Bergman kernel and stability of holomorphic vector bundles with sections

Abstract

In this thesis, we introduce a notion of asymptotic stability for a holomorphic vector bundle with a global holomorphic section on a projective manifold. We prove that the special metric on the bundle studied by Bradlow is the limit of a sequence of balanced metrics that are induced from the asymptotic stability. Conversely, assuming the convergence of a sequence of balanced metrics, we show that the sequence converge to a special metric in the sense of Bradlow. The proof uses the asymptotic expansion of the Bergman kernel for general holomorphic vector bundle and machineries about moment maps involving two group actions developed by Donaldson.