#### Volume 16, issue 3 (2016)

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 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Near-symplectic $2n$–manifolds

### Ramón Vera

Algebraic & Geometric Topology 16 (2016) 1403–1426
##### Abstract

We give a generalization of the concept of near-symplectic structures to $2n$ dimensions. According to our definition, a closed $2$–form on a $2n$–manifold $M$ is near-symplectic if it is symplectic outside a submanifold $Z$ of codimension $3$ where ${\omega }^{n-1}$ vanishes. We depict how this notion relates to near-symplectic $4$–manifolds and broken Lefschetz fibrations via some examples. We define a generalized broken Lefschetz fibration as a singular map with indefinite folds and Lefschetz-type singularities. We show that, given such a map on a $2n$–manifold over a symplectic base of codimension $2$, the total space carries such a near-symplectic structure whose singular locus corresponds precisely to the singularity set of the fibration. A second part studies the geometry around the codimension-$3$ singular locus $Z$. We describe a splitting property of the normal bundle ${N}_{Z}$ that is also present in dimension four. A tubular neighbourhood theorem for $Z$ is provided, which has a Darboux-type theorem for near-symplectic forms as a corollary.

##### Keywords
near-symplectic forms, broken Lefschetz fibrations, stable Hamiltonian structures, singular symplectic forms, folds, singularities
##### Mathematical Subject Classification 2010
Primary: 53D35, 57R17
Secondary: 57R45