Invariants of almost complex and almost Kähler manifolds

Given a compact almost complex manifold (M-2n,J), the almost complex invariant h(J)(p,q) is defined as the complex dimension of the cohomology space {[alpha] is an element of H-dR(p+q)(M-2n; C)|alpha is an element of A(p,q)(M-2n),d alpha = 0}. Its properties have been studied mainly when 2n = 4. If we endow (M2n,J) with an almost Hermitian metric g, then the number h(d)(p,q), i.e. the complex dimension of the space of Hodge-de Rham harmonic (p,q)-forms, does not depend on the choice of almost Kahler metrics when 2n = 4. In this paper, we study the relationship between h(J)(p,q) and h(d)(p,q) in dimension 2n >= 4. We prove hJn,0 = 0 if J is non-integrable and observe that h(d)(p,0) = h(J)(p,0) if the metric is almost Kahler. If M-2n is a compact quotient of a completely solvable Lie group and (J,g,omega) is a left-invariant almost Kahler structure on M, we prove h(d)(1,1) = h(J)(1,1). Finally, we study the C-infinity-pure and C-infinity-full properties of J on n-forms for the special dimension 2n = 4m.