Near-symplectic 2n–manifolds

Vera, Ramón¹

¹Department 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

Algebraic and Geometric Topology, Tome 16 (2016) no. 3, pp. 1403-1426

Résumé

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.

Détail

DOI : 10.2140/agt.2016.16.1403

Classification : 53D35, 57R17, 57R45
Keywords: near-symplectic forms, broken Lefschetz fibrations, stable Hamiltonian structures, singular symplectic forms, folds, singularities

Bibliographie

[1] S Akbulut, Ç Karakurt, Every 4–manifold is BLF, J. Gökova Geom. Topol. GGT 2 (2008) 83

[2] D Auroux, S K Donaldson, L Katzarkov, Singular Lefschetz pencils, Geom. Topol. 9 (2005) 1043

[3] R İ Baykur, Existence of broken Lefschetz fibrations, Int. Math. Res. Not. 2008 (2008)

[4] R İ Baykur, Topology of broken Lefschetz fibrations and near-symplectic four-manifolds, Pacific J. Math. 240 (2009) 201

[5] K Cieliebak, E Volkov, First steps in stable Hamiltonian topology, J. Eur. Math. Soc. (JEMS) 17 (2015) 321

[6] S K Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999) 205

[7] D T Gay, R Kirby, Constructing symplectic forms on 4–manifolds which vanish on circles, Geom. Topol. 8 (2004) 743

[8] D T Gay, R Kirby, Constructing Lefschetz-type fibrations on four-manifolds, Geom. Topol. 11 (2007) 2075

[9] M Golubitsky, V Guillemin, Stable mappings and their singularities, 14, Springer (1973)

[10] R E Gompf, A I Stipsicz, 4–manifolds and Kirby calculus, 20, Amer. Math. Soc. (1999)

[11] K Honda, Local properties of self-dual harmonic 2–forms on a 4–manifold, J. Reine Angew. Math. 577 (2004) 105

[12] K Honda, Transversality theorems for harmonic forms, Rocky Mountain J. Math. 34 (2004) 629

[13] Y Lekili, Wrinkled fibrations on near-symplectic manifolds, Geom. Topol. 13 (2009) 277

[14] T Perutz, Zero-sets of near-symplectic forms, J. Symplectic Geom. 4 (2006) 237

[15] T Perutz, Lagrangian matching invariants for fibred four-manifolds, I, Geom. Topol. 11 (2007) 759

[16] C H Taubes, The structure of pseudo-holomorphic subvarieties for a degenerate almost complex structure and symplectic form on S1 × B3, Geom. Topol. 2 (1998) 221

[17] K Wehrheim, C Woodward, Pseudoholomorphic quilts, preprint (2015)