# $L^2$ estimates and vanishing theorems for holomorphic vector bundles equipped with singular Hermitian metrics

## TL;DR

This paper develops $L^2$ estimates and vanishing theorems for singular Hermitian metrics on vector bundles, focusing on Griffiths positivity, which broadens the scope beyond Nakano positivity in singular contexts.

## Contribution

It introduces new $L^2$ estimates and vanishing theorems for singular Hermitian metrics assuming Griffiths positivity, bypassing the need for Nakano positivity in singular settings.

## Key findings

- Established $L^2$ estimates for Griffiths positive singular Hermitian vector bundles.
- Proved vanishing theorems under Griffiths positivity assumptions.
- Extended classical results to singular Hermitian metrics on projective manifolds.

## Abstract

We investigate singular Hermitian metrics on vector bundles, especially strictly Griffiths positive ones. $L^2$ esitimates and vanishing theorems usually require an assumption that vector bundles are Nakano positive. However there is no general definition of the Nakano positivity in singular settings. In this paper, we show various $L^2$ estimates and vanishing theromes by assuming that the vector bundle is strictly Griffiths positive and the base manifold is projective.