Closed orbits of Reeb fields on Sasakian manifolds and elliptic curves on Vaisman manifolds
| Autor: | Ornea, Liviu, Verbitsky, Misha |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Math. Z. 299, 2287-2296 (2021) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s00209-021-02776-w |
| Popis: | A compact complex manifold $V$ is called Vaisman if it admits an Hermitian metric which is conformal to a K\"ahler 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 K\"ahler 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 K\"ahler orbifold obtained as an $S^1$-quotient of $M$. Comment: 15 pages, version 2.2, added more reference and more stuff about the Vaisman case (Sasakian case is due to Rukimbra, 1995) |
| Databáze: | arXiv |
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