Skip to main content
SpringerLink
Log in

Moduli stacks of vector bundles and Frobenius morphisms

  • Chapter
Algebra and Number Theory

Abstract

We describe the action of the different Frobenius morphisms on the cohomology ring of the moduli stack of algebraic vector bundles of fixed rank and determinant on an algebraic curve over a finite field in characteristic p and analyse special situations like vector bundles on the projective line and relations with infinite Grassmannians.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 66.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Artin, A. Grothendieck, J. L. Verdier: Théorie des topos et cohomologie étale des schémas, SGA 4. Lect. Notes Math. vol. 269, 270, 305, Berlin-Heidelberg-New York, Springer 1972, 1973.

    MATH  Google Scholar 

  2. M. F. Atiyah, R. Bott: The Yang-Mills equation over Riemann surfaces, Philos. Trans. Roy. Soc. London, 308A (1982), 523–615.

    Article  MathSciNet  MATH  Google Scholar 

  3. K. A. Behrend: The Lefschetz trace formula for algebraic stacks, Inv. Math. 112 (1993), 127–149.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Beauville, Y. Laszlo: La lemme de descente, Comptes Rendus Acad. Sci. Paris 320 Serie I (1995), 335–340.

    MATH  Google Scholar 

  5. E. Bifet, F. Ghione, M. Latrizia: On the Abel-Jacobi map for divisors of higher rank on a curve, Math. Ann. 299 (1994), 641–672.

    Article  MathSciNet  MATH  Google Scholar 

  6. P. Deligne: Cohomologie étale, SGA4 1/2. Lect. Notes Math., vol. 569, Berlin-Heidelberg-New York, Springer, 1977.

    Book  MATH  Google Scholar 

  7. V. Drinfeld, C. Simpson: B-structures on G-bundles and local triviality, Math. Res. Lett. 2 (1995), 823–829.

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Faltings: Lectures on vector bundles on curves, Notes by M. Stoll, Mathematisches Institut der Universität Bonn 1995.

    Google Scholar 

  9. G. Faltings: Algebraic loop groups and moduli spaces of bundles, J. Eur. Math. Soc. (JEMS) 5, no. 1 (2003), 41–68.

    Article  MathSciNet  MATH  Google Scholar 

  10. D. Gaitsgory: Construction of central elements in the affine Hecke algebras via nearby cycles, Inv. Math. 144 (2001), 253–280.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Harder, M. S. Narasimhan: On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1973), 214–248.

    MathSciNet  Google Scholar 

  12. J. Heinloth: Über den Modulstack der Vektorbündel auf Kurven, Diplomarbeit, Mathematisches Institut der Universität Bonn, Juni 1998.

    Google Scholar 

  13. K. Joshi, S. Ramanan, E. Xia, J.-K. Yu: On vector bundles destabilized by Frobenius pullback, Preprint, August 2002, arXiv:math.AG/020809.

    MATH  Google Scholar 

  14. S. Kumar, M.S. Narasimhan, A. Ramanathan: Infinite Grassmannians and moduli spaces of G-bundles, Math. Ann. 300 (1993), 395–423.

    MathSciNet  MATH  Google Scholar 

  15. H. Lange, C. Pauly: On Frobenius-destabilized rank-2 vector bundles over curves, Preprint, September 2003, arXiv:math.AG/0309456.

    MATH  Google Scholar 

  16. Y. Laszlo, C. Pauly: On the Hitchin morphism in positive characteristic, Internat. Math. Res. Notices, 403 (2001), 129–143.

    Article  MathSciNet  MATH  Google Scholar 

  17. Y. Laszlo, C. Pauly: The action of the Frobenius map on rank-2 vector bundles in characteristic 2, J. Algebraic Geom. 11, no. 2 (2002), 219–243.

    Article  MathSciNet  MATH  Google Scholar 

  18. G. Laumon, L. Moret-Bailly: Champs algébriques, Ergebnisse der Mathematik, 3. Folge, vol. 39, Berlin-Heidelberg-New York, Springer 2000.

    MATH  Google Scholar 

  19. D. Mumford: Abelian varieties Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Oxford University Press 1970.

    MATH  Google Scholar 

  20. M. S. Ragunathan: On spaces of morphisms of curves in algebraic homogeneous spaces, Preprint, Tata Institute of Fund. Research Mumbai, 2001, 1–23.

    Google Scholar 

  21. Séminaire de Géométrie algébrique du Bois-Marie 1965–66, SGA5. dirigé par A. Grothendieck. Cohomologie Z-adique et fonctions L. Lect. Notes Math, vol 589, Berlin-Heidelberg-New York, Springer-Verlag 1977.

    Google Scholar 

  22. C. Teleman: Borel-Weil-Bott theory on the moduli stack of G-bundles over a curve, Inv. Math. 134 (1998), 1–57.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, England

    Frank Neumann

  2. Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstr. 3-5, D-37073, Göttingen, Germany

    Ulrich Stuhler

Authors
  1. Frank Neumann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ulrich Stuhler
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, University of Hyderabad P.O. Central University, Hyderabad, 500 046, India

    Rajat Tandon

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Hindustan Book Agency

About this chapter

Cite this chapter

Neumann, F., Stuhler, U. (2005). Moduli stacks of vector bundles and Frobenius morphisms. In: Tandon, R. (eds) Algebra and Number Theory. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-23-1_9

Download citation

Publish with us

Policies and ethics

Search

Navigation