Advertisement

Higgs Bundles and Characteristic Classes

  • Nigel HitchinEmail author
Chapter
Part of the Progress in Mathematics book series (PM, volume 319)

Abstract

Sixty years ago Hirzebruch observed how the vanishing of the Stiefel–Whitney class w2 led to integrality of the \(\hat{A}\)-genus of an algebraic variety [Hirz1]. This was one motivation for the Atiyah–Singer index theorem but also for my own thesis about Dirac operators and Kähler manifolds. Indeed the interaction between topology and algebraic geometry which he developed has been a constant theme in virtually all my work.

Keywords

Modulus Space Vector Bundle Line Bundle Abelian Variety Spectral Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [AB]
    M.F. Atiyah, R. Bott, A Lefschetz fixed point formula for elliptic complexes II. Applications. Ann. Math. 88, 451–491 (1968)MathSciNetzbMATHGoogle Scholar
  2. [AGH]
    E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris, Geometry of Algebraic Curves, vol. I (Springer, New York, 1985)CrossRefzbMATHGoogle Scholar
  3. [BNR]
    A. Beauville, M.S. Narasimhan, S. Ramanan, Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398, 169–179 (1989)MathSciNetzbMATHGoogle Scholar
  4. [Bon]
    J. Bonsdorff, Autodual connection in the Fourier transform of a Higgs bundle. Asian J. Math. 14, 153–173 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [Burg]
    M. Burger, A. Iozzi, A. Wienhard, Surface group representations with maximal Toledo invariant. Ann. Math. 172, 517–566 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [DP]
    R. Donagi, T. Pantev, Langlands duality for Hitchin systems. Invent. Math. 189, 653–735 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [EOP]
    A. Eskin, A. Okounkov, R. Pandharipande, The theta characteristic of a branched covering. Adv. Math. 217, 873–888 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [GGM]
    O. Garcia-Prada, P. Gothen, I. Mundet i Riera, Higgs bundles and surface group representations in the real symplectic group. J. Topol. 6, 64–118 (2013)Google Scholar
  9. [Hirz1]
    F. Hirzebruch, Problems on differentiable and complex manifolds. Ann. Math. 60, 213–236 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [Hit1]
    N.J. Hitchin, The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. 55, 59–126 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [Hit2]
    N.J. Hitchin, Stable bundles and integrable systems. Duke Math. J. 54, 91–114 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [Hit3]
    N.J. Hitchin, Lie groups and Teichmüller space. Topology 31, 449–473 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [Hit4]
    N.J. Hitchin, Langlands duality and G 2 spectral curves. Q. J. Math. Oxf. 58, 319–344 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [Hit5]
    N.J. Hitchin, The Dirac operator, in Invitations to Geometry and Topology, ed. by M. Bridson, S. Salamon. Oxford Graduate Texts in Mathematics (Oxford University Press, Oxford, 2002), pp. 208–232Google Scholar
  15. [HT]
    T. Hausel, M. Thaddeus, Mirror symmetry, Langlands duality and Hitchin systems. Invent. Math. 153, 197–229 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [LS]
    L.P. Schaposnik, Spectral data for G-Higgs bundles. D. Phil Thesis, Oxford (2013)Google Scholar
  17. [LS0]
    L.P. Schaposnik, Monodromy of the SL 2 Hitchin fibration. Int. J. Math. 24, 1350013 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [LS1]
    L.P. Schaposnik, Spectral data for U(m, m)-Higgs bundles. Int. Math. Res. Not. (2014). doi:10.1093/imrn/rnu029Google Scholar
  19. [MFA1]
    M.F. Atiyah, Riemann surfaces and spin structures. Ann. Sci. École Norm. Sup. 4, 47–62 (1971)MathSciNetzbMATHGoogle Scholar
  20. [PG]
    P. Gothen, The topology of Higgs bundle moduli spaces. Ph.D. thesis, Warwick (1995)Google Scholar
  21. [Sim]
    C. Simpson, Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. 75, 5–95 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [Tr]
    W. Trench, On the eigenvalue problem for Toeplitz band matrices. Linear Algebra Appl. 64, 199–214 (1985)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Mathematical InstituteRadcliffe Observatory QuarterOxfordUK

Personalised recommendations