Skip to main content
SpringerLink
Log in

Direct images of bundles under Frobenius morphism

  • Published:
Inventiones mathematicae Aims and scope

Abstract

Let X be a smooth projective variety of dimension n over an algebraically closed field k with char(k)=p>0 and F:X→X 1 be the relative Frobenius morphism. For any vector bundle W on X, we prove that instability of F * W is bounded by instability of W⊗Tℓ(Ω1 X ) (0≤ℓ≤n(p-1)) (Corollary 4.9). When X is a smooth projective curve of genus g≥2, it implies F * W being stable whenever W is stable.

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. Biswas, I., Holla, Y.: Comparison of fundamental group schemes of a projective variety and an ample hypersurface. arXiv:math.AG./0603299v1, 13 March (2006)

  2. Biswas, I., Holla, Y.: Comparison of fundamental group schemes of a projective variety and an ample hypersurface. J. Algebr. Geom. 16, 547–597 (2007)

    MATH  MathSciNet  Google Scholar 

  3. Doty, S., Walker, G.: Truncated symmetric powers and modular representations of GLn. Math. Proc. Camb. Philos. Soc. 119, 231–242 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. Joshi, K., Ramanan, S., Xia, E.Z., Yu, J.-K.: On vector bundles destabilized by Frobenius pull-back. Compos. Math. 142, 616–630 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Katz, N.: Nilpotent connection and the monodromy theorem: Application of a result of Turrittin. Publ. Math., Inst. Hautes Étud. Sci. 39, 175–232 (1970)

    Article  MATH  Google Scholar 

  6. Lange, H., Pauly, C.: On Frobenius-destabilized rank two vector bundles over curves. arXiv:math.AG./0309456 v2, 6 Oct. (2005)

  7. Mehta, V., Pauly, C.: Semistability of Frobenius direct images over curves. arXiv:math.AG./0607565 v1, 22 July, (2006)

  8. Sun, X.: Stability of direct images under Frobenius morphisms. arXiv:math.AG./0608043 v1, 2 Aug. (2006)

Download references

Author information

Authors and Affiliations

  1. Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing, P.R. China

    Xiaotao Sun

Authors
  1. Xiaotao Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaotao Sun.

Additional information

Dedicated to Professor Zhexian Wan on the occasion of his 80th birthday.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sun, X. Direct images of bundles under Frobenius morphism . Invent. math. 173, 427–447 (2008). https://doi.org/10.1007/s00222-008-0125-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-008-0125-y

Keywords

Search

Navigation