Mathematische Zeitschrift

, Volume 252, Issue 1, pp 19–25

Free circle actions with contractible orbits on symplectic manifolds

Article

Abstract

We prove that closed symplectic four-manifolds do not admit any smooth free circle actions with contractible orbits, without assuming that the actions preserve the symplectic forms. In higher dimensions such actions by symplectomorphisms do exist, and we give explicit examples based on the constructions of FGM.

Mathematics Subject Classification (2000)

57R17 57R57 57M60 57R91 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Allday, C., Oprea, J.: A c-symplectic free S1-manifold with contractible orbits and cat Open image in new window dim. Proc. Amer. Math. Soc. (to appear)Google Scholar
  2. 2.
    Amorós, J., Burger, M., Corlette, K., Kotschick, D., Toledo, D.: Fundamental Groups of Compact Kähler Manifolds. Amer. Math. Soc., Providence, R.I. 1996Google Scholar
  3. 3.
    Barth, W., Peters, C., Van de Ven, A.: Compact Complex Surfaces. Springer-Verlag, Berlin, 1984Google Scholar
  4. 4.
    Donaldson, S. K.: Polynomial invariants for smooth four-manifolds. Topology 29, 257–315 (1990)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Fernandez, M., Gray, A., Morgan, J. W.: Compact symplectic manifolds with free circle actions, and Massey products. Michigan Math. J. 38, 271–283 (1991)MATHMathSciNetGoogle Scholar
  6. 6.
    Hempel, J.: (3)-Manifolds. Princeton University Press, 1976Google Scholar
  7. 7.
    Kotschick, D.: Remarks on geometric structures on compact complex surfaces. Topology 31, 317–321 (1992)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kotschick, D.: The Seiberg-Witten invariants of symplectic four–manifolds. Séminaire Bourbaki, 48ème année, 1995-96, no. 812, Astérisque 241, 195–220 (1997)Google Scholar
  9. 9.
    Kotschick, D.: Entropies, volumes, and Einstein metrics. Preprint, 2004Google Scholar
  10. 10.
    Laudenbach, F.: Topologie de la dimension trois: homotopie et isotopie. Astérisque 12, Société Mathématique de France, Paris, 1974Google Scholar
  11. 11.
    Matumoto, T.: On diffeomorphisms of a K 3 surface. In: Algebraic and topological theories (Kinosaki, 1984), pages 616–621. Kinokuniya, Tokyo, 1986Google Scholar
  12. 12.
    McDuff, D., Salamon, D.: Introduction to Symplectic Topology. Clarendon Press, Oxford, 1994Google Scholar
  13. 13.
    Nakamura, I.: Towards classification of non-Kählerian complex surfaces. Sugaku Exp. 2, 209–229 (1989)Google Scholar
  14. 14.
    Taubes, C. H.: The Seiberg–Witten invariants and symplectic forms. Math. Research Letters 1, 809–822 (1994)MATHMathSciNetGoogle Scholar
  15. 15.
    Yau, S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math. 31, 339–411 (1978)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Mathematisches InstitutLudwig-Maximilians-Universität MünchenMünchenGermany

Personalised recommendations