Non-pluripolar products on vector bundles and Chern--Weil formulae

TL;DR

This paper introduces pluripotential techniques for singular metrics on vector bundles, defines a new class of singularities, and derives a Chern--Weil formula relating Chern numbers to b-divisors.

Contribution

It develops non-pluripolar products on vector bundles, introduces $ ext{I}$-good singularities, and establishes a Chern--Weil formula using intersection theory on the Riemann--Zariski space.

Findings

Defined non-pluripolar products for vector bundles

Introduced $ ext{I}$-good singularities for Hermitian metrics

Derived a Chern--Weil formula involving b-divisors

Abstract

In this paper, we develop several pluripotential-theoretic techniques for singular metrics on vector bundles. We first introduce the theory of non-pluripolar products on holomorphic vector bundles on complex manifolds. Then we define and study a special class of singularities of Hermitian metrics on vector bundles, called $\mathcal{I}$-good singularities, partially extending Mumford's notion of good singularities. Next, we derive a Chern--Weil type formula expressing the Chern numbers of Hermitian vector bundles with $\mathcal{I}$-good singularities in terms of the associated b-divisors. We also define an intersection theory on the Riemann--Zariski space and apply it to reformulate our Chern--Weil formula.