On the convergence rate of Bergman metrics

TL;DR

This paper investigates the rate at which Bergman metrics converge on certain polarized Kähler manifolds, establishing uniform $C^{1,eta}$ convergence and discussing the sharpness of these estimates.

Contribution

The paper provides a uniform convergence rate for Bergman metrics on polarized Kähler manifolds using Tian's peak section method, with insights into the optimality of these estimates.

Findings

Established uniform $C^{1,eta}$ convergence of Bergman metrics.

Provided bounds on convergence rates depending on geometric parameters.

Discussed the sharpness and potential improvements of the estimates.

Abstract

We study the convergence rate of Bergman metrics on the class of polarized pointed K\"ahler $n$-manifolds $(M,L,g,x)$ with $\mathrm{Vol}\left( B_1 (x) \right) >v $ and $|\sec |\leq K $ on $M$. Relying on Tian's peak section method [Tian, 1990], we show that the $C^{1,\alpha }$ convergence of Bergman metrics is uniform. At the end we discuss the sharpness of our estimates.