ALE gravitational instantons — asymptotically locally Euclidean solutions to the vacuum Einstein equations with self-dual curvature — come in families classified by Dynkin diagrams. The A-type instantons (multi-Eguchi-Hanson spaces) are the simplest: resolution of cyclic singularities, with a chain of 2-spheres as the exceptional set.
The paper studies their twistor spaces — complex 3-folds that encode the instanton geometry via holomorphic data. For A_{2n-1} type (odd-indexed), the twistor space has structure that the minitwistor space (a 2-dimensional reduction) makes visible. Distinguished twistor lines — special holomorphic curves in the twistor space corresponding to points of the instanton — map to hyperplane sections of the minitwistor space.
The central sphere in the instanton plays a recurring role. This is the middle sphere in the chain of exceptional curves — geometrically special because it sits at the center of the A-type resolution. In the twistor space, this sphere's image organizes the geometry: the base locus of the relevant linear system, the real structure, and the family of twistor lines all reference it.
The 3-dimensional family of real minitwistor lines — which encodes the metric data of the instanton — is characterized through these boundary conditions. The instanton's geometry, seemingly a Riemannian object, is fully captured by holomorphic data in the twistor space.
Geometry translated into complex analysis. The curvature condition (self-duality) becomes an integrability condition (holomorphic structure), and the physical space becomes a moduli space of curves.