Vanishing theorems for Dolbeault cohomology of log homogeneous varieties

TL;DR

This paper proves new vanishing theorems for Dolbeault cohomology on log homogeneous varieties, extending classical results and providing explicit conditions for cohomology groups to vanish.

Contribution

It introduces vanishing theorems for Dolbeault cohomology of log homogeneous varieties with explicit bounds, generalizing known results like Broer's theorem.

Findings

Vanishing of certain Dolbeault cohomology groups for nef line bundles.

Extension of Broer's vanishing theorem to log homogeneous varieties.

Explicit bounds depending on the geometry of (X,D,L).

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 $H^i\big(X, L^{-1} \otimes \Omega_X^j(\log D)\big) = 0$ 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.