Let $(M^{ 2n},J)$ be a compact almost complex manifold. The almost complex invariant $h^{ p,q}_J$ is defined as the complex dimension of the cohomology space $\left\{ \left[\alpha\right]\in H^{ p+q}_{ dR}(M^{ 2n};\mathbb{ C}) \,\vert\,\alpha\in A^{ p,q}(M^{ 2n}),\, d\alpha = 0 \right\}$. When $2n=4$, it has many interesting properties. Endow $(M^{ 2n},J)$ with an almost Hermitian metric $g$. The number $h^{ p,q}_d$, i.e., the complex dimension of the space of Hodge-de Rham harmonic $(p,q)$-forms, is almost K\"ahler invariant when $2n=4$. In this paper we study the relationship between $h^{ p,q}_J$ and $h^{ p,q}_d$ in dimension $2n\ge4$. We prove $h^{ n,0}_J=0$ if $J$ is non integrable and show that $h^{ p,0}_d$ is almost K\"ahler 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 $h^{ 1,1}_d$. Finally we study the $\mathcal{ C}^\infty$-pure and $\mathcal{ C}^\infty$-full properties of $J$ on $n$-forms for the special dimension $2n=4m$.
Categories: math.DG math.SG