A Calabi-Yau theorem for 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.