Toric Vaisman Manifolds

TL;DR

This paper introduces the concept of toric Vaisman manifolds, establishing their relationship with toric Sasaki and K"ahler geometries, and characterizes when such manifolds are toric based on their coverings and quotients.

Contribution

It extends the theory of Vaisman manifolds to the toric setting, showing the equivalence of toric structures between Vaisman, Sasaki, and K"ahler geometries in various cases.

Findings

Toric Vaisman manifolds correspond to toric Sasaki manifolds via their minimal coverings.

Complete toric Vaisman manifolds have toric Sasaki manifolds as their bases.

Strongly regular compact Vaisman manifolds are toric if and only if their K"ahler quotients are toric.

Abstract

Vaisman manifolds are strongly related to K\"ahler and Sasaki geometry. In this paper we introduce toric Vaisman structures and show that this relationship still holds in the toric context. It is known that the so-called minimal covering of a Vaisman manifold is the Riemannian cone over a Sasaki manifold. We show that if a complete Vaisman manifold is toric, then the associated Sasaki manifold is also toric. Conversely, a toric complete Sasaki manifold, whose K\"ahler cone is equipped with an appropriate compatible action, gives rise to a toric Vaisman manifold. In the special case of a strongly regular compact Vaisman manifold, we show that it is toric if and only if the corresponding K\"ahler quotient is toric.