Advertisement

Critical Loci for Higgs Bundles

  • Nigel HitchinEmail author
Open Access
Article

Abstract

The paper studies the locus in the moduli space of rank 2 Higgs bundles over a curve of genus g corresponding to points which are critical for d of the Poisson commuting functions defining the integrable system. These correspond to the Higgs field vanishing on a divisor D of degree d. The degree d critical locus is shown to have an induced integrable system related to K(−D)-twisted Higgs bundles. It is embedded in the singular part of the fibration and a description of these singular fibres using Hecke curves is given. The methods used are also applied to give information about the cohomology classes of components of the nilpotent cone and of the first critical locus. It is shown that whereas in the extreme case d = 2g − 2 the locus is a hyperkähler submanifold this does not hold in general. The example of genus 2 is studied concretely and the d = 1 integrable system is seen to be described by a pencil of Kummer surfaces.

Notes

Acknowledgements

The author wishes to thank A. Oliveira for useful conversations and the Engineering and Physical Sciences Research Council and the Instituto de Ciencias Matemáticas, Madrid for support during the preparation of this work.

References

  1. 1.
    Alexeev V.: Compactified Jacobians and the Torelli map. Publ. Res. Inst. Math. Sci. 40, 1241–1265 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beauville A., Narasimhan M.S., Ramanan S.: Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398, 169–179 (1989)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Birkenhake C., Lange H.: Complex Abelian Varieties, 2nd edition, Grundlehren der Mathematischen Wissenschaften, vol. 302. Springer, Berlin (2004)Google Scholar
  4. 4.
    Bolsinov, A.V., Oshemkov, A.A.: Singularities of integrable Hamiltonian systems. In: Bolsinov, A. V. et al. (eds.) Topological Methods in the Theory of Integrable Systems, pp. 1–67, Cambridge Scientific Publishers, Cambridge (2006)Google Scholar
  5. 5.
    Biswas I., Ramanan S.: An infinitesimal study of the moduli of Hitchin pairs. J. Lond. Math. Soc. 49, 219–231 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gawȩdzki K., Tran-Ngoc-Bich P.: Self-duality of the SL 2 Hitchin integrable system at genus 2. Commun. Math. Phys. 196, 641–670 (1998)ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Gothen P.B., Oliveira A.G.: The singular fiber of the Hitchin map. Int. Math. Res. Not. 2013, 1079–1121 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Griffiths P., Harris J.: Principles of Algebraic Geometry. Wiley, New York (1978)zbMATHGoogle Scholar
  9. 9.
    Gukov S.: Quantization via mirror symmetry. Jpn. J. Math. 6, 65–119 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hausel T., Thaddeus M.: Relations in the cohomology ring of the moduli space of rank 2 Higgs bundles. J. AMS 16, 303–327 (2003)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hitchin N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. 55, 59–126 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hitchin N.J.: Stable bundles and integrable systems. Duke Math. J. 54, 91–114 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hitchin, N.J.: Remarks on Nahm’s equations. In: Muñoz, V. et al. (eds.) Modern Geometry: A Celebration of the Work of Simon Donaldson, Proc of Symposia in Pure Mathematics, vol. 99, pp. 83–95. American Mathematical Society (2018)Google Scholar
  14. 14.
    Hwang J.-M.: Tangent vectors to Hecke curves on the moduli space of rank 2 bundles over an algebraic curve. Duke Math. J. 101, 179–187 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hwang J.-M., Ramanan S.: Hecke curves and Hitchin discriminant. Ann. Sci. École Norm. Sup. 37, 801–817 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ito H.: Action-angle coordinates at singularities for analytic integrable systems. Math. Z. 206, 363–407 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kapustin A., Witten E.: Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys. 1, 1–236 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Loray, F., Heu, V.: Hitchin Hamiltonians in genus 2. In: Aryasomayajula, A. et al. (eds.) Analytic and Algebraic Geometry, pp. 153–172. Springer Nature Singapore Pte Ltd. and Hindustan Book Agency (2017)Google Scholar
  19. 19.
    Narasimhan M.S., Ramanan S.: Moduli of vector bundles on a compact Riemann surface. Ann. Math. 89, 14–51 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Narasimhan, M.S., Ramanan, S.: Geometry of Hecke cycles I. In: C.P. Ramanujam—A tribute, Tata Inst. Fund. Res. Studies in Math. vol. 8, pp. 291–345. Springer, Berlin (1978)Google Scholar
  21. 21.
    Nitsure N.: Moduli space of semistable pairs on a curve. Proc. Lond. Math. Soc. 62, 275–300 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Oxbury, W.: Stable bundles and branched coverings over Riemann surfaces, DPhil thesis, University of Oxford (1987)Google Scholar
  23. 23.
    Sawon J.: On the discriminant locus of a Lagrangian fibration. Math. Ann. 341, 201–221 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Simpson C.T.: Moduli of representations of the fundamental group of a smooth projective variety II. Publ. Math. IHES 79, 47–129 (1994)ADSCrossRefzbMATHGoogle Scholar
  25. 25.
    Thaddeus, M.: Topology of the moduli space of stable bundles over a compact Riemann surface, Oxford dissertation (1990)Google Scholar
  26. 26.
    van Geemen B., Previato E.: On the Hitchin system. Duke Math. J. 85, 659–683 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Colin de Verdier̀e Y., Vu Ngoc S.: Singular Bohr–Sommerfeld rules for 2D integrable systems. Ann. Sci. École Norm. Sup. 36, 1–55 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Vey J.: Sur certain systèmes dynamiques séparables. Am. J. Math. 100, 591–614 (1978)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

OpenAccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Mathematical InstituteOxfordUK

Personalised recommendations