A Calabi-Yau theorem for Vaisman manifolds

TL;DR

This paper extends Calabi-Yau type results to Vaisman manifolds, showing that Vaisman metrics are uniquely determined by their volume form and Lee class, with existence results for prescribed data.

Contribution

It establishes a Calabi-Yau theorem analogue for Vaisman manifolds, characterizing Vaisman metrics via volume form and Lee class, and proving existence and uniqueness.

Findings

Vaisman metric is uniquely determined by volume form and Lee class.

Existence of Vaisman structures for given volume form and Lee class.

Analogue of Calabi-Yau theorem for non-Kähler Vaisman manifolds.

Abstract

A compact complex Hermitian manifold $(M, I, w)$ is called Vaisman if $dw=w\wedge \theta$ and the 1-form $\theta$, called the Lee form, is parallel with respect to the Levi-Civita connection. The volume form of $M$ is invariant with respect to the action of the vector field $X$ dual to $\theta$ (called the Lee field) and the vector field $I(X)$, called { the anti-Lee field}. The cohomology class of $\theta$, called the Lee class, plays the same role as the Kahler class in Kahler geometry. We prove that a Vaisman metric is uniquely determined by its volume form and the Lee class, and, conversely, for each Lee class $[\theta]$ and each Lee- and anti-Lee-invariant volume form $V$, there exists a Vaisman structure with the volume form $V$ and the Lee class $c[\theta]$. This is an analogue of the Calabi-Yau theorem claiming that the Kahler form is uniquely determined by its volume and the cohomology class.