An alternative approach on the singular metric on a vector bundle

TL;DR

This paper proposes a new way to define singular Hermitian metrics on vector bundles, explores their positivity properties, and extends Griffiths' vanishing theorem to this context.

Contribution

It introduces an alternative definition of singular Hermitian metrics and generalizes Griffiths' vanishing theorem for these metrics.

Findings

New definition of singular Hermitian metrics on vector bundles

Analysis of Griffiths and Nakano positivity under the new framework

Generalized Griffiths' vanishing theorem proved

Abstract

In this paper, we will provide an alternative definition for the singular Hermitian metric on a vector bundle. Moreover, we discuss the Griffiths and Nakano positivities under this circumstance and prove a generalised Griffiths' vanishing theorem.