On this page we record information about our book Holomorphic Morse Inequalities and Bergman Kernels, appeared in in the series Progress in Mathematics, vol. 254 at Birkhäuser, 2007.
In the present book we give a self-contained and unified approach of the holomorphic Morse inequalities and the asymptotic expansion of the Bergman kernel by using the heat kernel, and we present also various applications.
Our point of view is from the local index theory, especially from the analytic localization techniques developed by Bismut-Lebeau.
Basically, the holomorphic Morse inequalities are a consequence of the small time asymptotic expansion of the heat kernel. The Bergman kernel corresponds to the limit of the heat kernel when time goes to infinity, and is more sophisticate.
A simple principle in this book is that the existence of the spectral gap of the operators implies the existence of the asymptotic expansion of the corresponding Bergman kernel whether the manifold is compact or not, or singular, or with boundary.
Moreover, we present a general and algorithmic way to compute the coefficients in the expansion.
The first chapter can be used as material for an advanced course in geometry treating Dirac operators, Lichnerowicz formulas and heat kernels.
Before rushing to buy the book it is wise to have a look at the Contents and Introduction.
We are grateful for any comments or criticism. The list of inaccuracies can be found in the Errata.