nearsymplectic

Near-symplectic refers to a generalization of symplectic geometry in four dimensions, defined to extend the toolbox of symplectic techniques to manifolds that do not admit a global symplectic form. A near-symplectic form on a closed oriented 4-manifold X is a closed 2-form ω whose square ω ∧ ω is positive on X outside a specified singular set, and which vanishes along a smoothly embedded 1-dimensional submanifold Z ⊂ X. On X \ Z, ω is nondegenerate and hence symplectic, while along Z the degeneracy is controlled in a precise, transverse way.

The singular set Z consists of a disjoint union of circles, and near each component of Z the form has a prescribed local behavior that keeps the degeneracy manageable. In particular, the kernel of ω along Z is one-dimensional and tangent to Z, so the degeneracy is one-dimensional rather than appearing in higher codimensions. This structure allows one to retain many features of symplectic geometry away from Z while accommodating the degeneracy along Z.

Near-symplectic forms have been used to study smooth structures on 4-manifolds, linking symplectic methods to gauge theoretic invariants such as Seiberg–Witten invariants. They provide a framework for applying symplectic techniques in cases where a genuine symplectic form cannot exist globally, and they play a role in Taubes’s work relating Seiberg–Witten theory and Gromov-type invariants. Existence results typically involve constructive perturbations and considerations of the manifold’s topology, particularly in dimension four.

See also: symplectic geometry, Seiberg–Witten invariants, Taubes, four-manifolds.