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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.

Bergman kernel functions. --- Holomorphic functions. --- Morse theory. --- Symplectic manifolds. --- Variational inequalities (Mathematics) --- Inequalities, Variational (Mathematics) --- Calculus of variations --- Differential inequalities --- Functions, Holomorphic --- Functions of several complex variables --- Kernel functions, Bergman --- Holomorphic mappings --- Kernel functions --- Manifolds, Symplectic --- Geometry, Differential --- Manifolds (Mathematics) --- Critical point theory (Mathematical analysis) --- Morse, théorie de --- Inégalités variationnelles --- Fonctions holomorphes --- Variétés symplectiques --- EPUB-LIV-FT SPRINGER-B LIVMATHE --- Global differential geometry. --- Differential equations, partial. --- Global analysis. --- Differential Geometry. --- Several Complex Variables and Analytic Spaces. --- Global Analysis and Analysis on Manifolds. --- Global analysis (Mathematics) --- Partial differential equations --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Differential geometry. --- Functions of complex variables. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Complex variables --- Elliptic functions --- Functions of real variables --- Topology --- Differential geometry