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

TL;DR

This paper investigates the number of closed elliptic curves and Reeb orbits on Vaisman and Sasakian manifolds, revealing they are either infinite or related to Betti numbers of associated K"ahler orbifolds, and provides new proofs for existing results.

Contribution

It establishes a count for closed elliptic curves on Vaisman manifolds and offers a new proof for the number of Reeb orbits on Sasakian manifolds, linking these to Betti numbers of K"ahler orbifolds.

Findings

Number of closed elliptic curves is either infinite or equals the sum of Betti numbers of a K"ahler orbifold.

Number of Reeb orbits is either infinite or equals the sum of Betti numbers of a K"ahler orbifold.

Provides a new proof of Rukimbira's result on Reeb orbits.

Abstract

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 $M$.