Geodesic
Global 𝐹-regularity of Schubert varieties with applications to 𝒟-modules
Lauritzen, Niels1 ; Raben-Pedersen, Ulf1 ; Thomsen, Jesper1
1Institut for matematiske fag, Aarhus Universitet, Ny Munkegade, DK-8000 Århus, C Denmark
Journal of the American Mathematical Society, Tome 19 (2006) no. 2, pp. 345-355
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We prove that Schubert varieties are globally $F$-regular in the sense of Karen Smith. We apply this result to the category of equivariant and holonomic ${\mathcal {D}}$-modules on flag varieties in positive characteristic. Here recent results of Blickle are shown to imply that the simple ${\mathcal {D}}$-modules coincide with local cohomology sheaves with support in Schubert varieties. Using a local Grothendieck-Cousin complex, we prove that the decomposition of local cohomology sheaves with support in Schubert cells is multiplicity free.
DOI : 10.1090/S0894-0347-05-00509-6

Lauritzen, Niels  1   ; Raben-Pedersen, Ulf  1   ; Thomsen, Jesper  1

1 Institut for matematiske fag, Aarhus Universitet, Ny Munkegade, DK-8000 Århus, C Denmark 
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Lauritzen, Niels; Raben-Pedersen, Ulf; Thomsen, Jesper. Global 𝐹-regularity of Schubert varieties with applications to 𝒟-modules. Journal of the American Mathematical Society, Tome 19 (2006) no. 2, pp. 345-355. doi: 10.1090/S0894-0347-05-00509-6

[1] Beĭlinson, Alexandre, Bernstein, Joseph Localisation de 𝑔-modules C. R. Acad. Sci. Paris Sér. I Math. 1981 15 18

[2] Blickle, Manuel The intersection homology 𝐷-module in finite characteristic Math. Ann. 2004 425 450

[3] Bøgvad, Rikard Some results on 𝒟-modules on Borel varieties in characteristic 𝓅>0 J. Algebra 1995 638 667

[4] Bögvad, Rikard An analogue of holonomic 𝒟-modules on smooth varieties in positive characteristics Homology Homotopy Appl. 2002 83 116

[5] Brylinski, J.-L., Kashiwara, M. Kazhdan-Lusztig conjecture and holonomic systems Invent. Math. 1981 387 410

[6] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV Inst. Hautes Études Sci. Publ. Math. 1967 361

[7] Hartshorne, Robin Local cohomology 1967

[8] Emerton, Matthew, Kisin, Mark The Riemann-Hilbert correspondence for unit 𝐹-crystals Astérisque 2004

[9] Hochster, Melvin, Huneke, Craig Tight closure and strong 𝐹-regularity Mém. Soc. Math. France (N.S.) 1989 119 133

[10] Kashiwara, Masaki Representation theory and 𝐷-modules on flag varieties Astérisque 1989

[11] Kashiwara, Masaki, Lauritzen, Niels Local cohomology and 𝒟-affinity in positive characteristic C. R. Math. Acad. Sci. Paris 2002 993 996

[12] Kazhdan, David, Lusztig, George Representations of Coxeter groups and Hecke algebras Invent. Math. 1979 165 184

[13] Lakshmibai, V., Magyar, Peter Degeneracy schemes, quiver schemes, and Schubert varieties Internat. Math. Res. Notices 1998 627 640

[14] Lauritzen, Niels, Thomsen, Jesper Funch Line bundles on Bott-Samelson varieties J. Algebraic Geom. 2004 461 473

[15] Mehta, V. B., Ramanathan, A. Frobenius splitting and cohomology vanishing for Schubert varieties Ann. of Math. (2) 1985 27 40

[16] Ramanathan, A. Equations defining Schubert varieties and Frobenius splitting of diagonals Inst. Hautes Études Sci. Publ. Math. 1987 61 90

[17] Peskine, C., Szpiro, L. Dimension projective finie et cohomologie locale. Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck Inst. Hautes Études Sci. Publ. Math. 1973 47 119

[18] Smith, Karen E. Globally F-regular varieties: applications to vanishing theorems for quotients of Fano varieties Michigan Math. J. 2000 553 572

[19] Springer, T. A. Linear algebraic groups 1998

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