  In this paper we will discuss local coordinates canonically corresponding to
a Kahler metric. We will also discuss and prove the $C^\infty$ 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.

Differential Geometry;Algebraic Geometry;Algebraic Geometry;Differential Geometry