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 groups 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 groups of a compact Sasakian manifold. In the current paper, using the formalism of odd dimensional symplectic geometry, we prove a Hard Lefschetz theorem for compact K-contact manifolds, 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 ≥ 9, and answer an open question asked by Boyer and late Galicki concerning the existence of such examples.
"Lefschetz Contact Manifolds and Odd Dimensional Symplectic Geometry."