$L^2$-type Dolbeault isomorphisms and vanishing theorems for logarithmic sheaves twisted by multiplier ideal sheaves

TL;DR

This paper develops an $L^2$-type Dolbeault isomorphism for logarithmic differential forms twisted by multiplier ideal sheaves, leading to new vanishing theorems on compact Kähler manifolds with normal crossing divisors.

Contribution

It introduces an $L^2$-type Dolbeault isomorphism for logarithmic sheaves twisted by multiplier ideals and derives associated vanishing theorems using $L^2$ estimates.

Findings

Established an $L^2$-type Dolbeault isomorphism for logarithmic sheaves.

Derived logarithmic vanishing theorems involving multiplier ideal sheaves.

Applied the results to compact Kähler manifolds with simple normal crossing divisors.

Abstract

In this article, we first establish an $L^2$-type Dolbeault isomorphism for the sheaf of logarithmic differential forms twisted by the multiplier ideal sheaf. By using this isomorphism and $L^2$-estimates equipped with a singular Hermitian metric, we obtain logarithmic vanishing theorems involving multiplier ideal sheaves on compact K\"ahler manifolds with simple normal crossing divisors.