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 $\mathrm{Vol}\left(B_1(x)\right)>v$ and $|\sec |\le K$ 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.

arXiv comment31 pages, all comments are welcome. Peking Math J (2022)