Positivity of Singular Hermitian Metrics for Holomorphic Vector Bundles

TL;DR

This paper develops new notions of positivity for singular Hermitian metrics on holomorphic vector bundles, enabling advanced vanishing theorems and a sharp Ohsawa-Takegoshi extension theorem using Berndtsson and Lempert's methods.

Contribution

It introduces Nakano and Demailly positivity concepts for singular metrics and proves a Berndtsson-type positivity theorem, leading to a sharp L^2 extension result.

Findings

Established a notion of positivity supporting vanishing theorems.

Proved a sharp Ohsawa-Takegoshi type L^2 extension theorem.

Developed a Berndtsson-type positivity theorem for vector bundles.

Abstract

We introduce a notion of Nakano and Demailly positivity for singular Hermitian metrics of holomorphic vector bundles. Our definitions support the usual H\"ormander and Nadel type vanishing theorems with estimates, at least on essentially Stein manifolds. As an application, we establish a sharp Ohsawa-Takegoshi type $L^2$ extension theorem. We use the method of Berndtsson and Lempert to prove the latter theorem, and for this purpose we require a Berndtsson-type positivity theorem for holomorphic vector bundles, which we also prove.