Holomorphic Morse Inequalities and Bergman Kernels (eBook)

Contents 9
Introduction 14
Notation 20
Acknowledgments 21
Demailly’s Holomorphic Morse Inequalities 22
1.1 Connections on vector bundles 22
1.1.1 Hermitian connection 22
1.1.2 Chern connection 24
1.2 Connections on the tangent bundle 25
1.2.1 Levi–Civita connection 26
1.2.2 Chern connection 30
1.2.3 Bismut connection 34
1.3 Spinc Dirac operator 34
1.3.1 Clifford connection 35
1.3.2 Dirac operator and Lichnerowicz formula 37
1.3.3 Modified Dirac operator 39
1.3.4 Atiyah–Singer index theorem 41
1.4 Lichnerowicz formula for 42
1.4.1 The operator 43
1.4.2 Bismut’s Lichnerowicz formula for E 47
1.4.3 Bochner–Kodaira–Nakano formula 48
1.4.4 Bochner–Kodaira–Nakano formula with boundary term 53
1.5 Spectral gap 56
1.5.1 Vanishing theorem and spectral gap 56
1.5.2 Spectral gap of modified Dirac operators 60
1.6 Asymptotic of the heat kernel 62
1.6.1 Statement of the result 62
1.6.2 Localization of the problem 63
1.6.3 Rescaling of the operator 66
1.6.4 Uniform estimate on the heat kernel 68
1.6.5 Proof of Theorem 1.6.1 73
1.7 Demailly’s holomorphic Morse inequalities 74
1.8 Bibliographic notes 79
Characterization of Moishezon Manifolds 81
2.1 Line bundles, divisors and blowing-up 81
2.2 The Siu–Demailly criterion 92
2.2.1 Big line bundles 92
2.2.2 Moishezon manifolds 96
2.3 The Shi.man–Ji–Bonavero–Takayama criterion 109
2.3.1 Singular Hermitian metrics on line bundles 109
2.3.2 Bonavero’s singular holomorphic Morse inequalities 113
2.3.3 Volume of big line bundles 123
2.3.4 Some examples of Moishezon manifolds 129
2.4 Algebraic Morse inequalities 133
2.5 Bibliographic notes 136
Holomorphic Morse Inequalities on Non-compact Manifolds 139
3.1 L2-cohomology and Hodge theory 139
3.2 Abstract Morse inequalities for the L2-cohomology 146
3.2.1 The fundamental estimate 146
3.2.2 Asymptotic distribution of eigenvalues 149
3.2.3 Morse inequalities for the L2 cohomology 156
3.3 Uniformly positive line bundles 160
3.4 Siu–Demailly criterion for isolated singularities 164
3.5 Morse inequalities for q-convex manifolds 172
3.6 Morse inequalities for coverings 179
3.6.1 Covering manifolds, von Neumann dimension 179
3.6.2 Holomorphic Morse inequalities 181
3.7 Bibliographic notes 185
Asymptotic Expansion of the Bergman Kernel 187
4.1 Near diagonal expansion of the Bergman kernel 187
4.1.1 Diagonal asymptotic expansion of the Bergman kernel 188
4.1.2 Localization of the problem 190
4.1.3 Rescaling and Taylor expansion of the operator 191
4.1.4 Sobolev estimate on the resolvent 195
4.1.5 Uniform estimate on the Bergman kernel 199
4.1.6 Bergman kernel of L 201
4.1.7 Proof of Theorem 4.1.1 203
4.1.8 The coefficient b1: a proof of Theorem 4.1.2 206
4.1.9 Proof of Theorem 4.1.3 208
4.2 Off-diagonal expansion of the Bergman kernel 209
4.2.1 From heat kernel to Bergman kernel 209
4.2.2 Uniform estimate on the heat kernel and the Bergman kernel 211
4.2.3 Proof of Theorem 4.2.1 216
4.2.4 Proof of Theorem 4.2.3 216
4.3 Bibliographic notes 221
Kodaira Map 223
5.1 The Kodaira embedding theorem 223
5.1.1 Universal bundles 224
5.1.2 Convergence of the induced Fubini–Study metrics 225
5.1.3 Classical proof of the Kodaira embedding theorem 231
5.1.4 Grassmannian embedding 233
5.2 Stability and Bergman kernel 236
5.2.1 Extremal Kähler metrics 236
5.2.2 Scalar curvature and projective embeddings 238
5.2.3 Gieseker stability and Grassmannian embeddings 241
5.3 Distribution of zeros of random sections 243
5.4 Orbifold projective embedding theorem 247
5.4.1 Basic de.nitions on orbifolds 247
5.4.2 Complex orbifolds 250
5.4.3 Asymptotic expansion of the Bergman kernel 251
5.4.4 Projective embedding theorem 258
5.5 The asymptotic of the analytic torsion 262
5.5.1 Mellin transformation 262
5.5.2 Definition of the analytic torsion 263
5.5.3 Anomaly formula 265
5.5.4 The asymptotics of the analytic torsion 268
5.5.5 Asymptotic anomaly formula for the L2-metric 272
5.5.6 Uniform asymptotic of the heat kernel 274
5.6 Bibliographic notes 281
Bergman Kernel on Non-compact Manifolds 283
6.1 Expansion on non-compact manifolds 283
6.1.1 Complete Hermitian manifolds 283
6.1.2 Covering manifolds 287
6.2 The Shiffman–Ji–Bonavero–Takayama criterion revisited 288
6.3 Compactification of manifolds 293
6.3.1 Filling strongly pseudoconcave ends 293
6.3.2 The compacti.cation theorem 298
6.4 Weak Lefschetz theorems 302
6.5 Bibliographic notes 305
Toeplitz Operators 307
7.1 Kernel calculus on Cn 307
7.2 Asymptotic expansion of Toeplitz operators 310
7.3 A criterion for Toeplitz operators 314
7.4 Algebra of Toeplitz operators 322
7.5 Toeplitz operators on non-compact manifolds 324
7.6 Bibliographic notes 326
Bergman Kernels on Symplectic Manifolds 327
8.1 Bergman kernels of modified Dirac operators 327
8.1.1 Asymptotic expansion of the Bergman kernel 328
8.1.2 Toeplitz operators on symplectic manifolds 330
8.2 Bergman kernel: mixed curvature case 332
8.2.1 Spectral gap 332
8.2.2 Asymptotic expansion of the Bergman kernel 334
8.3 Generalized Bergman kernel 336
8.3.1 Spectral gap 336
8.3.2 Generalized Bergman kernel 339
8.3.3 Near diagonal asymptotic expansion 340
8.3.4 The second coeffcient 345
8.3.5 Symplectic Kodaira embedding theorem 350
8.4 Bibliographic notes 355
Sobolev Spaces 356
A.1 Sobolev spaces on 356
A.2 Sobolev spaces on 359
A.3 Sobolev spaces on manifolds 359
Elements of Analytic and Hermitian Geometry 362
B.1 Analytic sets and complex spaces 362
B.2 Currents on complex manifolds 365
B.3 q-convex and q-concave manifolds 373
B.4 L2 estimates for . 376
B.5 Chern-Weil theory 380
Spectral Analysis of Self- adjoint Operators 386
C.1 Quadratic forms and Friedrichs extension 386
C.2 Spectral theorem 390
C.3 Variational principle 392
Heat Kernel and Finite Propagation Speed 394
D.1 Heat kernel 394
D.2 Wave equation 399
Harmonic Oscillator 403
E.1 Harmonic oscillator on 403
E.2 Harmonic oscillator on vector spaces 407
Bibliography 409
Index 425