Moduli Space of Vector Bundles

Abstract

In this chapter, we recall Mumford’s construction of a moduli space of semistable holomorphic vector bundles over Riemann surfaces by using geometric invariant theory, and the notion of the Harder-Narasimhan filtration as the main tool to understand unstable bundles left out of the moduli space. We also give the basics of the analytical construction of Dolbeault’s moduli space of differential operators ∂¯ and Narasimhan-Seshadri relation with the moduli of representations of the fundamental group. We finish by exposing the main facts about Hitchin’s moduli space of Higgs bundles.