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摘要翻译：
设$x$是正特征代数闭域$k$上的光滑簇，${\rm D}_x$是pd-微分算子的簇，${\bar D}_x$是它的中心约简，是小微分算子的簇。本文证明了如果$X$是一个线-超平面关联簇($(1，n，n+1)$类型的部分标志簇）或任意维数的二次曲面（在这种情况下，特征为奇数），则${\rm H}^{i}(X,{\bar D}_X)=0$,对于$i>0$。利用这个消失的结果和导出的晶体微分算子局部化定理(\cite{BMR}）,我们证明了结构束的Frobenius推进是这些变体上的一个倾斜丛，条件是相应群的Coxeter数$P>H$。
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英文标题：
《A vanishing theorem for sheaves of small differential operators in
positive characteristic》
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作者：
Alexander Samokhin
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最新提交年份：
2010
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics 数学
二级分类：Representation Theory 表象理论
分类描述：Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra
代数和群的线性表示，李理论，结合代数，多重线性代数
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英文摘要：
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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PDF链接：
https://arxiv.org/pdf/0707.0913
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