Harmonic morphisms and the Penrose-Ward transform (Q2418840)

In this paper, the author gives a direct and general approach aiming to generalize the Gibbons-Hawking construction. At first, a quaternionic version of the Ward transform is given. The class of manifolds is that of \(\rho\)-quaternionic manifolds defined in a functorial way in Section 1. The construction is a bit too abstract for the flavour of the reviewer, however many explicit representations can be found in the proposed literature. After dealing with the Ward transform, the author shows that hypercomplex manifolds obtained through the Ward transform, admit compatible hyper-Kähler metrics and, such hyper-Kähler manifolds are domains of harmonic morphisms. In this context an Obata-type connection is introduced in relation, again, with the Ward transform.