Hard Lefschetz Theorem for Vaisman manifolds

TL;DR

This paper proves a Hard Lefschetz Theorem for the cohomology of compact Vaisman manifolds and introduces related notions for locally conformal symplectic manifolds, expanding understanding of their geometric structures.

Contribution

It establishes a Hard Lefschetz Theorem for Vaisman manifolds and introduces the concepts of Lefschetz and basic Lefschetz l.c.s. manifolds, clarifying their equivalence under certain metric conditions.

Findings

Hard Lefschetz Theorem proven for Vaisman manifolds' de Rham cohomology

Similar Lefschetz result established for basic cohomology with respect to Lee vector field

Examples of l.c.s. manifolds without compatible Vaisman metrics discussed

Abstract

We establish a Hard Lefschetz Theorem for the de Rham cohomology of compact Vaisman manifolds. A similar result is proved for the basic cohomology with respect to the Lee vector field. Motivated by these results, we introduce the notions of a Lefschetz and of a basic Lefschetz locally conformal symplectic (l.c.s.) manifold of the first kind. We prove that the two notions are equivalent if there exists a Riemannian metric such that the Lee vector field is unitary and parallel and its metric dual $1$-form coincides with the Lee $1$-form. Finally, we discuss several examples of compact l.c.s. manifolds of the first kind which do not admit compatible Vaisman metrics.