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 \(\overline{J}\)-invariant and the \(\overline{J}\)-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 \(\overline{J}\)-invariant and the \(\overline{J}\)-anti-invariant subgroups, together with their analogue in the Bott-Chern cohomology. For instance, in real dimension 8, we characterize the existence of hyperkähler with torsion metrics in terms of the dimension of the \(\overline{J}\)-invariant subgroup. We also study the existence of special hypercomplex structures on almost abelian solvmanifolds.