Toric structures on near-symplectic 4-manifolds

  • Margaret Symington

    Mercer University, Macon, United States
  • David T. Gay

    University of Cape Town, Rondebosch, South Africa


A near-symplectic structure on a 4-manifold is a closed 2-form that is symplectic away from the 1-dimensional submanifold along which it vanishes and that satisfies a certain transversality condition along this vanishing locus. We investigate near-symplectic 4-manifolds equipped with singular Lagrangian torus fibrations which are locally induced by effective Hamiltonian torus actions. We show how such a structure is completely characterized by a singular integral affine structure on the base of the fibration whenever the vanishing locus is nonempty. The base equipped with this geometric structure generalizes the moment map image of a toric 4-manifold in the spirit of earlier work by the second author on almost toric symplectic 4-manifolds. We use the geometric structure on the base to investigate the problem of making given smooth torus actions on 4-manifolds symplectic or Hamiltonian with respect to near-symplectic structures and to give interesting constructions of structures which are locally given by torus actions but have nontrivial global monodromy.

Cite this article

Margaret Symington, David T. Gay, Toric structures on near-symplectic 4-manifolds. J. Eur. Math. Soc. 11 (2009), no. 3, pp. 487–520

DOI 10.4171/JEMS/157