## 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

### Citations:

Zbl 1065.14006; Zbl 1016.14009
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### References:

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