On Junyan Cao's vanishing theorem for pseudoeffective line bundles

TL;DR

This paper presents a version of Junyan Cao's vanishing theorem for pseudoeffective line bundles on compact Kähler manifolds, relating it to the numerical dimension of the line bundle rather than a specific metric.

Contribution

It offers a new formulation of Cao's vanishing theorem based on the numerical dimension of the line bundle, broadening its applicability.

Findings

Vanishing theorem extended to numerical dimension context

Applicable to any compact Kähler manifold

Improves understanding of pseudoeffective line bundles

Abstract

Junyan Cao has obtained a very general vanishing theorem, valid on any compact K\"ahler manifold, for the cohomology groups with values in a pseudoeffective line bundle twisted by the associated multiplier ideal sheaf. In this note, we give a version of this result in terms of the numerical dimension of the line bundle, instead of the numerical dimension of a given singular metric, which can be smaller.