On the Anti-invariant Cohomology of Almost Complex Manifolds

We study the space of closed anti-invariant forms on an almost complex manifold, possibly non-compact. We construct families of (non-integrable) almost complex structures on R4, such that the space of closed J-anti-invariant forms is infinite dimensional, and also 0- or 1-dimensional. In the compact case, we construct 6-dimensional almost complex manifolds with arbitrary large anti-invariant cohomology and a 2-parameter family of almost complex structures on the Kodaira–Thurston manifold whose anti-invariant cohomology group has maximum dimension.