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 the (n-1)-th power of the 2-form 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 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 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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:620811 |
| Date | January 2014 |
| Creators | Vera-Sanchez, Ramon |
| Publisher | Durham University |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://etheses.dur.ac.uk/10778/ |
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