Logarithmic vanishing theorems for effective $q$-ample divisors

TL;DR

This paper proves new logarithmic vanishing theorems for the cohomology of log differential forms on compact Kähler manifolds, extending classical results to effective q-ample divisors.

Contribution

It establishes vanishing results for cohomology groups involving log differential forms when the divisor is the support of an effective q-ample divisor, generalizing previous theorems.

Findings

Vanishing of H^q(X,Ω^p_X(log D)) for p+q > n+k.

Extension of classical vanishing theorems to q-ample divisors.

Applicability to compact Kähler manifolds with simple normal crossing divisors.

Abstract

Let $X$ be a compact K\"ahler manifold and $D$ be a simple normal crossing divisor. If $D$ is the support of some effective $k$-ample divisor, we show $$ H^q(X,\Omega^p_X(\log D))=0,\quad \text{for}\quad p+q>n+k.$$