On the Convergence Rate of Bergman Metrics

Abstract

We study the convergence rate of Bergman metrics on the class of polarized pointed Kähler n-manifolds (M, L, g, x) with $\textrm{Vol}\left( B_1 (x) \right) >v $ and $|\!\sec \!|\le K $ on M. Relying on Tian’s peak section method (Tian in J Differ Geom 32(1):99–130, 1990), we show that the $C^{1,\alpha }$ convergence of Bergman metrics is uniform. In the end, we discuss the sharpness of our estimates.