Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 024, 16 pages      arXiv:1809.06021      https://doi.org/10.3842/SIGMA.2019.024
Contribution to the Special Issue on Geometry and Physics of Hitchin Systems

On Higgs Bundles on Nodal Curves

Marina Logares
School of Computing Electronics and Mathematics, University of Plymouth, Drake Circus, PL4 8AA, UK

Received October 16, 2018, in final form March 14, 2019; Published online March 28, 2019

Abstract
This is a review article on some applications of generalised parabolic structures to the study of torsion free sheaves and $L$-twisted Hitchin pairs on nodal curves. In particular, we survey on the relation between representations of the fundamental group of a nodal curve and the moduli spaces of generalised parabolic bundles and generalised parabolic $L$-twisted Hitchin pairs on its normalisation as well as on an analogue of the Hitchin map for generalised parabolic $L$-twisted Hitchin pairs.

Key words: Higgs bundles; nodal curves; generalised parabolic structures.

pdf (400 kb)   tex (23 kb)

References

  1. Balaji V., Barik P., Nagaraj D.S., A degeneration of moduli of Hitchin pairs, Int. Math. Res. Not. 2016 (2016), 6581-6625, arXiv:1308.4490.
  2. Bhosle U.N., Generalised parabolic bundles and applications to torsionfree sheaves on nodal curves, Ark. Mat. 30 (1992), 187-215.
  3. Bhosle U.N., Representations of the fundamental group and vector bundles, Math. Ann. 302 (1995), 601-608.
  4. Bhosle U.N., Generalized parabolic bundles and applications. II, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), 403-420.
  5. Bhosle U.N., Principal $G$-bundles on nodal curves, Proc. Indian Acad. Sci. Math. Sci. 111 (2001), 271-291.
  6. Bhosle U.N., Generalized parabolic Hitchin pairs, J. Lond. Math. Soc. 89 (2014), 1-23.
  7. Bhosle U.N., Biswas I., Hurtubise J., Grassmannian-framed bundles and generalized parabolic structures, Internat. J. Math. 24 (2013), 1350090, 49 pages, arXiv:1202.4239.
  8. Bhosle U.N., Logares M., Newstead P.E., Vector bundles on singular curves, in preparation.
  9. Bhosle U.N., Parameswaran A.J., Holonomy group scheme of an integral curve, Math. Nachr. 287 (2014), 1937-1953.
  10. Bhosle U.N., Parameswaran A.J., Singh S.K., Hitchin pairs on an integral curve, Bull. Sci. Math. 138 (2014), 41-62.
  11. Cook P.R., Local and global aspects of the module theory of singular curves, Ph.D. Thesis, University of Liverpool, 1993.
  12. Corlette K., Flat $G$-bundles with canonical metrics, J. Differential Geom. 28 (1988), 361-382.
  13. Donaldson S.K., Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. 55 (1987), 127-131.
  14. Hitchin N.J., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126.
  15. Mazzeo R., Swoboda J., Weiss H., Witt F., Ends of the moduli space of Higgs bundles, Duke Math. J. 165 (2016), 2227-2271, arXiv:1405.5765.
  16. Mehta V.B., Seshadri C.S., Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), 205-239.
  17. Narasimhan M.S., Seshadri C.S., Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. 82 (1965), 540-567.
  18. Oda T., Seshadri C.S., Compactifications of the generalized Jacobian variety, Trans. Amer. Math. Soc. 253 (1979), 1-90.
  19. Seshadri C.S., Moduli of $\pi $-vector bundles over an algebraic curve, in Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, 1970, 139-260.
  20. Seshadri C.S., Fibrés vectoriels sur les courbes algébriques, Astérisque, Vol. 96, Société Mathématique de France, Paris, 1982.
  21. Simpson C.T., Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990), 713-770.
  22. Simpson C.T., Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5-95.
  23. Simpson C.T., Moduli of representations of the fundamental group of a smooth projective variety. I, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 47-129.
  24. Simpson C.T., Moduli of representations of the fundamental group of a smooth projective variety. II, Inst. Hautes Études Sci. Publ. Math. 80 (1994), 5-79.
  25. Swoboda J., Moduli spaces of Higgs bundles on degenerating Riemann surfaces, Adv. Math. 322 (2017), 637-681, arXiv:1507.04382.
  26. Weil A., Généralisation des fonctions abéliennes, J. Math. Pure Appl. 17 (1938), 47-87.

Previous article  Next article   Contents of Volume 15 (2019)