Lefschetz contact manifolds and odd dimensional symplectic geometry

In the literature, there are two different versions of Hard Lefschetz theorems for a compact Sasakian manifold. The first version, due to Kacimi-Alaoui, asserts that the basic cohomology of a compact Sasakian manifold satisfies the transverse Lefschetz property. The second version, established far more recently by Cappelletti-Montano, De Nicola, and Yudin, holds for the De Rham cohomology of a compact Sasakian manifold. In the current paper, using the formalism of odd dimensional symplectic geometry, we prove a Hard Lefschetz theorem for a compact $K$-contact manifold, which implies immediately that the two existing versions of Hard Lefschetz theorems are mathematically equivalent to each other. Our method sheds new light on the Hard Lefschetz property of a Sasakian manifold. It enables us to give a simple construction of simply-connected $K$-contact manifolds without any Sasakian structures in any dimension $\geq 9$, and answers an open question asked by Boyer and late Galicki concerning the existence of such examples. It also allows us to establish a sufficient and necessary condition for a finitely presentable group to be the fundamental group of a compact Lefschetz contact five manifold. As an immediate application, we use it to produce first examples of compact Lefschetz contact manifolds which do not support any Sasakian structures. This provides an answer to another open question asked by Cappelletti-Montano, De Nicola, and Yudin in their recent work.