The Weil-Petersson current for moduli of vector bundles and applications to orbifolds

TL;DR

This paper constructs a generalized Weil-Petersson current on the moduli space of stable vector bundles over orbifolds, extending classical forms and line bundles, with applications to compactifications.

Contribution

It introduces a Weil-Petersson current for orbifold moduli spaces and relates it to determinant line bundles with Quillen metrics, extending classical results.

Findings

The Weil-Petersson form extends as a semi-positive current for degenerating families.

The determinant line bundle admits an extension with a Quillen metric.

The Weil-Petersson form coincides with the curvature of the determinant line bundle.

Abstract

We investigate stable holomorphic vector bundles on a compact complex K\"ahler manifold and more generally on an orbifold that is equipped with a K\"ahler structure. We use the existence of Hermite-Einstein connections in this set-up and construct a generalized Weil-Petersson form on the moduli space of stable vector bundles with fixed determinant bundle. We show that the Weil-Petersson form extends as a (semi-)positive closed current for degenerating families that are restrictions of coherent sheaves. Such an extension will be called a Weil-Petersson current. When the orbifold is of Hodge type, there exists a determinant line bundle on the moduli space; this line bundle carries a Quillen metric, whose curvature coincides with the generalized Weil-Petersson form. As an application we show that the determinant line bundle extends to a suitable compactification of the moduli space.