An Integrability Theorem for Almost-K\"ahler Structures using J-anti-invariant Two-Forms on Four-Manifolds

We establish a new criterion for a compatible almost complex structure on a symplectic four-manifold to be integrable and hence K\"ahler. Our main theorem shows that the existence of three linearly independent closed J-anti-invariant two-forms implies the integrability of the almost complex structure. This proves the conjecture of Draghici-Li-Zhang in the almost-K\"ahler case