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 ωon a 2n-manifold M is near-symplectic, if it is symplectic outside a submanifold Z of codimension 3, where ω^{n-1} vanishes. This extends the concept known in dimension 4. 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, or BLF, 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, then 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 for Z is provided, which has as a corollary a Darboux-type theorem for near-symplectic forms.