Lefschetz Contact Manifolds and Odd Dimensional Symplectic Geometry

<p> In the literature, there are two different versions of Hard Lefschetz theorems for a compact Sasakian manifold. The &filig;rst version, due to Kacimi-Alaoui, asserts that the basic cohomology groups of a compact Sasakian manifold satis&filig;es 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.</p><p> 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 &ge; 9, and answer an open question asked by Boyer and late Galicki concerning the existence of such examples.</p>