Variation of numerical dimension of singular hermitian line bundles

TL;DR

This paper explores the variation of numerical dimensions in singular hermitian line bundles and introduces new vanishing and injectivity theorems, advancing understanding in complex algebraic geometry.

Contribution

It provides a relative vanishing theorem based on numerical dimension variation and an analytic injectivity theorem for log canonical pairs, addressing conjectures in the field.

Findings

Established a relative Kawamata-Viehweg-Nadel type vanishing theorem.

Proved an analytic injectivity theorem for log canonical pairs on surfaces.

Connected numerical dimension variation to vanishing theorems.

Abstract

The purpose of this paper is to give two supplements for vanishing theorems: One is a relative version of the Kawamata-Viehweg-Nadel type vanishing theorem, which is obtained from an observation for the variation of the numerical dimension of singular hermitian line bundles. The other is an analytic injectivity theorem for log canonical pairs on surfaces, which can be seen as a partial answer for Fujino's conjecture.