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, Kahler 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 Kahler manifolds of pinched negative curvature,
Berezin-Toeplitz quantization, weak Lefschetz theorems, and
asymptotics of the Ray-Singer analytic torsion.
General
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