  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.
