Numerical dimension and a Kawamata-Viehweg-Nadel type vanishing theorem on compact K\"ahler manifolds

TL;DR

This paper introduces a new numerical dimension for pseudo-effective line bundles with singular metrics on compact Kähler manifolds and proves a general Kawamata-Viehweg-Nadel type vanishing theorem in this setting.

Contribution

It defines a novel numerical dimension for pseudo-effective line bundles with singular metrics and establishes a broad vanishing theorem on compact Kähler manifolds.

Findings

Defined a new numerical dimension for pseudo-effective line bundles

Proved a general Kawamata-Viehweg-Nadel type vanishing theorem

Extended vanishing results to arbitrary compact Kähler manifolds

Abstract

Let $X$ be a compact K\"ahler manifold and let $(L, \varphi)$ be a pseudo-effective line bundle on $X$. We first define a notion of numerical dimension of pseudo-effective line bundles with singular metrics, and then discuss the properties of this type numerical dimension. We finally prove a very general Kawamata-Viehweg-Nadel type vanishing theorem on an arbitrary compact K\"ahler manifold.