Periodica Mathematica Hungarica

, Volume 73, Issue 1, pp 16–26 | Cite as

Symplectic fillability of toric contact manifolds

  • Aleksandra MarinkovićEmail author


According to Lerman, compact connected toric contact 3-manifolds with a non-free toric action whose moment cone spans an angle greater than \(\pi \) are overtwisted, thus non-fillable. In contrast, we show that all compact connected toric contact manifolds in dimension greater than three are weakly symplectically fillable and many of them are strongly symplectically fillable. The proof is based on Lerman’s classification of toric contact manifolds and on our observation that the only contact manifolds in higher dimensions that admit free toric action are the cosphere bundle of \(T^d, d\ge 3\,(T^d\times S^{d-1})\) and \(T^2\times L_k,\,k\in \mathbb {N}\), with the unique contact structure.


Contact manifold Toric action Symplectic fillability 



I would like to thank Professors Miguel Abreu, Gustavo Granja, Klaus Niederkrüger, Milena Pabiniak and Branislav Prvulović for many useful disscussions. Moreover I thank the anonymous referees for their comments which have significantly improved an earlier version of this paper. My research was supported by the Fundação para a Ciência e a Tecnologia (FCT, Portugal) Grant SFRH/BD/77639/2011 and project PTDC/MAT/117762/2010.


  1. 1.
    M. Abreu, L. Macarini, Contact homology of good toric contact manifolds. Compos. Math. 148, 304–334 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    M.S. Borman, Y. Eliashberg, E. Murphy, Existence and classification of overtwisted contact structures in all dimensions. Acta Math. (to appear), arXiv:1404.6157
  3. 3.
    F. Bourgeois, Odd dimensional tori are contact manifolds. Int. Math. Res. Not. 30, 1571–1574 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    C.P. Boyer, Maximal tori in contactomorphism groups. Diff. Geom. Appl. 31, 190–216 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    C.P. Boyer, K. Galicki, A note on toric contact geometry. J. Geom. Phys. 35, 288–298 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Y. Eliashberg, Contact 3-manifolds twenty years since J. Martinets work. Ann. Inst. Fourier (Grenoble) 42, 165–192 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Y. Eliashberg, Unique holomorphically fillable contact structure on the 3torus. Int. Math. Res. Not. 2, 77–82 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Y. Eliashberg, in Filling by holomorphic discs and its applications, Geometry of Low-Dimensional Manifolds, Durham, 1989. London Mathematical Society Lecture Note Series 151, vol. 2 (Cambridge University Press, 1990), pp. 45–67Google Scholar
  9. 9.
    E. Giroux, Une structure de contact, même tendue, est plus ou moins tordue. Ann. Sci. E cole Norm. Sup. (4) 27, 697–705 (1994)MathSciNetGoogle Scholar
  10. 10.
    E. Lerman, Contact toric manifolds. J. Symplectic Geom. 1, 785–828 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    E. Lerman, Contact cuts. Israel J. Math. 124, 77–92 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    E. Lerman, Maximal tori in the contactomorphism groups of circle bundles over Hirzebruch surfaces. Math. Res. Lett. 10, 133–144 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    R. Lutz, Sur la géométrie des structures de contact invariantes. Ann. Inst. Fourier (Grenoble) 29, 283–306 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    P. Massot, K. Niederkrüger, C. Wendl, Weak and strong fillability of higher dimensional contact manifolds. Invent. Math. 192, 287–373 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    K. Niederkrüger, F. Pasquotto, Resolution of symplectic cyclic orbifold singularities. J. Symplectic Geom. 7, 337–355 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  1. 1.Center for Mathematical Analysis, Geometry and Dynamical Systems, Mathematics Department, Instituto Superior TécnicoUniversidade de Lisboa Av. Rovisco PaisLisboaPortugal

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