Vanishing theorems for Dolbeault cohomology of log homogeneous varieties

Details Vanishing theorems for Dolbeault cohomology of log homogeneous varieties Vanishing theorems for Dolbeault cohomology of log homogeneous varieties

2008-12-14

arxiv.org/abs/0812.2658v1

Mathematics

Algebraic Geometry

Scientific paper

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.

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