Local algebraicity and localization of the Bergman kernel on Stein spaces with finite type boundaries

TL;DR

This paper investigates the relationship between the algebraic properties of the Bergman kernel and the boundary geometry of Stein spaces with finite type boundaries, providing new estimates and characterizations.

Contribution

It introduces a boundary type estimate based on the local algebraic degree of the Bergman kernel and characterizes certain Stein spaces as ball quotients.

Findings

Boundary type estimate in terms of algebraic degree

Characterization of two-dimensional ball quotients

Localization result for the Bergman kernel

Abstract

On a two dimensional Stein space with isolated, normal singularities, smooth finite type boundary, and locally algebraic Bergman kernel, we establish an estimate on the type of the boundary in terms of the local algebraic degree of the Bergman kernel. As an application, we characterize two dimensional ball quotients as the only Stein spaces with smooth finite type boundary and locally rational Bergman kernel. A key ingredient in the proof of the degree estimate is a new localization result for the Bergman kernel of a pseudoconvex, finite type domain in a complex manifold.