Chern forms of singular metrics on vector bundles

TL;DR

This paper develops a method to define Chern forms as closed currents for singular hermitian metrics on vector bundles, under certain codimension restrictions, extending the understanding of curvature in singular settings.

Contribution

It introduces a way to define Chern forms as closed currents for singular metrics, overcoming previous limitations on curvature as a measure.

Findings

Chern forms can be defined as closed currents with locally finite mass

The approach applies under specific codimension conditions on the singular set

This extends the theory of characteristic classes to singular hermitian metrics

Abstract

We study singular hermitian metrics on holomorphic vector bundles, following Berndtsson-P{\u{a}}un. Previous work by Raufi has shown that for such metrics, it is in general not possible to define the curvature as a current with measure coefficients. In this paper we show that despite this, under appropriate codimension restrictions on the singular set of the metric, it is still possible to define Chern forms as closed currents of order 0 with locally finite mass, which represent the Chern classes of the vector bundle.