Logarithmic vanishing theorems on compact K\"{a}hler manifolds I

TL;DR

This paper proves new vanishing theorems for logarithmic differential forms on compact Kähler manifolds, extending classical results using $L^2$-techniques and Hermitian metrics.

Contribution

It establishes an $L^2$-type Dolbeault isomorphism for logarithmic forms and derives several generalized vanishing theorems.

Findings

New vanishing theorems for logarithmic differential forms.

Generalization of classical vanishing theorems.

Application of $L^2$-estimates to complex geometry.

Abstract

In this paper, we first establish an $L^2$-type Dolbeault isomorphism for logarithmic differential forms by H\"{o}rmander's $L^2$-estimates. By using this isomorphism and the construction of smooth Hermitian metrics, we obtain a number of new vanishing theorems for sheaves of logarithmic differential forms on compact K\"ahler manifolds with simple normal crossing divisors, which generalize several classical vanishing theorems, including Norimatsu's vanishing theorem, Gibrau's vanishing theorem, Le Potier's vanishing theorem and a version of the Kawamata-Viehweg vanishing theorem.