On the twistor spaces of ALE gravitational instantons of type $A_{\rm odd}$

Abstract

We study the twistor spaces of toric ALE gravitational instantons of type $A_{2n-1}$ and the associated non-standard minitwistor spaces introduced by Hitchin. By analyzing the base locus of the linear system that induces the quotient meromorphic map from the compactified twistor space, we explicitly determine the images of certain distinguished twistor lines as hyperplane sections of the minitwistor space. Using this family of special minitwistor lines as boundary data, we describe the $3$-dimensional family of real minitwistor lines arising from the instanton. The central sphere in the gravitational instanton appears naturally throughout the analysis.