TY - JOUR

T1 - Near-symplectic 2n-manifolds

AU - Vera, Ramón

N1 - Publisher Copyright:
© 2016, Mathematical Sciences Publishers. All rights reserved.

PY - 2016/7/1

Y1 - 2016/7/1

N2 - 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 ωn-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 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.

AB - 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 ωn-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 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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U2 - 10.2140/agt.2016.16.1403

DO - 10.2140/agt.2016.16.1403

M3 - Article

AN - SCOPUS:84978966812

SN - 1472-2747

VL - 16

SP - 1403

EP - 1426

JO - Algebraic and Geometric Topology

JF - Algebraic and Geometric Topology

IS - 3

ER -