Consider a smooth compact oriented 4-manifold $X$. Although not all 4-manifolds admit a spin structure, they do admit spin-c structures. And if $X$ does admit a spin structure, then there is a canonical spin-c structure. Now if instead $X$ is equipped with a symplectic form $\omega$ then there is also a canonical spin-c structure (corresponding to the spinor bundle which splits with summand $K^{-1}$, the canonical determinant line bundle). This is nice, turning the bijection $Spin^c(X)\simeq H^2(X;\mathbb{Z})$ into a canonical isomorphism.

I would like to question what goes on when I relax $\omega$ to be a *near-symplectic* form, i.e. a closed 2-form which is symplectic away from its zero-set $Z$ (a finite disjoint union of embedded circles). I still get a bijection $Spin^c(X)\simeq \lbrace \sigma\in H_2(X,Z)\;|\;\partial\sigma=[Z]\in H_1(Z)\rbrace$, but not necessarily canonical.

**Are there any conditions I can put on $(X,\omega)$ in order to have a natural origin?**