Parameter dependence of the Bergman kernels

TL;DR

This paper investigates how the Bergman kernel varies continuously and Hölder continuously with respect to a parameter in families of pseudoconvex domains, with applications to singularity theory and a new proof of the openness theorem.

Contribution

It provides new insights into the parameter dependence of Bergman kernels and introduces applications to the singularity theory of plurisubharmonic functions, including a novel proof of the openness theorem.

Findings

Established continuity and Hölder continuity of Bergman kernels in parameter families.

Applied results to singularity theory of psh functions.

Provided a new proof of the openness theorem.

Abstract

Let $\{\Omega_t:-1<t<1\}$ be a family of bounded pseudoconvex domains and $\varphi_t\in PSH(\Omega_t)$. Let $K_t(z,w)$ denote the Bergman kernel with weight $\varphi_t$ on $\Omega_t$. We study the continuity and H\"older continuity of $K_t(z,w)$ in $t$. Several applications to singularity theory of psh functions are given, including a new proof of the openness theorem.