This book examines holomorphic Morse inequalities and the asymptotic expansion of the Bergman kernel on manifolds by using the heat kernel. The point of view comes from local index theory. The book 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.

Most of the material appears for the first time in book form New approach to the holomorphic Morse inequalities and Bergman kernel expansions Exploits the analytic localization techniques in local index theory developed by Bismut-Lebeau Includes supplementary material: sn.pub/extras

Klappentext

This book gives for the first time a self-contained and unified approach to holomorphic Morse inequalities and the asymptotic expansion of the Bergman kernel on manifolds by using the heat kernel, and presents also various applications.

The main analytic tool is the analytic localization technique in local index theory developed by Bismut-Lebeau. The book includes the most recent results in the field and therefore opens perspectives on several active areas of research in complex, Kähler and symplectic geometry. A large number of applications are included, e.g., an analytic proof of the Kodaira embedding theorem, a solution of the Grauert-Riemenschneider and Shiffman conjectures, a compactification of complete Kähler manifolds of pinched negative curvature, the Berezin-Toeplitz quantization, weak Lefschetz theorems, and the asymptotics of the Ray-Singer analytic torsion.

Inhalt
Demailly'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.