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 ω 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, 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 nearsymplectic 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. arXiv:1211.5859v3 [math.SG]

Poisson and near-symplectic structures on generalized wrinkled fibrations in dimension 6

“…Then in section 4 we prove that the local models for the symplectic forms on the leaves are also the claimed ones.Finally, in section 5 we turn to near-symplectic geometry. Explicit models of near-symplectic forms have appeared in previous work [11,1,14,19]. Here we show a further construction of local near-symplectic forms that follows a general scheme for all the singularities of a generalized wrinkled fibration.…”

“…We proceed now to higher dimensions. To start, consider the definition of broken Lefschetz fibrations in higher dimensions [19]. Definition 2.4.…”

“…The latter are closed 2-forms that are non-degenerate outside a collection of circles where they vanish, and they are known to exist on any 4-manifold with b + 2 > 0. Recently, near-symplectic structures and generalized broken Lefschetz fibrations have been studied in higher dimensions [19].…”

“…57R17, 53D17. after combining the definition of generalized bLf [19] together with the results of [6] and [10] on Poisson structures. Theorem 1.1.…”

“…Then there exist a near-symplectic form ω on M with singular locus Z such that it restricts to a symplectic form on the smooth fibres of the fibration.The proof of this theorem appears in section 5.2. The global construction of the near-symplectic structure on the total space follows as in dimension 4 [1,14] with the modifications for higher dimensions introduced by the third named author [19]. The most substantial difference concerns the construction of the local forms around the new singularities.…”