An Analytic Application of Geometric Invariant Theory II: Coarse Moduli Spaces

TL;DR

This paper demonstrates that a classifying space for polystable holomorphic vector bundles, constructed via analytic GIT theory, functions as a coarse moduli space in complex geometry when considering Hermite-Einstein connections.

Contribution

It establishes the coarse moduli space property of the classifying space in the weakly normal category under the topology induced by Hermite-Einstein connections.

Findings

The classifying space is a coarse moduli space in the specified setting.

The construction applies to polystable holomorphic vector bundles on compact Kähler manifolds.

The work connects analytic GIT theory with complex geometric moduli spaces.

Abstract

In [arXiv:2008.04625] the authors constructed a classifying space for polystable holomorphic vector bundles on a compact K\"ahler manifold using analytic GIT theory. The aim of this article is to show that this classifying space taken in the weakly normal category is a coarse moduli space in the sense of complex geometry when the topology is fixed as induced by the space of Hermite-Einstein connections modulo the group of unitary gauge transformations.