On the Bergman kernel of polarized abelian varieties

TL;DR

This paper derives an explicit formula for the Bergman kernel of polarized abelian varieties and explores its implications for bundle isomorphisms, localization, and decay estimates.

Contribution

It provides a new explicit formula for the Bergman kernel and demonstrates its applications in bundle isomorphism criteria and kernel localization.

Findings

Explicit Bergman kernel formula for polarized abelian varieties

Criteria for bundle isomorphism based on Bergman kernels

Exponential decay estimates for the Bergman kernel off-diagonal

Abstract

We prove an explicit formula for the Bergman kernel of polarized abelian varieties. As applications, we show that if two positive line bundles represent the same first Chern class and have identical Bergman kernel functions for some tensor power, then the corresponding powers of the bundles are isomorphic. We also obtain localization results for the maxima and minima of the Bergman kernel function and establish exponential off-diagonal decay estimates.