Symplectic representatives in higher dimensions

Determine whether a cohomology class a\in H^2(M,\mathbb{R}) satisfying a^m\neq 0 on a closed almost complex manifold (M,J) of dimension 2m>4 can be represented by a symplectic form.

Background

The paper considers the fundamental symplectic-topology problem of deciding when a prescribed degree-two cohomology class on an even-dimensional manifold admits a symplectic representative. For a closed almost complex manifold of dimension 2m, the paper identifies the existence of an almost complex structure and the nonvanishing condition a^m\neq 0 as necessary conditions.

Although four-dimensional counterexamples are known through Seiberg–Witten theory, the paper states that the corresponding existence problem in dimensions higher than four remains unresolved. The proposed higher-dimensional gauge-theoretic equations are motivated in part by the goal of producing obstructions to compatible symplectic forms.