Skip to main content
SpringerLink
Log in

A vanishing theorem for differential operators in positive characteristic

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

Let X be a smooth variety over an algebraically closed field k of characteristic p, and let F: X → X be the Frobenius morphism. We prove that if X is an incidence variety (a partial flag variety in type A n ) or a smooth quadric (in this case p is supposed to be odd) then \( {H^i}\left( {X,\mathcal{E}nd\left( {{\sf{F}_*}{\mathcal{O}_X}} \right)} \right) = 0 \) for i > 0. Using this vanishing result and the derived localization theorem for crystalline differential operators [3], we show that the Frobenius direct image \( {\sf{F}_*}{\mathcal{O}_X} \) is a tilting bundle on these varieties provided that p > h, the Coxeter number of the corresponding group.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. H. Andersen, M. Kaneda, On the D-affinity of the flag variety in type B2, Manuscripta Math. 103 (2000), 393–399.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Beilinson, J. Bernstein, Localisation de \( \mathfrak{g} \) -modules, C. R. Acad. Sci. Paris Ser. I 292 (1981), 15–18.

    MATH  MathSciNet  Google Scholar 

  3. R. Bezrukavnikov, I. Mirković, D. Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic, Ann. of Math. (2) 167 (2008), no. 3, 945–991.

    Article  MATH  MathSciNet  Google Scholar 

  4. R. Bezrukavnikov, I. Mirković, D. Rumynin, Singular localization and intertwining functors for reductive Lie algebras in prime characteristic, Nagoya Math. J. 184 (2006), 1–55.

    MATH  MathSciNet  Google Scholar 

  5. W. Borho, H. Kraft, Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen, Comment. Math. Helv. 54 (1979), no. 1, 61–104.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Brion, S. Kumar, Frobenius Splitting Methods in Geometry and Representation Theory, Progress in Mathematics, Vol. 231, Birkhäuser Boston, Boston, MA, 2005.

    MATH  Google Scholar 

  7. S. Donkin, The normality of closures of conjugacy classes of matrices, Invent. Math. 101 (1990), no. 3, 717–736.

    Article  MATH  MathSciNet  Google Scholar 

  8. A. Grothendieck, EGA IV, Etude locale des schémas et des morphismes de schémas IV, Inst. Hautes Etudes Sci. Publ. Math., Vol. 32, 1967, 361 pp.

  9. B. Haastert, Über Differentialoperatoren und \( \mathbb{D} \) -Moduln in positiver Charakterik, Manuscripta Math. 58 (1987), no. 4, 385–415.

    Article  MATH  MathSciNet  Google Scholar 

  10. W. J. Haboush, A short proof of the Kempf vanishing theorem, Invent. Math. 56 (1980), no. 2, 109–112.

    Article  MATH  MathSciNet  Google Scholar 

  11. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, Berlin, 1977. Russian transl.: Р. Хартсхорн, Алгебраическая геометрия, Мир, М., 1981.

    MATH  Google Scholar 

  12. Y. Hashimoto, M. Kaneda, D. Rumynin, On localization of \( \overline D \)-modules, in: Representations of Algebraic Groups, Quantum Groups, and Lie Algebras, Contemporary Mathematics, Vol. 413, Amer. Math. Soc., Providence, RI, 2006, pp. 43–62.

    Google Scholar 

  13. J. C. Jantzen, Representations of Algebraic Groups, 2nd ed., Amer. Math. Soc., Providence, RI, 2003.

    MATH  Google Scholar 

  14. D. Kaledin, Derived equivalences by quantization, Geom. Funct. Anal. 17 (2008), no. 6, 1968–2004.

    Article  MATH  MathSciNet  Google Scholar 

  15. N. Katz, Nilpotent connections and the monodromy theorem: applications of a result of Turritin, Publ. Math. Inst. Hautes Etud. Sci., 39 (1970), 175–232.

    Article  MATH  Google Scholar 

  16. M. Kaneda, J. Ye, Equivariant localization of \( \overline D \) -modules on the flag variety of the symplectic group of degree 4, J. Algebra 309 (2007), no. 1, 236–281.

    Article  MATH  MathSciNet  Google Scholar 

  17. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), 479–508.

    Article  MATH  MathSciNet  Google Scholar 

  18. M. Kashiwara, N. Lauritzen, Local cohomology and \( \mathcal{D} \) -affinity in positive characteristic, C. R. Acad. Sci. Paris Ser. I 335 (2002), 993–996.

    MATH  MathSciNet  Google Scholar 

  19. S. Kumar, N. Lauritzen, J. F. Thomsen, Frobenius splitting of cotangent bundles of flag varieties, Invent. Math. 136 (1999), 603–621.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Langer, D-affinity and Frobenius morphism on quadrics, Int. Math. Res. Not. (2008), no. 1, Art. ID rnm 145, 26 pp.

  21. N. Lauritzen, V. Mehta, Differential operators and the Frobenius pull back of the tangent bundle on G/P, Manuscripta Math. 98 (1999), no. 4, 507–510.

    Article  MATH  MathSciNet  Google Scholar 

  22. V. Mehta, W. Van der Kallen, A simultaneous Frobenius splitting for closures of conjugacy classes of nilpotent matrices, Compositio Math. 84 (1992) no. 2, 211–221.

    MATH  MathSciNet  Google Scholar 

  23. A. Samokhin, On the D -affinity of quadrics in positive characteristic, C. R. Acad. Sci. Paris Ser. I 344 (2007), 377–382.

    MATH  MathSciNet  Google Scholar 

  24. A. Samokhin, A vanishing theorem in positive characteristic and tilting equivalences, IHES preprint, March 2007, available at www.ihes.fr.

  25. A. Samokhin, Tilting bundles via the Frobenius morphism, arXiv:0904.1235v2.

  26. A. Samokhin, On the D -affinity of flag varieties in positive characteristic, arXiv:0906.1555v4.

  27. A. Samokhin, The Frobenius morphism on homogeneous spaces, in preparation.

Download references

Author information

Authors and Affiliations

  1. Institute for Information, Transmission Problems, B. Karetnyǐ per., 19, 127994, GSP-4, Moscow, Russia

    A. Samokhin

Authors
  1. A. Samokhin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Samokhin.

Additional information

Supported in part by an RFBR grant 07-01-00051a.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Samokhin, A. A vanishing theorem for differential operators in positive characteristic. Transformation Groups 15, 227–242 (2010). https://doi.org/10.1007/s00031-010-9083-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-010-9083-8

Keywords

Search

Navigation