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

TL;DR

This paper investigates the structure of twistor spaces associated with toric ALE gravitational instantons of type A_{2n-1}, providing explicit descriptions of minitwistor lines and their geometric properties.

Contribution

It offers a detailed analysis of the base locus and images of twistor lines, and describes a family of real minitwistor lines for these instantons, extending understanding of their geometric structure.

Findings

Explicit determination of images of twistor lines as hyperplane sections.

Description of a 3-dimensional family of real minitwistor lines.

Natural appearance of the central sphere in the analysis.

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.