It is well known that in any compact Kaehler manifold the exterior multiplication by a suitable power of the symplectic form induces isomorphisms between the de Rham cohomology spaces in complementary degrees. This is the celebrated Hard Lefschetz Theorem.
In my talk I will present a version of the Hard Lefschetz theorem for compact locally conformal Kaehler manifolds with parallel Lee vector field, known as Vaisman manifolds. Our result is based on the Hard Lefschetz theorem for Sasakian manifolds and the fact that any compact Vaisman manifold is the mapping torus of a compact Sasakian manifold.