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

Abstract

Let $X$ be a smooth variety over an algebraically closed field $k$ of positive characteristic, ${\mathrm{D}}_X$ 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 ${\mathrm{H}}^i(X,{\bar{D}}_X)=0$ 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.

Revised version. A vanishing statement added allowing to remove previous restrictions on the characteristic in the case of quadrics. 15 pages

References in corpus (2)

• Localization of modules for a semisimple Lie algebra in prime characteristic
• Frobenius splitting of cotangent bundles of flag varieties and geometry of nilpotent cones

Cited by in corpus (3)

• Some observations on Karoubian complete strongly exceptional posets on the projective homogeneous varieties
• Tilting bundles via the Frobenius morphism
• On the D-affinity of flag varieties in positive characteristic