Toric structures on near-symplectic 4-manifolds

Gay, David T., and Symington, Margaret. "Toric structures on near-symplectic 4-manifolds." Journal of the European Mathematical Society 011.3 (2009): 487-520. <http://eudml.org/doc/277606>.

@article{Gay2009,
abstract = {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.},
author = {Gay, David T., Symington, Margaret},
journal = {Journal of the European Mathematical Society},
keywords = {symplectic; near-symplectic; toric; torus action; four-manifold; Hamiltonian; Lagrangian fibration; symplectic; near-symplectic; toric; torus action; 4-manifold; Hamiltonian; Lagrangian fibration},
language = {eng},
number = {3},
pages = {487-520},
publisher = {European Mathematical Society Publishing House},
title = {Toric structures on near-symplectic 4-manifolds},
url = {http://eudml.org/doc/277606},
volume = {011},
year = {2009},
}

TY - JOUR
AU - Gay, David T.
AU - Symington, Margaret
TI - Toric structures on near-symplectic 4-manifolds
JO - Journal of the European Mathematical Society
PY - 2009
PB - European Mathematical Society Publishing House
VL - 011
IS - 3
SP - 487
EP - 520
AB - 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.
LA - eng
KW - symplectic; near-symplectic; toric; torus action; four-manifold; Hamiltonian; Lagrangian fibration; symplectic; near-symplectic; toric; torus action; 4-manifold; Hamiltonian; Lagrangian fibration
UR - http://eudml.org/doc/277606
ER -