Lee classes on LCK manifolds with potential

TL;DR

This paper characterizes the set of Lee classes on LCK manifolds with potential, showing it forms an open half-space in cohomology, and provides a new proof for Vaisman manifolds.

Contribution

It determines the Lee classes on LCK manifolds with potential and offers a new proof for the Vaisman case from 1994.

Findings

The set of Lee classes is an open half-space in cohomology.

A new self-contained proof for Vaisman manifolds is provided.

The structure of Lee classes on LCK manifolds with potential is characterized.

Abstract

An LCK (locally conformally Kahler) manifold is a complex manifold $(M,I)$ equipped with a Hermitian form $\omega$ and a closed 1-form $\theta$, called the Lee form, such that $d\omega=\theta\wedge\omega$. An LCK manifold with potential is an LCK manifold with a positive Kahler potential on its cover, such that the deck group multiplies the Kahler potential by a constant. A Lee class of an LCK manifold is the cohomology class of the Lee form. We determine the set of Lee classes on LCK manifolds admitting an LCK structure with potential, showing that it is an open half-space in $H^1(M,{\mathbb R})$. For Vaisman manifolds, this theorem was proven in 1994 by Tsukada; we give a new self-contained proof of his result.