A vanishing theorem for sheaves of small differential operators in positive characteristic

Details A vanishing theorem for sheaves of small differential operators in positive characteristic A vanishing theorem for sheaves of small differential operators in positive characteristic

2007-07-06

arxiv.org/abs/0707.0913v5

Transform. Groups, 15 (2010), 227-242

Mathematics

Algebraic Geometry

Revised version. A vanishing statement added allowing to remove previous restrictions on the characteristic in the case of qua

Scientific paper

Let $X$ be a smooth variety over an algebraically closed field $k$ of positive characteristic, ${\rm D}_X$ the sheaf of PD-differential operators, and ${\bar D}_X$ its central reduction, the sheaf of small differential operators. In this paper we show that if $X$ is a line-hyperplane incidence variety (a partial flag variety of type $(1,n,n+1)$) or a quadric of arbitrary dimension (in this case the characteristic is supposed to be odd) then ${\rm H}^{i}(X,{\bar D}_X)=0$ for $i>0$. Using this vanishing result and the derived localization theorem for crystalline differential operators (\cite{BMR}) we show that the Frobenius pushforward of the structure sheaf is a tilting bundle on these varieties, provided that $p>h$, the Coxeter number of the corresponding group.

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