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

Mathematische Zeitschrift. 2021. Vol. 299. P. 2287–2296.

Ornea L., Verbitsky M.

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 C. It is called quasi-regular if the 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 S1-quotient of M.

Keywords: elliptic curve, Vaisman manifold, Sasakian manifolds, Reeb field, CR manifold, Projective orbifold, Quasi-regular

Publication based on the results of: Categorical, classical and differential geometry of algebraic varieties (2021)