Invariants of almost complex and almost Kähler manifolds

Given a compact almost complex manifold (M2n, J), the almost complex invariant hp,qJ is defined as the complex dimension of the cohomology space {[α] ∈ Hp+q dR (M2n; C) | α ∈ Ap,q(M2n), dα = 0}. Its properties have been studied mainly when 2n= 4. If we endow (M2n, J) with an almost Hermitian metric g, then the number hp,qd , i.e. the complex dimension of the space of Hodge-de Rham harmonic (p, q)-forms, does not depend on the choice of almost Kähler metrics when 2n= 4. In this paper, we study the relationship between hp,qJ and hp,qd in dimension 2n≥4. We prove hn,0 J = 0 if J is non-integrable and observe that hp,0 d = hp,0 J if the metric is almost Kähler. If M2nis a compact quotient of a completely solvable Lie group and (J, g, ω) is a left-invariant almost Kähler structure on M, we prove h1,1d = h1,1J . Finally, we study the C∞-pure and C∞-full properties of J on n-forms for the special dimension 2n= 4m.