Existence Criteria for LCK Metrics

TL;DR

This paper explores conditions under which complex manifolds of LCK type admit special metrics, linking group actions to the existence of Vaisman and LCK metrics with positive potential, and classifies certain LCK manifolds.

Contribution

It establishes new criteria connecting torus actions to the existence of Vaisman and positive potential LCK metrics, and classifies LCK manifolds with torus bundle structures.

Findings

Presence of a non-real compact torus implies Vaisman metric existence.

A torus of half the manifold's dimension guarantees an LCK metric with positive potential.

New non-existence results for LCK metrics on certain product manifolds.

Abstract

We investigate the relation between holomorphic torus actions on complex manifolds of LCK type and the existence of special LCK metrics. We show that if the group of biholomorphisms of such a manifold $(M,J)$ contains a non-real compact torus, then there exists a Vaisman metric on the manifold. Moreover, we show that if the group of biholomorphisms contains a compact torus whose dimension is half the real dimension of $M$, then $(M,J)$ admits an LCK metric with positive potential. As an application, we obtain a classification of manifolds of LCK type among all the manifolds having the structure of a holomorphic principal torus bundle. Moreover, we obtain new non-existence results for LCK metrics on certain products of complex manifolds.