Logarithmic vanishing theorems for effective $q$-ample divisors

Abstract

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