Skip to main content
SpringerLink
Log in

Stability of rank 2 vector bundles along isomonodromic deformations

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We are interested in the stability of holomorphic rank 2 vector bundles of degree 0 over compact Riemann surfaces, which are provided with irreducible meromophic tracefree connections. In the case of a logarithmic connection on the Riemann sphere, such a vector bundle will be trivial up to the isomonodromic deformation associated to a small move of the poles, according to a result of A. Bolibruch. In the general case of meromorphic connections over Riemann surfaces of arbitrary genus, we prove that the vector bundle will be semi-stable, up to a small isomonodromic deformation. More precisely, the vector bundle underlying the universal isomonodromic deformation is generically semi-stable along the deformation, and even maximally stable. For curves of genus g ≥ 2, this result is non-trivial even in the case of non-singular connections.

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. Beilinson, A., Drinfeld, V.: Opers. arxiv : math/0501398v1(2005)

  2. Bolibruch A.A.: The Riemann-Hilbert problem. Russian Math. Surv. 45, 1–58 (1990)

    Article  Google Scholar 

  3. Brunella, M.: Birational geometry of foliations. Publicações Matemáticas do IMPA (IMPA Mathematical Publications). Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro (2004)

  4. Esnault H., Hertling C.: Semistable bundles on curves and reducible representations of the fundamental group. Intern. J. Math. 12(7), 847–855 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Esnault, H., Viehweg, E.: Semistable bundles on curves and irreducible representations of the fundamental group. In: Algebraic Geometry: Hirzebruch 70 (Warsaw, 1998), Contemp. Math., vol. 241, pp. 129–138. Am. Math. Soc., Providence (1999)

  6. Friedman R.: Algebraic surfaces and holomorphic vector bundles. Universitext. Springer, New York (1998)

    Google Scholar 

  7. Hakim, M.: Topos annelés et schémas relatifs. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, Band, vol. 64. Springer, Berlin (1972)

  8. Heu, V.: Isomonodromic deformations and maximally stable bundles. hal : 00308586-v1 (2008)

  9. Ince E.L.: Ordinary Differential Equations. Dover Publications, New York (1944)

    MATH  Google Scholar 

  10. Iwasaki, K., Kimura, H., Shimomura, S., Yoshida, M.: From Gauss to Painlevé. Aspects of Mathematics, E16. Friedr. In: A Modern Theory of Special Functions. Vieweg & Sohn, Braunschweig (1991)

  11. Krichever I.: Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations. Mosc. Math. J. 2(4), 717–752, 806 (2002)

    MATH  MathSciNet  Google Scholar 

  12. Loray, F.: Okamoto symmetry of Painlevé VI equation and isomonodromic deformation of Lamé connections. In: Algebraic, Analytic and Geometric Aspects of Complex Differential Equations and Their Deformations. Painlevé hierarchies, RIMS Kôkyûroku Bessatsu, B2, pp. 129–136. Res. Inst. Math. Sci. (RIMS), Kyoto (2007)

  13. Loray F., Pereira J.V.: Transversely projective foliations on surfaces: existence of minimal form and prescription of monodromy. Intern. J. Math. 18(6), 723–747 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Machu, F.X.: Monodromy of a class of logarithmic connections on an elliptic curve. SIGMA Symmetry Integr. Geom. Methods Appl. 3, Paper 082, 31 (2007)

    Google Scholar 

  15. Malgrange, B.: Sur les déformations isomonodromiques. I. Singularités régulières. In: Mathematics and Physics (Paris, 1979, 1982), Prog. Math., vol. 37, pp. 401–426. Birkhäuser Boston, Boston (1983)

  16. Malgrange, B.: Sur les déformations isomonodromiques. II. Singularités irrégulières. In: Mathematics and Physics (Paris, 1979, 1982), Progr. Math., vol. 37, pp. 427–438. Birkhäuser Boston, Boston (1983)

  17. Maruyama, M.: On classification of ruled surfaces. In: Lectures in Mathematics, Department of Mathematics, Kyoto University, vol. 3. Kinokuniya Book-Store Co. Ltd., Tokyo (1970)

  18. Nagata M.: On self-intersection number of a section on a ruled surface. Nagoya Math. J. 37, 191–196 (1970)

    MATH  MathSciNet  Google Scholar 

  19. Palmer J.: Zeros of the Jimbo, Miwa, Ueno tau function. J. Math. Phys. 40(12), 6638–6681 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  20. Weil A.: Généralisation des fonctions abéliennes. J. Math. Pures Appl. 40(IX), 47–87 (1938)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Insitut de Recherche Mathématique de Rennes, Rennes, France

    Viktoria Heu

Authors
  1. Viktoria Heu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Viktoria Heu.

Additional information

The author was partially supported by ANR SYMPLEXE BLAN06-3-137237.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Heu, V. Stability of rank 2 vector bundles along isomonodromic deformations. Math. Ann. 344, 463–490 (2009). https://doi.org/10.1007/s00208-008-0316-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-008-0316-2

Mathematics Subject Classification (2000)

Search

Navigation