Existence of HKT metrics on hypercomplex manifolds of real dimension 8

A hypercomplex manifold M is a manifold equipped with three complex structures I,J,K satisfying quaternionic relations. Such a manifold admits a canonical torsion-free connection preserving the quaternion action, called the Obata connection. A quaternionic Hermitian metric is a Riemannian metric which is invariant with respect to unitary quaternions. Such a metric is called hyperkähler with torsion (HKT for short) if it is locally obtained as the Hessian of a function averaged with quaternions. An HKT metric is a natural analogue of a Kähler metric on a complex manifold. We push this analogy further, proving a quaternionic analogue of the result of Buchdahl and of Lamari that a compact complex surface M admits a Kähler structure if and only if b1(M) is even. We show that a hypercomplex manifold M with the Obata holonomy contained in SL(2,H) admits an HKT structure if and only if H1(O(M,I)) is even-dimensional.