Invariants of almost complex and almost Kähler manifolds

Let $(M^{2n},J)$ be a compact almost complex manifold. The almost complex invariant $h_J^{p,q}$ is defined as the complex dimension of the cohomology space $\{\left[\alpha \right]\in H_{dR}^{p+q}(M^{2n};\mathbb{C})\mathrm{\mid }\alpha \in A^{p,q}(M^{2n}),d\alpha =0\}$. When $2n=4$, it has many interesting properties. Endow $(M^{2n},J)$ with an almost Hermitian metric $g$. The number $h_d^{p,q}$, i.e., the complex dimension of the space of Hodge-de Rham harmonic $(p,q)$-forms, is almost Kähler invariant when $2n=4$. In this paper we study the relationship between $h_J^{p,q}$ and $h_d^{p,q}$ in dimension $2n\ge 4$. We prove $h_J^{n,0}=0$ if $J$ is non integrable and show that $h_d^{p,0}$ is almost Kähler invariant. If $M^{2n}$ is a compact quotient of a completely solvable Lie group and $(J,g,\omega )$ is left invariant, we find information also on $h_d^{1,1}$. Finally we study the ${\mathcal{C}}^{\mathrm{\infty }}$-pure and ${\mathcal{C}}^{\mathrm{\infty }}$-full properties of $J$ on $n$-forms for the special dimension $2n=4m$.