Segre forms of singular metrics on vector bundles and Lelong numbers

TL;DR

This paper studies singular Hermitian metrics on vector bundles, defining associated Segre and Chern forms, and proves their limits and properties of Lelong numbers, extending classical notions to singular settings.

Contribution

It introduces a framework for defining Segre and Chern forms for singular metrics, proving their limits and properties of Lelong numbers in singular contexts.

Findings

Segre forms are limits of smooth metrics' Segre forms.

Lelong numbers of these forms are integers or non-negative.

The framework extends classical characteristic classes to singular metrics.

Abstract

Let $E\to X$ be a holomorphic vector bundle. We consider a class of a singular Hermitian metrics on $E$ with analytic singularities that contains all Griffiths negative such metrics. One can define, given a smooth reference metric $h_0$, a current $s(E,h,h_0)$ called the associated Segre form, which defines the expected Bott-Chern class and coincides with the usual Segre form of $h$ where it is smooth. We prove that $s(E,h,h_0)$ is the limit of the Segre forms of a sequence of smooth metrics if the metric is smooth outside the degeneracy locus, and in general as a limit of Segre forms of metrics with empty degeneracy locus. One can also define an associated Chern form $c(E,h,h_0)$. We prove that the Lelong numbers of $s(E,h,h_0)$ and $c(E,h,h_0)$ are integers if the singularities are integral, and non-negative for $s(E,h,h_0)$.