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Near-symplectic $2n$–manifolds

Ramón Vera

Algebraic & Geometric Topology 16 (2016) 1403–1426

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 ωn1 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 NZ 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.

near-symplectic forms, broken Lefschetz fibrations, stable Hamiltonian structures, singular symplectic forms, folds, singularities
Mathematical Subject Classification 2010
Primary: 53D35, 57R17
Secondary: 57R45
Received: 12 August 2014
Revised: 19 August 2015
Accepted: 3 October 2015
Published: 1 July 2016
Ramón Vera
Department of Mathematics
The Pennsylvania State University
University Park
State College, PA 16802
United States
Department of Mathematical Sciences
Durham University
Science Laboratories
South Rd
Durham DH1 3LE United Kingdom