Journal of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6834(e) ISSN 0894-0347(p)

Global -regularity of Schubert varieties with applications to $\mathcal{D}$ -modules

Authors: Niels Lauritzen, Ulf Raben-Pedersen and Jesper Funch Thomsen
Journal: J. Amer. Math. Soc. 19 (2006), 345-355
MSC (2000): Primary 32C38, 14B15
Posted: December 2, 2005
MathSciNet review: 2188129
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that Schubert varieties are globally -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.

References

1. A. Beilinson and J. Bernstein, Localisation de ${\mathfrak{g}}$ -modules, C. R. Acad. Sci. Paris 292(1981), 15-18. MR 0610137 (82k:14015)
2. Manuel Blickle, The intersection homology -module in finite characteristic, Math. Ann. 328 (2004), 425-450. MR 2036330 (2005a:14005)
3. R. Bögvad, Some results on -modules on Borel varieties, J. Algebra 173 (1995), 638-667. MR 1327873 (97a:14015)
4. -, An analogue of holonomic -modules on smooth varieties in positive characteristics, Homology Homotopy Appl. 4 (2002), 83-116. MR 1918185 (2003h:14030)
5. J. L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), 387-410. MR 0632980 (83e:22020)
6. A. Grothendieck, EGA IV, Étude locale des schemas et des morphismes de schemas, Inst. Hautes Études Sci. Publ. Math. 32 (1967).MR 0238860 (39:220)
7. -, Local Cohomology, Springer-Verlag, 1966. MR 0224620 (37:219)
8. M. Emerton and M. Kisin, A Riemann-Hilbert correspondence for unit -crystals, Astérisque 293 (2004). MR 2071510 (2005e:14027)
9. M. Hochster and C. Huneke, Tight closure and strong -regularity, Mémoires de la Société Mathématique de France 38 (1989), 119-133. MR 1044348 (91i:13025)
10. M. Kashiwara, Representation theory and -modules on flag varieties, Astérisque 173-174 (1989), 55-109. MR 1021510 (90k:17029)
11. M. Kashiwara and N. Lauritzen, Local cohomology and $\mathcal{D}$ -affinity in positive characteristic, C. R. Acad. Sci. Paris 335 (2002), 993-996. MR 1955575 (2004a:14020)
12. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 0560412 (81j:20066)
13. V. Lakshmibai and P. Magyar, Degeneracy schemes and Schubert varieties, Int. Math. Res. Notices 12 (1998), 627-640. MR 1635873 (99g:14065)
14. N. Lauritzen and J. F. Thomsen, Line bundles on Bott-Samelson varieties, J. Alg. Geom. 13 (2004), 461-473. MR 2047677 (2005b:14087)
15. V. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. Math. 122 (1985), 27-40.MR 0799251 (86k:14038)
16. A. Ramanathan, Equations defining Schubert varieties and Frobenius splitting of diagonals, Pub. Math. IHES 65 (1987), 61-90.MR 0908216 (88k:14032)
17. C. Peskine, L. Szpiro, Dimension projective finie et cohomologie locale, Pub. Math. IHES 42 (1973), 47-119. MR 0374130 (51:10330)
18. K. E. Smith, Globally -regular varieties: Applications to vanishing theorems for quotients of Fano varieties, Mich. J. Math. 48 (2000), 553-572. MR 1786505 (2001k:13007)
19. T. Springer, Linear Algebraic Groups, Springer Verlag (2nd edition), 1998.MR 1642713 (99h:20075)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|