Hard Lefschetz Theorem for Vaisman manifolds

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.

18 pages

References in corpus (4)

• LCK rank of locally conformally Kahler manifolds with potential
• On locally conformal symplectic manifolds of the first kind
• Locally conformally Kähler manifolds. A selection of results
• A non-Sasakian Lefschetz K-contact manifold of Tievsky type

Cited by in corpus (1)

• The hard Lefschetz duality for locally conformally almost Kähler manifolds