Bergman kernel and complex singularity exponent

Abstract

We give a precise estimate of the Bergman kernel for the model domain defined by ${\Omega }_F=\{(z,w)\in {\mathbb{C}}^{n+1}:\mathrm{Im}w-|F(z){|}^2>0\},$ where $F=(f_1,...,f_m)$ is a holomorphic map from ${\mathbb{C}}^n$ to ${\mathbb{C}}^m$, in terms of the complex singularity exponent of $F$.

to appear in Science in China, a special issue dedicated to Professor Zhong Tongde's 80th birthday