Near-symplectic $2n$–manifolds

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.