Vanishing theorems for Dolbeault cohomology of log homogeneous varieties

Abstract

We consider a complete nonsingular variety $X$ over $\bC$, having a normal crossing divisor $D$ such that the associated logarithmic tangent bundle is generated by its global sections. We show that $H^i(X,L^{-1}\otimes {\Omega }_X^j(\log D))=0$ for any nef line bundle $L$ on $X$ and all $i<j-c$, where $c$ is an explicit function of $(X,D,L)$. This implies e.g. the vanishing of $H^i(X,L\otimes {\Omega }_X^j)$ for $L$ ample and $i>j$, and gives back a vanishing theorem of Broer when $X$ is a flag variety.