HODGE THEORY ON TRANSVERSELY SYMPLECTIC FOLIATIONS.

In this paper, we develop symplectic Hodge theory on transversely symplectic foliations. In particular, we establish the symplectic dδ-lemma for any such foliations with the (transverse) s-Lefschetz property. As transversely symplectic foliations include many geometric structures, such as contact manifolds, co-symplectic manifolds, symplectic orbifolds and symplectic quasifolds as special examples, our work provides a unifying treatment of symplectic Hodge theory in these geometries. As an application, we show that on compact K-contact manifolds, the s-Lefschetz property implies a general result on the vanishing of cup products, and that the cup length of a 2n + 1 dimensional compact K-contact manifold with the (transverse) s-Lefschetz property is at most 2n - s. For any even integer s≥2, we also apply our main result to produce examples of K-contact manifolds that are s-Lefschetz, but not (s + 1)-Lefschetz. [ABSTRACT FROM AUTHOR]