LAUSR

Abstract 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 hyperkahler 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 Kahler 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 Kahler structure if and only if b 1 ( M ) is even. We show that a hypercomplex manifold M with the Obata holonomy contained in S L ( 2 , H ) admits an HKT structure if and only if H 1 ( O ( M , I ) ) is even-dimensional.