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 quasi-folds 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 2nΣs. 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.