If M is a quaternionic manifold and P is an S 1-instanton over M, then Joyce constructed a hypercomplex manifold we call(M) over M. These hypercomplex manifolds admit a U (2)-action of a special type permuting the complex structures. We show that up to double covers, all such hypercomplex manifolds arise in this way. Examples, including that of a hypercomplex structure on SU (3), show the necessity of including double covers of(M).

It is proved that if the twistor space of a self-dual four-manifold of positive scalar curvature contains a real effective divisor of degree two, then the four-manifold is diffeomorphic to the connected sum n\mathbb {C}{P^ 2} of n complex projective planes for some n. It follows that if the four-manifold is known to be homeomorphic to n\mathbb {C}{P^ 2} , then it is also diffeomorphic to n\mathbb {C}{P^ 2} .

We develop deformation theory for abelian invariant complex structures on a nilmanifold, and prove that in this case the invariance property is preserved by the Kuranishi process. A purely algebraic condition characterizes the deformations leading again to abelian structures, and we prove that such deformations are unobstructed. Various examples illustrate the resulting theory, and the behavior possible in three complex dimensions.

A geometric classification of the compact four-dimensional Einstein-Weyl manifolds with at least four-dimensional symmetry group is given. Our results also sharpen previous results on four-dimensional Einstein metrics and correct Parker's topological classification of cohomogeneity-one four-manifolds.

The internal space of a N = 4 supersymmetric model with WessZumino term has a connection with totally skew-symmetric torsion and holonomy in SP(<i>n</i>). We study the mathematical background of this type of connection. In particular, we relate it to classical Hermitian geometry, construct homogeneous as well as inhomogeneous examples, characterize it in terms of holomorphic data, develop its potential theory and reduction theory.