Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry

A nilmanifold is a quotient of a nilpotent group G by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a G-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of G-invariant complex structures which satisfy quaternionic relations). We prove that a hypercomplex nilmanifold admits an HKT (hyperkhler with torsion) metric if and only if the underlying hypercomplex structure is abelian. Moreover, any G-invariant HKT-metric on a nilmanifold is balanced with respect to all associated complex structures.