Annals of Global Analysis and Geometry

, Volume 39, Issue 1, pp 45–82 | Cite as

Deformations of symplectic vortices

Original Paper

Abstract

We prove a gluing theorem for a symplectic vortex on a compact complex curve and a collection of holomorphic sphere bubbles. Using the theorem we show that the moduli space of regular strongly stable symplectic vortices on a fixed curve with varying markings has the structure of a stratified-smooth topological orbifold. In addition, we show that the moduli space has a non-canonical C1-orbifold structure.

Keywords

Symplectic vortices Pseudo-holomorphic curves Gauged Gromov-Witten invariants 

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References

  1. 1.
    Abouzaid, M.: Framed bordism and Lagrangian embeddings of exotic spheres (2008). arXiv.org: 0812.4781Google Scholar
  2. 2.
    Adams, R.A.: Sobolev Spaces. Pure Appl. Math., vol. 65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York, London, (1975)Google Scholar
  3. 3.
    Aubin T.: Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics. Springer-Verlag, Berlin (1998)Google Scholar
  4. 4.
    Cieliebak, K., Gaio, A.R., Salamon, D.A.: J-holomorphic curves, moment maps, and invariants of Hamiltonian group actions. Int. Math. Res. Not. (16), 831–882 (2000)Google Scholar
  5. 5.
    Cieliebak K., Rita G.A., Mundet i Riera I., Salamon D.A.: The symplectic vortex equations and invariants of Hamiltonian group actions. J. Symplectic Geom. 1(3), 543–645 (2002)MATHMathSciNetGoogle Scholar
  6. 6.
    Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. (36), 75–109 (1969)Google Scholar
  7. 7.
    Douady, A.: Le problème des modules locaux pour les espaces C-analytiques compacts. Ann. Sci. École Norm. 7(4), 569–602 (1975) 1974Google Scholar
  8. 8.
    Earle C.J., Eells J.: A fibre bundle description of Teichmüller theory. J. Differ. Geom. 3, 19–43 (1969)MATHMathSciNetGoogle Scholar
  9. 9.
    Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Intersection Floer Theory: Anomaly and Obstruction. AMS/IP Studies in Advanced Mathematics, vol. 46. American Mathematical Society, Providence RI (2009)Google Scholar
  10. 10.
    Fulton, W., Pandharipande, R.: Notes on stable maps and quantum cohomology. In: Algebraic geometry—Santa Cruz 1995, pp. 45–96. American Mathematical Society, Providence RI (1997)Google Scholar
  11. 11.
    Gonzalez, E., Woodward, C.: Area-dependence in gauged Gromov-Witten theory (2008). arXiv: 0811.3358Google Scholar
  12. 12.
    Harris, J., Morrison, I.: Moduli of Curves. Graduate Texts in Mathematics, vol. 187. Springer-Verlag, New York (1998)Google Scholar
  13. 13.
    Hofer H., Wysocki K., Zehnder E.: sc-smoothness, retractions and new models for smooth spaces. Discret. Contin. Dyn. Syst. 28, 665–788 (2010)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Ionel E.-N., Parker T.H.: Relative Gromov-Witten invariants. Ann. Math. (2) 157(1), 45–96 (2003)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    McDuff, D., Salamon, D.: J-holomorphic curves and symplectic topology. American Mathematical Society Colloquium Publications, vol. 52. American Mathematical Society, Providence, RI (2004)Google Scholar
  16. 16.
    Mundet i Riera I.: Hamiltonian Gromov-Witten invariants. Topology 42(3), 525–553 (2003)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Ott, A.: Non-local vortex equations and gauged Gromov-Witten invariants. PhD thesis, ETH Zurich (2010).Google Scholar
  18. 18.
    Palais R.S.: C 1 actions of compact Lie groups on compact manifolds are C 1-equivalent to C actions. Am. J. Math. 92, 748–760 (1970)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Robbin J.W., Ruan Y., Salamon D.A.: The moduli space of regular stable maps. Math. Z. 259(3), 525–574 (2008)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Robbin J.W., Salamon D.A.: A construction of the Deligne-Mumford orbifold. J. Eur. Math. Soc. (JEMS) 8(4), 611–699 (2006)MATHMathSciNetGoogle Scholar
  21. 21.
    Ruan Y., Tian G.: Higher genus symplectic invariants and sigma models coupled with gravity. Invent. Math. 130(3), 455–516 (1997)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Siebert, B.: Gromov-Witten invariants of general symplectic manifolds. arXiv:dg-ga/9608005Google Scholar
  23. 23.
    Wehrheim, K., Woodward, C.T.: Orientations for pseudoholomorphic quilts. (2010) (preprint).Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Massachusetts BostonBostonUSA
  2. 2.Mathematics-Hill CenterRutgers UniversityPiscatawayUSA

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