$L^2$-Dolbeault resolutions and Nadel vanishing on weakly pseudoconvex complex spaces with singular Hermitian metrics

TL;DR

This paper develops an advanced $L^2$-theory for the $ar{ abla}$-operator on complex spaces with singular Hermitian metrics, leading to generalized vanishing theorems.

Contribution

It introduces $L^2$-Dolbeault resolutions and estimates on complex spaces with singular metrics, extending classical results to more general settings.

Findings

Established $L^2$-Dolbeault resolutions for singular Hermitian metrics

Derived generalized Nadel vanishing theorems

Provided new $L^2$-estimates for the $ar{ abla}$-operator

Abstract

In this paper, in order to develop a more general $L^2$-theory for the $\overline{\partial}$-operator on complex spaces, we provide $L^2$-Dolbeault fine resolutions and isomorphisms, and $L^2$-estimates, for holomorphic line bundles on complex spaces equipped with singular Hermitian metrics. As applications, we obtain several generalizations of the Nadel vanishing theorem.