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Global \( F\)-regularity of Schubert varieties with applications to \( \mathcal{D}\)-modules. (English) Zbl 1098.14038

In this nice article the authors prove that Schubert varieties are globally \(F\)-regular. As a consequence they obtain that local rings of Schubert varieties are strongly \(F\)-regular, and thereby \(F\)-rational, and that the local rings of varieties that can be identified with open subsets of Schubert varieties, like determinantal varieties, are strongly \(F\)-regular. The term globally \(F\)-regularity was introduced by Karen Smith, and means that the section ring \(S({\mathcal L}) =\bigoplus_{n\geq 0}H^0(X,{\mathcal L}^n)\) of an ample line bundle \(\mathcal L\) on a projective algebraic variety \(X\) over an algebraically closed field of positive characteristic, is strongly \(F\)-regular in the sense of Hochster and Huneke.
Let \(X\) denote a flag variety and \(Y\) a Schubert variety. As a consequence of recent results by M. Blickle [Math. Ann. 328, No. 3, 425–450 (2004; Zbl 1065.14006)] the authors prove that the simple objects in the category of equivariant and holonomic \({\mathcal D}_X\)-modules are precisely the local cohomology sheaves \({\mathcal H}_Y^c({\mathcal O}_X)\), where \(c\) is the codimension of \(Y\) in \(X\). Using a local Grothendieck-Cousin complex [M. Kashiwara and N. Lauritzen, C. R., Math., Acad. Sci. Paris 335, No. 12, 993–996 (2002; Zbl 1016.14009)] they prove that the decomposition of the local cohomology modules with support in Bruhat cells is multiplicity free.

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14B15 Local cohomology and algebraic geometry
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