[Paper Review] Compact anti-self-dual orbifolds with torus actions

Dominic Wright|ArXiv.org|May 15, 2008

Geometry and complex manifolds13 references4 citations

TL;DR

This paper classifies compact anti-self-dual orbifolds with torus actions and positive orbifold Euler characteristic using twistor theory, showing they arise from Joyce's construction via ALE scalar-flat Kähler metrics. It establishes that such orbifolds are toric resolutions of $\mathbb{C}^2/\Gamma$ for cyclic $\Gamma$, and proves that ALE scalar-flat Kähler toric 4-orbifolds extend to compact anti-self-dual conformal structures via point compactification.

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 Kähler orbifold. We use this observation to classify ALE scalar-flat Kähler 4-orbifolds whose isometry group contain a 2-torus.

Motivation & Objective

• To classify compact anti-self-dual conformal structures on 4-orbifolds with torus actions and positive orbifold Euler characteristic.
• To extend Fujiki's classification of smooth anti-self-dual manifolds to the orbifold setting.
• To characterize ALE scalar-flat Kähler toric 4-orbifolds as arising from compact anti-self-dual orbifolds via point compactification.
• To establish a correspondence between meromorphic data on twistor lines and the conformal structure of the base orbifold.
• To show that scalar-flat Kähler metrics on toric 4-orbifolds are biholomorphic to toric resolutions of $\mathbb{C}^2/\Gamma$ for cyclic $\Gamma \subset U(2)$.

Proposed method

• Uses twistor theory to analyze the complex geometry of the twistor space associated with anti-self-dual conformal structures.
• Identifies complex torus orbits in the twistor space and uses their closure to define meromorphic data.
• Constructs a holomorphic involution and a pair of meromorphic functions on principal twistor lines to encode the conformal structure.
• Applies a correspondence between real torus-invariant divisors in the twistor space and torus orbits in the 4-orbifold.
• Uses moment map convexity and polytope geometry to relate the intersection form to the orbifold's topology.
• Applies the Chen-LeBrun-Weber procedure to compactify ALE scalar-flat Kähler metrics by adding a fixed point at infinity.

Experimental results

Research questions

• RQ1 Which compact 4-orbifolds with torus actions and positive orbifold Euler characteristic admit anti-self-dual conformal structures?
• RQ2 How do the meromorphic data on twistor lines reconstruct the conformal structure of an anti-self-dual 4-orbifold?
• RQ3 What is the relationship between ALE scalar-flat Kähler toric 4-orbifolds and compact anti-self-dual orbifolds via point compactification?
• RQ4 Under what conditions does a toric anti-self-dual orbifold arise from Joyce's construction?
• RQ5 Are scalar-flat Kähler metrics on toric 4-orbifolds biholomorphic to resolutions of $\mathbb{C}^2/\Gamma$ for cyclic $\Gamma \subset U(2)$?

Key findings

• Compact anti-self-dual orbifolds with torus actions and positive orbifold Euler characteristic are diffeomorphic to quotients of simply-connected 4-orbifolds by finite subgroups of the torus action.
• The intersection form on such orbifolds is negative-definite, a condition equivalent to the convexity of the moment polytope and the vanishing of the self-dual Weyl curvature.
• The conformal structure on the orbifold is equivalent to one constructed by Joyce via solutions to a system of PDEs on the half-plane.
• ALE scalar-flat Kähler toric 4-orbifolds that are ALE to order $l > 3/2$ arise as the complement of a fixed point in a compact anti-self-dual orbifold.
• Such ALE metrics are biholomorphic to toric resolutions of $\mathbb{C}^2/\Gamma$ for cyclic $\Gamma \subset U(2)$, and are isometric to those constructed by Calderbank and Singer.
• The anti-self-dual conformal structure on the compact orbifold is parameterized by the conformal structure of a surface obtained by gluing four copies of a manifold along boundary components.