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The Quarterly Journal of Mathematics






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 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.


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