On the convergence rate of Bergman metrics

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.