Covering stability of Bergman kernels on K\"ahler hyperbolic manifolds

TL;DR

This paper provides estimates for the Bergman kernel on Kähler hyperbolic manifolds using $L^2$ techniques and Bochner formulas, leading to criteria for the ampleness of the canonical bundle.

Contribution

It introduces new $L^2$ and Bochner formula-based estimates for Bergman kernels and generalizes criteria for the canonical bundle's very ampleness.

Findings

Derived effective Bergman kernel estimates for Kähler hyperbolic manifolds.

Established a generalized criterion for the very ampleness of the canonical line bundle.

Extended previous results to broader classes of Kähler hyperbolic manifolds.

Abstract

This paper is a sequel to \cite{Xu}. In this paper, an estimation of the Bergman Kernel of K\"ahler hyperbolic manifold is given by the $L^2$ estimate and the Bochner formula. As an application, an effective criterion of the very ampleness of the canonical line bundle of K\"ahler hyperbolic manifold is given, which is a generalization of Yeung's result.