This book examines holomorphic Morse inequalities and the asymptotic expansion of the Bergman kernel on manifolds by using the heat kernel. It opens perspectives on several active areas of research in complex, Kähler and symplectic geometry. A large number of applications are also included, such as an analytic proof of Kodaira's embedding theorem, a solution of the Grauert-Riemenschneider and Shiffman conjectures, compactification of complete Kähler manifolds of pinched negative curvature, Berezin-Toeplitz quantization, weak Lefschetz theorems, and asymptotics of the Ray-Singer analytic torsion.

InhaltsangabeDemailly's Holomorphic Morse Inequalities.- Characterization of Moishezon Manifolds.- Holomorphic Morse Inequalities on Non-compact Manifolds.- Asymptotic Expansion of the Bergman Kernel.- Kodaira Map.- Bergman Kernel on Non-compact Manifolds.- Toeplitz Operators.- Bergman Kernels on Symplectic Manifolds.

0. Introduction.- 1. Demailly''s Holomorphic Morse Inequalities.- 2. Characterization of Moishezon Manifolds.- 3. Holomorphic Morse Inequalities on Non-compact Manifolds.- 4. Asymptotic Expansion of the Bergman Kernel.- 5. Kodaira Map.- 6. Bergman Kernel on Non-compact Manifolds.- 7. Toeplitz Operators.- 8. Bergman Kernels on Symplectic Manifolds.- Appendix.- A. Sobolev Spaces - B. Elements of Analytic and Hermitian Geometry - C. Spectral Analysis of Self-adjoint Operators - D. Heat Kernel and Finite Propagation Speed - E. Harmonic Oscillator.

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