Hodge decomposition and Hard Lefschetz Condition on almost Kähler manifolds

In this article, we discuss the spaces of harmonic forms ${\mathcal{H}}_d^{\bullet }$ over a closed almost Kähler manifold $(X,J,\omega )$. We show that if the almost complex structure $J$ on the almost Kähler manifold $X$ is not too non-integrable in some sense, then the spaces ${\mathcal{H}}_d^{\bullet }$ have the Hodge decomposition ${\mathcal{H}}_d^k={\oplus }_{p+q=k}{\mathcal{H}}_d^{p,q}$. As a consequence, the not too non-integrable almost complex structure $J$ is complex $C^{\mathrm{\infty }}$-pure-and-full, and the Hard Lefschetz Condition (HLC) on ${\mathcal{H}}_d^{\bullet }$ is satisfied. Moreover, we can prove a rigidity result for the closed $4$-dimensional almost Kähler manifold with $b_2^{+}(X)\ge 2$.