Nakano positivity of singular Hermitian metrics and vanishing theorems of Demailly-Nadel-Nakano type

TL;DR

This paper introduces a new definition of Nakano semi-positivity for singular Hermitian metrics on vector bundles, establishing $L^2$-estimates and vanishing theorems that generalize existing results, with applications to constructing semi-positive metrics.

Contribution

It proposes a novel Nakano semi-positivity concept for singular metrics and proves generalized vanishing theorems, extending classical results in complex geometry.

Findings

Established $L^2$-estimates for Nakano positive singular metrics

Proved generalized Nakano and Demailly-Nadel vanishing theorems

Constructed globally Nakano semi-positive singular Hermitian metrics

Abstract

In this article, we propose a definition of Nakano semi-positivity of singular Hermitian metrics on holomorphic vector bundles. By using this positivity notion, we establish $L^2$-estimates for holomorphic vector bundles with Nakano positive singular Hermitian metrics. We also show vanishing theorems, which generalize both Nakano type and Demailly-Nadel type vanishing theorems. As applications, we specifically construct globally Nakano semi-positive singular Hermitian metrics for several bundles, and prove vanishing theorems associated with them.