Harmonic morphisms and the Penrose–Ward transform

Abstract

We show that, in quaternionic geometry, the Ward transform is a manifestation of the functoriality of the basic correspondence between the $\rho$-quaternionic manifolds and their twistor spaces. We apply this fact, together with the Penrose transform, to obtain existence results for quaternionic manifolds and for harmonic morphisms from hyper-Kähler manifolds.