On the invariant and anti-invariant cohomologies of hypercomplex manifolds

A hypercomplex structure $(I,J,K)$ on a manifold $M$ is said to be $C^\infty$-pure-and-full if the Dolbeault cohomology $H^{2,0}_{\partial}(M,I)$ is the direct sum of two natural subgroups called the $\bar{J}$-invariant and the $\bar{J}$-anti-invariant subgroups. We prove that a compact hypercomplex manifold that satisfies the quaternionic version of the $dd^c$-Lemma is $C^\infty$-pure-and-full. Moreover, we study the dimensions of the $\bar{J}$-invariant and the $\bar{J}$-anti-invariant subgroups, together with their analogue in the Bott-Chern cohomology. For instance, in real dimension 8, we characterize the existence of hyperk\"ahler with torsion metrics in terms of the dimension of the $\bar{J}$-invariant subgroup. We also study the existence of special hypercomplex structures on almost abelian solvmanifolds.