Homogeneous locally conformally K\"ahler manifolds

TL;DR

This paper investigates the structure of automorphism groups of compact homogeneous locally conformally K"ahler manifolds, establishing bounds on their centers and characterizing the Vaisman case as a mapping torus of a Sasakian manifold.

Contribution

It proves the center of the automorphism group is at most 2-dimensional and characterizes the Vaisman case as a mapping torus of a Sasakian manifold.

Findings

The automorphism group's center is at most 2-dimensional.

If the center is 2-dimensional, the manifold is Vaisman.

Vaisman manifolds are isometric to a mapping torus of a Sasakian manifold.

Abstract

It is known that automorphism group $G$ of a compact homogeneous locally conformally K\"ahler manifold $M=G/H$ has at least a 1-dimensional center. We prove that the center of $G$ is at most 2-dimensional, and that if its dimension is 2, then $M$ is Vaisman and isometric to a mapping torus of an isometry of a homogeneous Sasakian manifold.