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-infinity-pure-and-full if the Dolbeault cohomology H-partial derivative(2,0) (M, I) is the direct sum of two natural subgroups called the (J) over bar -invariant and the (J) over bar -anti-invariant subgroups. We prove that a compact hypercomplex manifold that satisfies the quaternionic version of the dd(c)-Lemma is C-infinity-pure-and-full. Moreover, we study the dimensions of the (J) over bar -invariant and the (J) over bar -anti-invariant subgroups, together with their analogue in the Bott-Chern cohomology. For instance, in real dimension 8, we characterize the existence of hyperkahler with torsion metrics in terms of the dimension of the (J) over bar -invariant subgroup. We also study the existence of special hypercomplex structures on almost abelian solvmanifolds.