Hypercomplex structures associated to quaternionic manifolds (Q1295472)

If \(M\) is a quaternionic manifold and \(P\) is an \(S^1\)-instanton over \(M\), then \textit{D. D. Joyce} [J. Differ. Geom. 35, 743-761 (1992; Zbl 0735.53050)] constructed a hypercomplex manifold we call \({\mathcal P}(M)\) 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 \({\mathcal P}(M)\).