Hodge decomposition and Hard Lefschetz Condition on almost K\"{a}hler manifolds

TL;DR

This paper investigates harmonic forms on almost Kähler manifolds, showing that under certain conditions, they admit a Hodge decomposition and satisfy the Hard Lefschetz Condition, with implications for the structure and rigidity of these manifolds.

Contribution

It establishes conditions under which harmonic forms on almost Kähler manifolds admit a Hodge decomposition and satisfy the Hard Lefschetz Condition, extending classical results to a broader setting.

Findings

Hodge decomposition holds under certain non-integrability conditions.

Almost complex structures are shown to be complex pure-and-full.

Rigidity results for 4-dimensional almost Kähler manifolds with specific Betti number conditions.

Abstract

In this article, we discuss the spaces of harmonic forms $\mathcal{H}^{\bullet}_{d}$ over a closed almost K\"{a}hler manifold $(X, J,\omega)$. We show that if the almost complex structure $J$ on the almost K\"{a}hler manifold $X$ is not too non-integrable in some sense, then the spaces $\mathcal{H}^{\bullet}_{d}$ have the Hodge decomposition $\mathcal{H}^{k}_{d}=\oplus_{p+q=k}\mathcal{H}^{p,q}_{d}$. As a consequence, the not too non-integrable almost complex structure $J$ is complex $C^{\infty}$-pure-and-full, and the Hard Lefschetz Condition (HLC) on $\mathcal{H}^{\bullet}_{d}$ is satisfied. Moreover, we can prove a rigidity result for the closed $4$-dimensional almost K\"{a}hler manifold with $b^{+}_{2}(X)\geq2$.