Chern forms of hermitian metrics with analytic singularities on vector bundles

TL;DR

This paper introduces a method to define Chern and Segre currents for vector bundles with singular hermitian metrics, extending classical forms to singular settings using Monge-Ampère products.

Contribution

It constructs Chern and Segre currents for singular hermitian metrics with analytic singularities, generalizing classical differential forms to singular contexts.

Findings

Chern and Segre currents represent the bundle's characteristic classes.

Currents coincide with classical forms when metrics are smooth.

Method extends characteristic classes to singular hermitian metrics.

Abstract

We define Chern and Segre forms, or rather currents, associated with a Griffiths positive singular hermitian metric $h$ with analytic singularities on a holomorphic vector bundle $E$. The currents are constructed as pushforwards of generalized Monge-Amp\`ere products on the projectivization of $E$. The Chern and Segre currents represent the Chern and Segre classes of $E$, respectively, and coincide with the Chern and Segre forms of $E$ and $h$, where $h$ is smooth. Moreover, our currents coincide with the Chern and Segre forms constructed by the first three authors and Ruppenthal in the cases when these are defined.