An Analytic Application of Geometric Invariant Theory

TL;DR

This paper applies Geometric Invariant Theory to construct local analytic models for moduli spaces of stable holomorphic vector bundles on compact Kähler manifolds, linking symplectic forms and determinant line bundles.

Contribution

It introduces analytic GIT-quotients as local models for classifying spaces of stable bundles, connecting invariant Weil-Petersson forms with determinant line bundles and Quillen metrics.

Findings

Existence of invariant Weil-Petersson forms on parameter spaces.

Construction of holomorphic line bundles with hermitian metrics on GIT spaces.

Identification of Weil-Petersson form as Chern form of determinant line bundle.

Abstract

Given a compact K\"ahler manifold, Geometric Invariant Theory is applied to construct analytic GIT-quotients that are local models for a classifying space of (poly)stable holomorphic vector bundles containing the coarse moduli space of stable bundles as an open subspace. For local models invariant generalized Weil-Petersson forms exist on the parameter spaces, which are restrictions of symplectic forms on smooth ambient spaces. If the underlying K\"ahler manifold is of Hodge type, then the Weil-Petersson form on the moduli space of stable vector bundles is known to be the Chern form of a certain determinant line bundle equipped with a Quillen metric. It gives rise to a holomorphic line bundle on the classifying GIT space together with a continuous hermitian metric.