Compact homogeneous locally conformally Kahler manifolds are Vaisman. A new proof

TL;DR

This paper proves that all compact homogeneous locally conformally Kahler (LCK) manifolds are Vaisman, providing a new proof based on the existence of an LCK potential on such manifolds.

Contribution

It introduces a new proof that compact homogeneous LCK manifolds are Vaisman by demonstrating they admit a metric with an LCK potential.

Findings

All compact homogeneous LCK manifolds are Vaisman.

Homogeneous LCK manifolds admit a metric with an LCK potential.

Abstract

An LCK manifold with potential is a complex manifold with a Kahler potential on its cover, such that any deck transformation multiplies the Kahler potential by a constant multiplier. We prove that any homogeneous LCK manifold admits a metric with LCK potential. This is used to give a new proof that any compact homogeneous LCK manifold is Vaisman.