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

Abstract

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 $M$.