In this paper we will discuss local coordinates canonically corresponding to a Kähler metric. We will also discuss the C ∞ convergence of Bergman metrics following Tian's result on C 2 convergence of Bergman metrics. At the end we present an interesting characterization of ample line bundle that could be useful in arithmetic geometry.

Abstract We study the Bergman metric of a finite ball quotient $\mathbb{B}^n/\Gamma $, where $n \geq 2$ and $\Gamma \subseteq{\operatorname{Aut}}({\mathbb{B}}^n)$ is finite, fixed point free, abelian group. prove that this Kähler–Einstein if only $ trivial, is, when unit ${\mathbb{B}}^n$ itself. As consequence, we characterize among normal Stein spaces with isolated singularities fundamental gr...

The analysis of holomorphic sections of high powers L of holomorphic ample line bundles L → M over compact Kähler manifolds has been widely applied in complex geometry and mathematical physics. Any polarized Kähler metric g with respect to the ample line bundle L corresponds to the Ricci curvature of a hermitian metric h on L. Any orthonormal basis {SN 0 , ..., S dN} of H(M,L ) induces a holomo...

We introduce a family of weighted BMO spaces in the Bergman metric on the unit ball of C and use them to characterize complex functions f such that the big Hankel operators Hf and Hf̄ are both bounded or compact from a weighted Bergman space into a weighted Lesbegue space with possibly different exponents and different weights. As a consequence, when the symbol function f is holomorphic, we char...

In the paper Gaussian curvature of Bergman metric on the unit disc and the dependence of this curvature on the weight function has been studied.