Advertisement

Geometriae Dedicata

, Volume 179, Issue 1, pp 187–195 | Cite as

Symplectic form on hyperpolygon spaces

  • Indranil Biswas
  • Carlos Florentino
  • Leonor Godinho
  • Alessia MandiniEmail author
Original paper

Abstract

In Godinho and Mandini (Adv Math 244:465–532, 2013), a family of parabolic Higgs bundles on \({{\mathbb {C}}}{\mathbb {P}}^1\) was constructed and the parameter space for the family was identified with a moduli space of hyperpolygons. Our aim here is to give a canonical alternative construction of this family. This enables us to compute the Higgs symplectic form for this family and show that the isomorphism of Godinho and Mandini (Adv Math 244:465–532, 2013) is actually a symplectomorphism.

Keywords

Hyperpolygon space Parabolic Higgs bundles Symplectic form  Liouville form 

Mathematics Subject Classification (2000)

14D20 14H60 

Notes

Acknowledgments

We thank the referee for helpful comments. The first author wishes to thank Instituto Superior Técnico, where the work was carried out, for its hospitality; his visit to IST was funded by the FCT project PTDC/MAT/099275/2008. He also acknowledges the support of J. C. Bose Fellowship. This work was partially supported by the FCT projects PTDC/MAT/108921/2008, PTDC/MAT/120411/2010 and the FCT Grant SFRH/BPD/44041/2008.

References

  1. 1.
    Biswas, I., Ramanan, S.: An infinitesimal study of the moduli of Hitchin pairs. J. Lond. Math. Soc. 49, 219–231 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Biswas, I.: A criterion for the existence of a parabolic stable bundle of rank two over the projective line. Int. J. Math. 9, 523–533 (1998)zbMATHCrossRefGoogle Scholar
  3. 3.
    Godinho, L., Mandini, A.: Hyperpolygon spaces and moduli spaces of parabolic Higgs bundles. Adv. Math. 244, 465–532 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Griffiths, P.B., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1978)zbMATHGoogle Scholar
  5. 5.
    Hitchin, N.: Stable bundles and integrable systems. Duke Math. J. 54, 91–114 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Konno, H.: On the cohomology ring of the Hyper-Kähler analogue of the polygon spaces, Integrable systems, topology and physics (Tokyo, 2000), pp. 129–149, Contemp. Math. 309, Am. Math. Soc., Providence, RI (2002)Google Scholar
  7. 7.
    Maruyama, M.: Openness of a family of torsion free sheaves. J. Math. Kyoto Univ. 16, 627–637 (1976)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Nakajima, H.: Quiver varieties and Kac-Moody algebras. Duke Math. J. 91, 515–560 (1998)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Indranil Biswas
    • 1
  • Carlos Florentino
    • 2
  • Leonor Godinho
    • 2
  • Alessia Mandini
    • 3
    Email author
  1. 1.School of MathematicsTata Institute of Fundamental ResearchBombayIndia
  2. 2.Departamento Matemática, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal
  3. 3.Departamento de MatemáticaPUC–RioRio de JaneiroBrasil

Personalised recommendations