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2016 Near-symplectic $2n$–manifolds
Ramón Vera
Algebr. Geom. Topol. 16(3): 1403-1426 (2016). DOI: 10.2140/agt.2016.16.1403


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.


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Ramón Vera. "Near-symplectic $2n$–manifolds." Algebr. Geom. Topol. 16 (3) 1403 - 1426, 2016.


Received: 12 August 2014; Revised: 19 August 2015; Accepted: 3 October 2015; Published: 2016
First available in Project Euclid: 28 November 2017

zbMATH: 1342.53110
MathSciNet: MR3523044
Digital Object Identifier: 10.2140/agt.2016.16.1403

Primary: 53D35 , 57R17
Secondary: 57R45

Keywords: broken Lefschetz fibrations , folds , near-symplectic forms , singular symplectic forms , singularities , stable Hamiltonian structures

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.16 • No. 3 • 2016
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