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Near-symplectic
$2n$–manifolds
Ramón Vera |
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Algebraic & Geometric Topology 16 (2016) 1403–1426
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Abstract |
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We give a generalization of the concept of near-symplectic structures to dimensions. According to our definition, a closed –form on a –manifold is near-symplectic if it is symplectic outside a submanifold of codimension where vanishes. We depict how this notion relates to near-symplectic –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 –manifold over a symplectic base of codimension , 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- singular locus . We describe a splitting property of the normal bundle that is also present in dimension four. A tubular neighbourhood theorem for is provided, which has a Darboux-type theorem for near-symplectic forms as a corollary. |
Keywords
near-symplectic forms, broken Lefschetz fibrations, stable
Hamiltonian structures, singular symplectic forms, folds,
singularities
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Mathematical Subject Classification 2010
Primary: 53D35, 57R17
Secondary: 57R45
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References |
Publication
Received: 12 August 2014
Revised: 19 August 2015
Accepted: 3 October 2015
Published: 1 July 2016
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Authors |
| 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 |
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