2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/147622In 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.44 pages, LaTexDifferential GeometryAlgebraic Geometrytext