Journal of the European Mathematical Society

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Volume 11, Issue 3, 2009, pp. 487–520
DOI: 10.4171/JEMS/157

Published online: 2009-06-30

Toric structures on near-symplectic 4-manifolds

David T. Gay[1] and Margaret Symington[2]

(1) University of Cape Town, Rondebosch, South Africa
(2) Mercer University, Macon, United States

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.

Keywords: Symplectic, near-symplectic, toric, torus action, four-manifold, Hamiltonian, Lagrangian fibration

Gay David, Symington Margaret: Toric structures on near-symplectic 4-manifolds. J. Eur. Math. Soc. 11 (2009), 487-520. doi: 10.4171/JEMS/157