Hodge theory on transversely symplectic foliations

TL;DR

This paper develops symplectic Hodge theory for transversely symplectic foliations, establishing key lemmas and properties that unify various geometric structures and lead to new results on cup products and examples of $K$-contact manifolds.

Contribution

It introduces symplectic Hodge theory on transversely symplectic foliations, including the $d\delta$-lemma and the $s$-Lefschetz property, unifying multiple geometric contexts.

Findings

Proves the symplectic $d\delta$-lemma for transversely symplectic foliations with the $s$-Lefschetz property.

Shows that the $s$-Lefschetz property implies vanishing cup products on compact $K$-contact manifolds.

Constructs examples of $K$-contact manifolds that are $s$-Lefschetz but not $(s+1)$-Lefschetz.

Abstract

In this paper, we develop symplectic Hodge theory on transversely symplectic foliations. In particular, we establish the symplectic $d\delta$-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 $s\geq 2$, we also apply our main result to produce examples of $K$-contact manifolds that are $s$-Lefschetz but not $(s+1)$-Lefschetz.