(Pluri)harmonic morphisms and the Penrose-Ward transform

TL;DR

This paper explores the relationship between quaternionic geometry, the Ward transform, and the Penrose transform, demonstrating how these tools can be used to establish the existence of hypercomplex manifolds and harmonic morphisms.

Contribution

It reveals the functorial nature of the Ward transform in quaternionic geometry and applies it alongside the Penrose transform to derive new existence results.

Findings

Establishes the functoriality of the Ward transform in quaternionic geometry.

Demonstrates the application of the Penrose transform to hypercomplex manifolds.

Provides existence results for harmonic morphisms from hyper-Kaehler manifolds.

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 hypercomplex manifolds and for harmonic morphisms from hyper-Kaehler manifolds.