Bergman kernel and complex singularity exponent

TL;DR

This paper provides a detailed estimate of the Bergman kernel for a class of model domains in complex analysis, linking it explicitly to the complex singularity exponent of a holomorphic map.

Contribution

It introduces a precise relationship between the Bergman kernel estimates and the complex singularity exponent for a specific class of domains defined by holomorphic maps.

Findings

Derived explicit estimates of the Bergman kernel for the model domain

Connected the Bergman kernel behavior to the complex singularity exponent of the defining map

Enhanced understanding of the geometric analysis of complex domains

Abstract

We give a precise estimate of the Bergman kernel for the model domain defined by $\Omega_F=\{(z,w)\in \mathbb{C}^{n+1}:{\rm Im}w-|F(z)|^2>0\},$ 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 $F$.