Compact anti-self-dual orbifolds with torus actions

TL;DR

This paper classifies compact toric anti-self-dual conformal structures on 4-orbifolds using twistor theory, and also classifies certain ALE scalar-flat K"ahler orbifolds with torus symmetry.

Contribution

It provides a twistor-theoretic classification of toric anti-self-dual orbifolds and extends this to classify ALE scalar-flat K"ahler orbifolds with torus symmetry.

Findings

Classification of toric anti-self-dual conformal structures on compact 4-orbifolds.

Construction of compact anti-self-dual orbifolds from ALE scalar-flat K"ahler orbifolds.

Identification of ALE scalar-flat K"ahler 4-orbifolds with 2-torus symmetry.

Abstract

We give a classification of toric anti-self-dual conformal structures on compact 4-orbifolds with positive Euler characteristic. Our proof is twistor theoretic: the interaction between the complex torus orbits in the twistor space and the twistor lines induces meromorphic data, which we use to recover the conformal structure. A compact anti-self-dual orbifold can also be constructed by adding a point at infinity to an asymptotically locally Euclidean (ALE) scalar-flat K\"ahler orbifold. We use this observation to classify ALE scalar-flat K\"ahler 4-orbifolds whose isometry group contain a 2-torus.