Closed orbits of Reeb fields on Sasakian manifolds and elliptic curves on Vaisman manifolds
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866911883510939648 |
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| author | Ornea, Liviu Verbitsky, Misha |
| author_facet | Ornea, Liviu Verbitsky, Misha |
| contents | A compact complex manifold $V$ is called Vaisman if it admits an 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 $M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_05962 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Closed orbits of Reeb fields on Sasakian manifolds and elliptic curves on Vaisman manifolds Ornea, Liviu Verbitsky, Misha Differential Geometry Algebraic Geometry Symplectic Geometry 53C25, 53C55, 32V99 A compact complex manifold $V$ is called Vaisman if it admits an 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 $M$. |
| title | Closed orbits of Reeb fields on Sasakian manifolds and elliptic curves on Vaisman manifolds |
| topic | Differential Geometry Algebraic Geometry Symplectic Geometry 53C25, 53C55, 32V99 |
| url | https://arxiv.org/abs/2102.05962 |