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Closed orbits of Reeb fields on Sasakian manifolds and elliptic curves on Vaisman manifolds

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A compact complex manifold V is called Vaisman if it admits a Hermitian metric which is conformal to a Kähler one, and a non-isometric conformal action by \(\mathbb {C}\). It is called quasi-regular if the \(\mathbb {C}\)-action has closed orbits. In this case the corresponding leaf space is a projective orbifold, called the quasi-regular quotient of V. It is known that the set of all quasi-regular Vaisman complex structures is dense in the appropriate deformation space. We count the number of closed elliptic curves on a Vaisman manifold, proving that their number is either infinite or equal to the sum of all Betti numbers of a Kähler orbifold obtained as a quasi-regular quotient of V. We also give a new proof of a result by Rukimbira showing that the number of Reeb orbits on a Sasakian manifold M is either infinite or equal to the sum of all Betti numbers of a Kähler orbifold obtained as an \(S^1\)-quotient of M.

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Acknowledgements

We are grateful to Gianluca Bande and Charles Boyer for drawing our attention to the reference [20] and to Oliver Goertsches and Hiraku Nozawa for clarifying us a part of their work [9]. We thank the anonymous referee for very careful reading our paper and for her or his useful remarks.

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Authors and Affiliations

  1. Faculty of Mathematics and Informatics, University of Bucharest, 14 Academiei str., 70109, Bucharest, Romania

    Liviu Ornea

  2. Institute of Mathematics ”Simion Stoilow” of the Romanian Academy, 21, Calea Grivitei Str., 010702, Bucharest, Romania

    Liviu Ornea

  3. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina, 110 Jardim Botânico, Rio de Janeiro, RJ, CEP 22460-320, Brazil

    Misha Verbitsky

  4. Laboratory of Algebraic Geometry, Faculty of Mathematics, National Research University Higher School of Economics, 6 Usacheva Str., Moscow, Russia

    Misha Verbitsky

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  1. Liviu Ornea
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Correspondence to Liviu Ornea.

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Liviu Ornea is partially supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDI - UEFISCDI, project number PN-III-P4-ID-PCE-2020-0025, within PNCDI III. Misha Verbitsky is partially supported by the Russian Academic Excellence Project ’5-100’, FAPERJ E-26/202.912/2018 and CNPq-Process 313608/2017-2.

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Ornea, L., Verbitsky, M. Closed orbits of Reeb fields on Sasakian manifolds and elliptic curves on Vaisman manifolds. Math. Z. 299, 2287–2296 (2021). https://doi.org/10.1007/s00209-021-02776-w

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  • DOI: https://doi.org/10.1007/s00209-021-02776-w

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