 2L2 = (2k + 2)−1 a+ aϕk , ϕk = 1.

E.1. Harmonic oscillator on R

395

Lemma E.1.2. The inversion of (E.1.11) is k/2

xk =

j=0

x ck,j 2−(k−2j)/2 Hk−2j √ . 2

(E.1.13)

Proof. We prove (E.1.13) by induction. At first (E.1.13) is true for k = 0. Suppose that (E.1.13) is true for k l. Relation (E.1.9) yields l/2 l+1

x

=

j=0

l/2

=

j=0

x cl,j 2−(l−2j)/2 xHl−2j √ 2

x ∂ cl,j 2−(l−2j+1)/2 Hl+1−2j + ( ∂x Hl−2j ) √ . 2

(E.1.14)

By the induction hypothesis, the terms containing a derivative sum to lxl−1 . Using again the induction hypothesis for xl−1 , we infer from (E.1.14) that x lcl−1,j−1 2−(l−2j+1)/2 Hl+1−2j √ . 2 j=0 j=1 (E.1.15) Note that cl+1,j = cl,j + lcl−1,j−1 , for j l/2 , and cl+1,j = lcl−1,j−1 if l is odd and j = (l + 1)/2. Thus we get (E.1.13) for k = l + 1 from (E.1.15). l/2

xl+1 =

(l+1)/2

cl,j 2−(l−2j+1)/2 Hl+1−2j +

Lemma E.1.3. The normalized Hermite functions ϕk (x) introduced in (E.1.12) form a complete orthonormal set in L2 (R). Proof. From (E.1.12), (E.1.13), we get k/2 √ ( 2x)k ϕ0 = ck,j ((k − 2j)!)1/2 ϕk−2j (x).

(E.1.16)

j=0

Thus k/2

xk ϕ0 2L2 =

k/2

c2k,j 2−k (k − 2j)! =

j=0

j=0

(k!)2 (k!)2 . (k − 2j)!(j!)2 22j+k k/3 ! 2k−1 (E.1.17)

In view of (E.1.17), we see that for any ν ∈ R, eiνx ϕ0 =

∞

(iνx)k (k!)−1 ϕ0

(E.1.18)

k=0

converges in L2 (R). Thus eiνx ϕ0 lies in the span of the {ϕk }k . But {eiνx ϕ0 }ν∈R span L2 (R). In fact, if v ∈ L2 (R) is orthogonal to eiνx ϕ0 , then the Fourier

396


transform (vϕ0 )∼ (ν) of vϕ0 is zero on R. Thus vϕ0 L2 = (vϕ0 )∼ L2 = 0, by Plancherel’s theorem. This means that v = 0. The proof of Lemma E.1.3 is complete. Let e−tH (x, x ) be the kernel of the heat operator e−tH with respect to dx . Then we have ∞ e−tH (x, y) = e−2kt ϕk (x)ϕk (y). (E.1.19) k=0

Recall that sinh(t) =

1 t (e − e−t ), 2

cosh(t) =

1 t (e + e−t ), 2

tanh(t) =

sinh(t) . cosh(t)

(E.1.20)

Now we prove Mehler’s formula. Theorem E.1.4. For t > 0, we have e

−tH

(x, y) = (π(1 − e

−4t

−1/2

))

6 xy x2 + y 2 + exp − . 2 tanh(2t) sinh(2t)

(E.1.21)

Proof. We can prove (E.1.21) from (E.1.19) by Fourier transformation, but here we take a different route. We denote by Pt (x, y) the right side of (E.1.21) and for u ∈ L2 (R), set v(t, x) = R

Pt (x, y)u(y)dy.

(E.1.22)

Since Pt (x, y) is rapidly decreasing for |x| + |y| → ∞, and t > 0, we deduce that v ∈ C ∞ (]0, ∞[, S(R)), provided u ∈ L2 (R). Here S(R) denotes the Schwartz space of rapidly decreasing functions. We verify by direct computation that ∂v = −Hv. ∂t

(E.1.23)

v(t + λ, ·) − e−tH v(λ, ·) = 0.

(E.1.24)

We claim now that for each λ > 0,

In fact, if we denote the left-hand side of (E.1.24) by w(t, ·), then we have w(0, ·) = ∞ 2 0, ∂w ∂t ∈ C (]0, ∞[, L (R)), and by (E.1.4), ∂ w(t, ·)2L2 = −2Hw, w 0. ∂t

(E.1.25)

Thus w(t, ·) = 0 for any t > 0. Finally, from (E.1.21), we know that v(t, ·) → u in L2 (R) as t → 0. Thus letting λ → 0 in (E.1.24), we get (E.1.21).

E.2. Harmonic oscillator on vector spaces

397

E.2 Harmonic oscillator on vector spaces Let V be a real vector space with complex structure J and dimR V = 2n. We V (1,0)√and V (0,1) are the have the decomposition V ⊗R C = V (1,0) ⊕ V (0,1) where √ eigenspaces of J corresponding to the eigenvalues −1 and − −1 respectively. Let g V be a Euclidean metric on V which is compatible with J. Then we V consider V√ h = (V, J) as a complex vector space√with Hermitian metric h (·, ·) = 1 V V g (·, ·) − −1g (J·, ·); moreover, Y → 2 (Y − −1JY ) induces a natural identification of Vh with V (1,0) . Let A be an invertible self-adjoint complex endomorphism of the complex vector space V (1,0) √. We extend A to V ⊗R C by defining Av = −Av for any v ∈ V (1,0) , then −1A induces an anti-symmetric endomorphism on the real vector space (V, g V ). We will denote by det the determinant on V (1,0) . V Let {ei }2n i=1 be an orthonormal basis of (V, g ). For Y ∈ V , let ∇Y be the ordinary differential operator on V in the direction Y . Consider the differential operators on V ,

1 (∇ei )2 + A2 Z, Z − TrV (1,0) [A], Z ∈ V ; 4 i 2 1 H =− ∇ei + AZ, ei − TrV (1,0) [A] = H − ∇AZ , 2 i H=−

(E.2.1) Z ∈ V.

Let e−tH (Z, Z ) and e−tH (Z, Z ), (Z, Z ∈ V ), be the smooth kernels of the heat operators e−tH and e−tH associated to the volume form dZ. Then by Theorem E.1.4, we know that e−tH (Z, Z ) = (2π)−n det

Since

A 1 A/2 Z, Z exp − 1 − exp(−2tA) 2 tanh(tA) 6 A/2 1 A/2 Z , Z + Z, Z − . (E.2.2) 2 tanh(tA) sinh(tA)

√ −1A ∈ End(V ) is anti-symmetric, thus [H, ∇AZ ] = 0.

(E.2.3)

From (E.2.2) and (E.2.3), we get that e−tH (Z, Z ) = e−tH (etA Z, Z ) A 1 A/2 = (2π)−n det Z, Z exp − 1 − exp(−2tA) 2 tanh(tA) 6 A/2 A/2 1 Z , Z + etA Z, Z − . 2 tanh(tA) sinh(tA)

(E.2.4)

398


If A = 0, then H is the Laplace operator ∆ = − classically

2 i (∇ei ) ,

1 e−t∆ (Z, Z ) = (4πt)−n exp − |Z − Z |2 . 4t

hence we have

(E.2.5)

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Index (∇E )1,0 , (∇E )0,1 , 12, 31 B X (x, ε), 14 B Tx X (0, ε), 14 Dc , 24 Dpc,A , 48 Dp , 46 Ep , 49, 134 H • (X, C), 370 H • (X, R), 370 H •,• (X, E), 30, 367 H • (X, E), H • (X, OX (E)), 367 r,q q H(2) (X, E), H(2) (X, E), 130 IC⊗E , 191 KX , 38, 64, 73 L2 (U ), 345 L2loc (X, E), 349 L2•,• (X, E), 365 Lp , 46 Lt2 , 54 O(n), 7 RT X , 13 Rp , 226 RE,p , 230 SL(n, C), 7 S B , 21 T X , 18 T torsion of ∇ T X, 13 T (1,0)X, T (0,1) X, 11 T ∗(1,0) X, T ∗(0,1)X, 11 Th X, 12, 355 X(q), X( q), X( q), 61, 103 X(q, ω), X( q, ω), 110, 154 CPn , 64 C, C∗ , 7 ∆ contour, 183

∆, ∆F Laplacian, 25 G(k, V ), G(k, m), 212 c,A ), 29 Ind(D+ Λ•,• (T ∗ X), 355 Λω (RE ), 177 N, N∗ , 7 Ω• (X), Ω•0 (X), Ω•,• 0 (X), 355 Ω• (X, E), 10 Ω•,• (X, E), 11 Ω • (X), Ω •,• (X), 356 Ω•,• 0 (X, E), 365 P(V ), 212 Q, Q+ , 7 R, R∗ , R+ , R∗+ , 7 , 8 Z, 7 α!, |α|, 7 C ∞ (X, E), 8 D , 356 H 0,• (X, E), 130 L , 189 L2t , L20 , 181 O(−1), O(k), 65, 212 OX (E), 353, 367 ch(F ), ch(F, ∇F ), 29 χ(X, E), 33 ∂-Neumann problem, 161 E E,∗ ∂ p , ∂ p , 134 ∂ b , 284 δ contour, 183 det(E), 8, 229 Div, 71, 352 Div(X), 71 DivQ (X), 114

416

Index

R˙ L , 46 ∂ α ∂ |α| ∂xα , ∂xα , 345 ·, ·Λ0,• ⊗E , 24 , 7 R radial vector field, 14, 179 S(Rn ), 346 S (Rm ), 356 T , 35 µ0 , 47 (1,0) X , 18 ∇T B ∇ , 21 ∇T X , 13 ∇Cl , 22, 24 ∇0,· , 189 ·L2 , 24, 345 ωd , 46 ωF S , 65, 212, 213 ordV , 71, 352 ∂-Lemma, 367 0,• H (2) (X, E), 130 E

E,∗

∂ , ∂ , 29 E , 35 rX , 226 ∂∂-Lemma, 43, 361 rk(E) rank of E, 8, 61 Xreg , Xsing , 353 Ricω , 64 σess , 381 sing supp, 357 E , 30, 35 p , 46 supp, 356 τ , 46 Td(F ), Td(F, ∇F ), 29 ∧, 14 { , } Poisson bracket, 310 a(X), 85 bi , b+ i , 190 c(·), 22 c1 (F ), c1 (F, ∇F ), 29, 239 c1 (L, hL ), 98 c1 (X), 224 dp , 226

dE,p , 230 dvX , 13 iW , 14 rωX , 226 wj ,w j , 17 H k (X, E), 349 H k (X, E, loc), 349 H k0 (X, E), H k0 (U, E), 349 analytic hypersurface, 352 analytic subset, 352 analytic torsion of Ray–Singer, 253 anomaly formula of ∼, 256 Andreotti, A., 293, 365 Andreotti–Vesentini density lemma, 148 Atiyah–Singer index theorem, 29, 66 Aubin, T., 225 Baily, W., 248 base point locus, 82 Berezin–Toeplitz quantization, 299 Bergman kernel, 31, 175, 189, 197, 315, 316 generalized ∼, 327 Bertini theorem, 82 bimeromorphically equivalent, 78 Bismut, J.-M., 21, 34, 66, 210, 270 blow-up of a manifold, 75 Bochner–Kodaira–Nakano formula, 35 with boundary term, 40, 163 Bonavero, L., 101 Bordemann, M., 314 Bouche, T., 66, 174, 209, 269 Boucksom, S., 116 boundary condition ∂-Neumann ∼, 160 Dirichlet ∼, 388 Neumann ∼, 388 Boutet de Monvel, L., 209, 293, 314, 343 Brian¸con–Skoda theorem, 104, 125

Index

Calabi, E., 117, 225 canonical section, 71 Cartan, H., 246 Cartan–Serre–Grothendieck lemma, 218 Catlin, D., 209, 269 Chern class, 29 ith ∼, 373 first ∼, 29, 46, 239 total ∼, 373 Chern-Simons class, 372 functional, 374 Chow’s theorem, 217, 267 Clifford action, 22 connection, 22 complete pluripolar set, 152 complex Hessian, 362 complex space, 353 irreducible ∼, 353 connection, 10 Bismut ∼, 21 Chern ∼, 11 Clifford ∼, 22 Hermitian ∼, 11 holomorphic ∼, 11 holomorphic Hermitian ∼, 11, 18 Levi–Civita ∼, 13 current, 355, 356 closed ∼, 356 curvature ∼, 98 integral ∼, 361 K¨ ahler ∼, 100, 361 Lebesgue decomposition of a ∼, 360 of integration on an analytic set, 360 positive ∼, 359 push-forward of a ∼, 357 singular support of a ∼, 357 strictly positive ∼, 359 with analytic singularities, 101 curvature

417

Hermitian scalar ∼, 328 pinched negative ∼, 282 Ricci ∼, 13 scalar ∼, 13 sectional ∼, 282 Dai, X., 210, 270, 343 de Rham cohomology, 369 Demailly’s current approximation theorem, 102, 115 Demailly, J.-P., 66, 96, 368 dimension algebraic ∼, 85 Γ-∼, 169 Kodaira–Iitaka ∼, 83 von Neumann ∼, 169 Dirac operator modified ∼, 26, 47, 316 spinc ∼, 24, 316 distributions, 356 tempered ∼, 356 divisor, 71, 352 approximate Zariski decomposition of a ∼, 115 effective ∼, 71, 114 exceptional ∼, 72, 75 Q-∼, 114 with normal crossings, 71 Zariski decomposition of a ∼, 114 Dolbeault cohomology, 30, 367 complex, 30, 367 isomorphism, 30, 367 L2 ∼ cohomology, 130 operator, 30 reduced L2 ∼ cohomology, 130 Dolbeault-Grothendieck Lemma, 367 domain pseudoconvex ∼, 364 strongly pseudoconvex ∼, 364 Donaldson, S., 228, 343

418

elliptic estimate, 347, 350 embedding CR ∼, 284 Grassmannian ∼, 221 projective ∼, 214 theorem, see theorem end, 281 cusp ∼, 282 hyperconcave ∼, 281 strongly pseudoconcave ∼, 281 extension Friedrichs ∼, 128, 378 Gaffney ∼, 128 maximal ∼, 128 minimal ∼, 128 Fefferman, C., 209, 314 finite propagation speed, 52, 54, 179, 200, 329, 389 form Fubini–Study ∼, 65, 212 harmonic ∼, 130 Hermitian ∼, 20 holomorphic r- ∼, 367 K¨ ahler ∼, 20, 239 Levi ∼, 41 positive ∼, 358 quadratic ∼, 377 quadratic ∼ associated to a positive self-adjoint operator, 377 Riemannian volume ∼, 13 semipositive ∼, 358 formula Bochner–Kodaira–Nakano ∼, 35 with boundary term, 40, 163 Lichnerowicz ∼, 25, 34 Mehler’s ∼, 396 Poincaré–Lelong ∼, 98, 360 Friedrichs’ lemma, 129 function Busemann ∼, 282 exhaustion ∼, 281, 362, 364 holomorphic ∼, 353

Index

meromorphic ∼, 70 plurisubharmonic ∼, 359 quasi-plurisubharmonic ∼, 102, 360 rational ∼, 84 spectrum counting ∼, 381 strictly plurisubharmonic ∼, 360 subharmonic ∼, 359 with analytic singularities, 101 fundamental estimate, 132, 134 G˚ arding’s inequality, 350 Gillet, H., 270 Grauert tube, 364 Grauert’s ampleness criterion, 364 Grauert, H., 293, 362 Grauert–Riemenschneider conjecture, 96 Siu–Demailly’s solution of the ∼, 96 Guillemin, V., 314, 343 H¨ ormander, L., 163, 365 harmonic oscillator, 393 heat kernel, 49, 55, 197, 384 heat operator, 50, 384 Hermite polynomials, 191, 394 Hermitian metric, 10 local weight of a ∼, 97 singular ∼, 97 with analytic singularities, 101 Hermitian torsion operator, 35 Hironaka’s resolution of singularities theorem, 76, 77 Hodge strong ∼ decomposition, 132 theory, 30, 239 weak ∼ decomposition, 131 Hodge ∗-operator, 253 holomorphic tangent bundle, 355 Hurwitz theorem, 84 Jacobi identity, 27 jet, 80

Index

K¨ ahler csc ∼ metric, 224 current, 100 –Einstein metric, 224 form, 20, 239 identities, 36 manifold, 20 metric, 20 orbifold, 239 Kodaira embedding theorem, 219 –Iitaka dimension, 83 map, 82, 214 Laplace transform, 142 inverse ∼, 142, 173 Laplacian ∂-∼, 37 ∂-∼, 37 Bochner ∼, 25 generalized ∼, 349 Kodaira ∼, 30, 35 Kohn ∼, 285 Lebeau, G., 66, 210 Lelong number, 361 Lelong, P., 360 lemma Andreotti–Vesentini density ∼, 148 Cartan–Serre–Grothendieck ∼, 218 Friedrichs’ ∼, 129 Siegel’s ∼, 83 Levi form, 41, 363 Levi’s removable singularity theorem, 154 Lichnerowicz formula, 25, 34 Lie bracket, 10 graded ∼, 27 Lie derivative, 13 line bundle ample ∼, 214 associated cocycle ∼, 70 big ∼, 83

419

canonical ∼, 64, 73 nef ∼, 113 negative ∼, 44 numerically effective ∼, 113 positive ∼, 44 pseudo-effective ∼, 115 semi-ample ∼, 214 semi-positive ∼, 44 very ample ∼, 214 volume of a ∼, 112 Liu, K., 210, 270, 343 Lu, Z., 209 manifold q-complete ∼, 362 q-concave ∼, 362 q-convex ∼, 362 algebraic reduction of a Moishezon ∼, 88 Andreotti pseudoconcave ∼, 152 blow-up of a ∼, 75 CR ∼, 283 Hermitian ∼, 20 hyper 1-concave ∼, 152 hyperconcave ∼, 152 K¨ ahler ∼, 20 Moishezon ∼, 87, 154 projective ∼, 75 strongly pseudoconvex CR ∼, 284 weakly 1-complete ∼, 364 map biholomorphic ∼, 354 bimeromorphic ∼, 78 finite holomorphic ∼, 354 holomorphic ∼, 354 Kodaira ∼, 214 meromorphic ∼, 77 moment ∼, 208, 341 proper holomorphic ∼, 354 Mehler’s formula, 396 Meinrenken, E., 314 Mellin transformation, 251 metric balanced ∼, 227, 230

420

csc Kähler ∼, 224 Fubini–Study ∼, 65, 212, 213 generalized Poincaré ∼, 276 Hermitian ∼, 10 K¨ ahler ∼, 20 K¨ ahler–Einstein ∼, 224 Quillen ∼, 253 Moishezon B.G., 87 moment map, 208, 341 Morse inequalities, 67 algebraic ∼, 145 Bonavero’s singular holomorphic ∼, 101 Demailly’s algebraic ∼, 121 Demailly’s holomorphic ∼, 61 multiplicity of an analytic set, 352 Nadel multiplier ideal sheaf, 102 Nadel vanishing theorem, 369 Nadel, A., 294 Nakano inequality, 37 Napier, T., 294 operator adjoint ∼, 375 Dolbeault ∼, 30 essentially self-adjoint ∼, 376 heat ∼, 50, 384 Hermitian torsion ∼, 35 Hilbert-Schmidt ∼, 386 modified Dirac ∼, 26, 47, 316 positive ∼, 376 resolvent set of an ∼, 379 self-adjoint ∼, 376 smoothing ∼, 358 spectrum of an ∼, 379 spinc Dirac ∼, 24, 316 tangential Cauchy-Riemann ∼, 284 Toeplitz ∼, 298, 318, 323 trace class ∼, 386 orbifold, 235 K¨ ahler ∼, 239

Index

peak section, 216 Picard group, 70 Poincaré–Lelong formula, 98, 360 points of indeterminacy, 79 Poisson bracket, 310 principal symbol, 298, 319, 347 projective algebraic variety, 75 projectivized normal bundle, 74 proper modification, 77 quadratic form, 377 associated to a positive self-adjoint operator, 377 Ramachandran, M., 294 ramified covering, 87 Ray, D.B., 253 Rellich’s theorem, 349 Remmert reduction, 363 Remmert’s proper mapping theorem, 354 resolvent set of an operator, 379 Riemann’s second extension theorem, 353 Riemann–Roch–Hirzebruch theorem, 33, 66, 178 Riemannian volume form, 13 Riesz representation theorem, 346 Ruan, W., 209, 216 Schlichenmaier, M., 314 Schwartz kernel theorem, 31, 357 Schwartz kernel with respect to dvX , 358 Schwartz space, 346, 356 Schwarz inequality, 123 set sublevel ∼, 362 superlevel ∼, 362 sheaf analytic inverse image ∼, 351 direct image ∼, 108, 351 quasi-positive ∼, 96 Shiffman, B., 101, 269

Index

Shiffman–Ji–Bonavero–Takayama criterion, 97, 276 Siegel’s lemma, 83 Singer, I.M., 253 singular support, 357 Siu, Y.-T., 96, 122, 293 Sj¨ ostrand, J., 209, 314, 343 Sobolev embedding theorem, 347–349 Sobolev space, 346 Soulé, C., 270 space complex ∼, 353 normal complex ∼, 246 of currents, 355 reduced complex ∼, 353 Schwartz ∼, 346 Sobolev ∼, 346 Stein ∼, 362 weighted projective ∼, 268 Spec, 47 spectral gap, 43, 181, 240, 316, 322, 325, 329 spectral measure, 379 spectral resolution, 381 spectral sequence, 90 Fr¨ ohlicher ∼, 90 spectrum counting function, 381 discrete ∼, 381 essential ∼, 381 of an operator, 379 speed finite propagation ∼, 52, 54, 179, 200, 329, 389 stable Chow poly-∼, 228 Gieseker ∼, 230 Hilbert poly-∼, 228 Mumford ∼, 230 Stein factorization, 354 strict transform, 75 supercommutator, 27 support

421

singular ∼, 357 Szegö kernel, 209, 314 theorem Andreotti–Grauert’s coarse vanishing ∼, 343 Atiyah–Singer index ∼, 29, 66 Bertini ∼, 82 Brian¸con–Skoda ∼, 104, 125 Chow’s ∼, 217 Demailly’s current approximation ∼, 102, 115 Grauert vanishing ∼, 266 Hironaka’s flattening ∼, 78 Hironaka’s resolution of singularities ∼, 76 Hurwitz ∼, 84 Kodaira embedding ∼, 219 Kodaira vanishing ∼, 45 Kodaira–Serre vanishing ∼, 47 Levi’s removable singularity ∼, 154 Nadel vanishing ∼, 369 Nakano vanishing ∼, 45, 65, 166 Rellich’s ∼, 349 Riemann–Roch–Hirzebruch ∼, 33, 66, 178 Riesz representation ∼, 346 Schwartz kernel ∼, 31 Serre vanishing ∼, 218 Sobolev embedding ∼, 347–349 Tian, G., 209, 216, 225 Todd class, 29 Toeplitz operator, 298, 318, 323 Tsuji, H., 294 vanishing theorem, see theorem Vasserot, E., 66, 270 vector bundle ample ∼, 364 Grauert positive ∼, 266, 364 Griffiths positive ∼, 12, 65 Griffiths semi-positive ∼, 12 Hermitian ∼, 10 holomorphic Hermitian ∼, 11

422

Nakano positive ∼, 12, 65 Nakano semi-positive ∼, 12 orbifold ∼, 237 proper orbifold ∼, 237 vector field radial ∼ R, 14, 179 Wang, X., 209, 231 weighted projective space, 268 Witten, E., 67 Yau, S.-T., 117, 209, 225, 293 Zariski’s main theorem, 354 Zelditch, S., 209, 269 Zhang, S., 228 Zhang, W., 343

Index

Holomorphic Morse Inequalities And Bergman Kernels

Holomorphic Morse inequalities and Bergman kernels

Holomorphic Morse Inequalities and Bergman Kernels

Heat Kernels and Dirac Operators

Heat kernels and Dirac operators

Heat kernels and spectral theory

Theory of Bergman spaces

Morse Theory

Inequalities

Holomorphic Dynamics

Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincare Conjecture

Theory of Bergman spaces

Bergman Jr, Peter - Arcade

Kernels for structured data

Sobolev inequalities, heat kernels under Ricci flow, and the Poincare conjecture

Inequalities and Applications

Polynomials and Polynomial Inequalities

Polynomials And Polynomial Inequalities

Order and Potential Resolvent Families of Kernels

Holomorphic maps and invariant distances

Circle-Valued Morse Theory

Social Inequalities and Cancer

Polynomials and polynomial inequalities

Topological Vector Spaces, Distributions And Kernels

Old and New Inequalities

Polynomials and Polynomial Inequalities

Polynomials and Polynomial Inequalities

Integral inequalities and applications

Circle-valued Morse theory

Lectures on Morse homology

Ingmar Bergman: A Reference Guide

Holomorphic Morse Inequalities And Bergman Kernels