A Calabi-Yau theorem for Vaisman manifolds

TL;DR

The paper extends Calabi-Yau type results to Vaisman manifolds by proving that a Vaisman metric is uniquely determined by its volume form and Lee class, and that for any Lee class and Lee- and anti-Lee-invariant volume form with matching total volume there exists a unique Vaisman structure realizing them. The approach leverages the transversally Kähler foliation intrinsic to Vaisman geometry and reduces the problem to a transversal complex Monge-Ampère equation, drawing on a transversal Calabi-Yau theorem. A key ingredient is the identification of the Lee field direction as determined by the complex structure, allowing a complete parametrization of Vaisman metrics by cohomological data and transversal volume forms. Collectively, these results provide a Calabi-Yau type classification in the non-Kähler Vaisman setting and establish a precise link between volume data, Lee class, and the underlying foliation geometry.

Abstract

A compact complex Hermitian manifold $(M, I, w)$ is called Vaisman if $dw=w\wedge θ$ and the 1-form $θ$, 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 $θ$ (called the Lee field) and the vector field $I(X)$, called { the anti-Lee field}. The cohomology class of $θ$, 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 $[θ]$ and each Lee- and anti-Lee-invariant volume form $V$, there exists a Vaisman structure with the volume form $V$ and the Lee class $c[θ]$. This is an analogue of the Calabi-Yau theorem claiming that the Kahler form is uniquely determined by its volume and the cohomology class.