Hard Leschetz Theorem for K-Contact Manifolds

Recently, Cappelletti-Montano, De Nicola, and Yudin proved a Hard Lefschetz theorem for the De Rham cohomology of compact Sasakian manifolds, and proposed an associated notion of Lefschetz contact manifolds. In this talk, we discuss a new approach to the Hard Lefschetz theorem for Sasakian manifolds using the formalism of odd dimensional symplectic geometry. This leads to a more general Hard Lefschetz theorem for K-contact manifolds, and provides us a sufficient and necessary condition for a finitely presentable group to be the fundamental group of a Lefschetz contact five manifolds. As an application, we show how to use our methods to construct simply-connected K-contact manifolds which do not support any Sasakian structures. This in particular answers an open question asked by Boyer and late Galicki.