Document Type

Preprint

Publication Date

12-14-2017

Publication Title

The Quarterly Journal of Mathematics

DOI

10.1093/qmath/hax051

ISSN

1464-3847

Abstract

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.

Comments

This version of the paper was obtained from arXIV.org. In order for the work to be deposited in arXIV.org, the authors must hold the rights or the work must be under Creative Commons Attribution license, Creative Commons Attribution-Noncommercial-ShareAlike license, or Create Commons Public Domain Declaration. The publisher's final edited version of this article is available at The Quarterly Journal of Mathematics.

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