The harmonic theory on a vector bundle with singular Hermitian metrics and positivity

TL;DR

This paper develops harmonic theory for holomorphic vector bundles with singular Hermitian metrics, extending key results and generalizing Nakano's vanishing theorem to broader contexts.

Contribution

It introduces harmonic theory for vector bundles with singular metrics and extends foundational theorems in complex geometry to this setting.

Findings

Extension of harmonic theory to singular Hermitian metrics

Generalization of Kollár and Takegoshi's results

Nakano's vanishing theorem applied to singular metrics

Abstract

Let $E$ be a holomorphic vector bundle endowed with a singular Hermitian metric $H$. In this paper, we develop the harmonic theory on $(E,H)$. Then we extend several canonical results of J. Koll\'{a}r and K. Takegoshi to this situation. In the end, we generalise Nakano's vanishing theorem.