On the Bergman Kernel of complex hyperbolic manifolds

TL;DR

This paper derives a formula for the Bergman kernel on complex hyperbolic manifolds, expressing it via geodesic loops, and investigates its extremal values and off-diagonal estimates.

Contribution

It introduces a new formula for the Bergman kernel involving geodesic loops and applies it to analyze its extremal values and off-diagonal behavior.

Findings

Bergman kernel expressed as a sum over geodesic loops

Maximum and minimum of the Bergman kernel function characterized

Off-diagonal Bergman kernel estimated

Abstract

We prove a formula for the Bergman kernel of polarized complex hyperbolic manifolds. The formula expresses the Bergman kernel as a sum over the geodesic loops in the manifold. As an application, we prove a result about the maximum and minimum of the Bergman kernel function. We also prove an estimate of the off-diagonal Bergman kernel.