Motivated by work of Fine and Panov, and of Lindsay and Panov, we prove that every closed symplectic complexity one space that is positive (e.g., positive monotone) enjoys topological properties that Fano varieties with a complexity one holomorphic torus action possess. In particular, such spaces are simply connected, have Todd genus equal to one and vanishing odd Betti numbers.

This is a preview of subscription content, log in to check access.


  1. [1]

    M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1–15.

    MathSciNet  Article  Google Scholar 

  2. [2]

    Y. Cho, M. K. Kim, Log-concavity of complexity one Hamiltonian torus actions, C. R. Math. Acad. Sci. Paris 350 (2012) no. 17–18, 845–848.

    MathSciNet  MATH  Google Scholar 

  3. [3]

    Y. Cho, Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions I, Internat. J. Math. 30 (2019), no. 6, 1950032, 71 pp.

  4. [4]

    Y. Cho, Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions II, arXiv:1904.10962v1 (2019).

  5. [5]

    Y. Cho, Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions III, arXiv:1905.07292v1 (2019).

  6. [6]

    J. J. Duistermaat, G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), no. 2, 259–268.

    MathSciNet  Article  Google Scholar 

  7. [7]

    J. Fine, D. Panov, Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle, Geom. Top. 14 (2010), no. 3, 1723–1763.

    Article  Google Scholar 

  8. [8]

    J. Fine, D. Panov, Circle invariant fat bundles and symplectic Fano 6-manifolds, J. London Math. Soc. 91 (2015), no. 3, 709–730.

    MathSciNet  Article  Google Scholar 

  9. [9]

    L. Godinho, F. von Heymann, S. Sabatini, 12, 24 and beyond, Adv. Math. 319 (2017), 472–521.

    MathSciNet  Article  Google Scholar 

  10. [10]

    M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307–347.

    MathSciNet  Article  Google Scholar 

  11. [11]

    V. Guillemin, S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), no. 3, 491–513.

    MathSciNet  Article  Google Scholar 

  12. [12]

    V. Guillemin, S. Sternberg, A normal form for the moment map, in: Differential Geometric Methods in Mathematical Physics (Jerusalem, 1982), Math. Phys. Stud., 6, Reidel, Dordrecht, 1984, pp. 161–175.

  13. [13]

    V. Guillemin, S. Sternberg, Birational equivalence in the symplectic category, Invent. Math. 97 (1989), no. 3, 485–522.

    MathSciNet  Article  Google Scholar 

  14. [14]

    F. Hirzebruch, T. Berger, R. Jung, Manifolds and Modular Forms, Aspects of Mathematics, E20, With appendices by Nils-Peter Skoruppa and by Paul Baum, Friedr. Vieweg & Sohn, Braunschweig, 1992.

  15. [15]

    V. A. Iskovskikh, Yu. G. Prokhorov, Fano varieties, in: Algebraic Geometry, V, Encyclopaedia Math. Sci., Vol. 47, Springer, Berlin, 1999, pp. 1–247.

  16. [16]

    Y. Karshon, Periodic Hamiltonian Flows on Four Dimensional Manifolds, Mem. Amer. Math. Soc. 141 (1999), no. 672, viii + 71.

  17. [17]

    Y. Karshon, S. Tolman, Centered complexity one Hamiltonian torus actions, Trans. Amer. Math. Soc. 353 (2001), no. 12, 4831–4861.

    MathSciNet  Article  Google Scholar 

  18. [18]

    Y. Karshon, S. Tolman, Complete invariants for Hamiltonian torus actions with two dimensional quotients, J. Symplectic Geom. 2 (2003), no. 1, 25–82.

    MathSciNet  Article  Google Scholar 

  19. [19]

    Y. Karshon, S. Tolman, Classification of Hamiltonian torus actions with two-dimensional quotients, Geom. Topol. 18 (2014), no. 2, 669–716.

    MathSciNet  Article  Google Scholar 

  20. [20]

    F. C. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Mathematical Notes, 31, Princeton University Press, Princeton, NJ, 1984, i+211.

  21. [21]

    H. Li, The fundamental group of symplectic manifolds with Hamiltonian Lie group actions, J. Symplectic Geom. 4 (2006), no. 3, 345–372.

    MathSciNet  Article  Google Scholar 

  22. [22]

    N. Lindsay, D. Panov, S1-invariant symplectic hypersurfaces in dimension 6 and the Fano condition, J. Topol. 12 (2019), no.1, 221–285.

  23. [23]

    N. Lindsay, Hamiltonian Circle Actions on Symplectic Fano Manifolds, PhD thesis, King’s College London, 2018.

  24. [24]

    C.-M. Marle, Modèle d’action hamiltonienne d’un groupe de Lie sur une variété symplectique, Rend. Sem. Mat. Univ. Politec. Torino 43 (1985), no. 2, 227–251.

    MathSciNet  MATH  Google Scholar 

  25. [25]

    D. McDuff, Some 6-dimensional Hamiltonian S1-manifolds, J. Topol. 2 (2009), no. 3, 589–623.

    MathSciNet  Article  Google Scholar 

  26. [26]

    D. McDuff, The structure of rational and ruled symplectic 4-manifolds, J. Amer. Math. Soc. 3 (1990), 679–712.

    MathSciNet  MATH  Google Scholar 

  27. [27]

    D. McDuff, D. Salamon, Introduction to Symplectic Topology, Oxford Mathematical Monographs, 2nd Edition, The Clarendon Press, Oxford University Press, New York, 1998, x+486.

  28. [28]

    D. McDuff, D. Salamon, J-holomorphic Curves and Symplectic Topology, AMS Colloquium Publications, 52, American Mathematical Society, Providence, RI, 2004, xii+669.

  29. [29]

    A. G. Reznikov, Symplectic twistor spaces, Ann. Global Ann. Geom. 11 (1993), no. 2, 109–118.

    MathSciNet  Article  Google Scholar 

  30. [30]

    C. H. Taubes, SeibergWitten and Gromov Invariants for Symplectic 4-Manifolds International Press, Somerville, 2000.

  31. [31]

    W. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), 467–468.

    MathSciNet  MATH  Google Scholar 

  32. [32]

    S. Tolman, A symplectic generalization of Petrie’s conjecture, Trans. Amer. Math. Soc. 362 (2010), no. 8, (2010), 3963–3996.

Download references

Author information



Corresponding author

Correspondence to D. SEPE.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

S. Sabatini is partially supported by Deutsche Forschungsgemeinschaft.

D. Sepe is partially supported by Deutsche Forschungsgemeinschaft, CNPq and CAPES.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

SABATINI, S., SEPE, D. ON TOPOLOGICAL PROPERTIES OF POSITIVE COMPLEXITY ONE SPACES. Transformation Groups (2020). https://doi.org/10.1007/s00031-020-09588-y

Download citation