Deformation theory of scalar-flat K\"ahler ALE surfaces

TL;DR

This paper develops a deformation theory for scalar-flat K"ahler ALE surfaces, establishing a local moduli space of such metrics and proving stability of scalar-flat K"ahler structures under small complex deformations.

Contribution

It proves a Kuranishi-type theorem for ALE K"ahler surfaces and constructs a universal local moduli space of scalar-flat K"ahler ALE metrics.

Findings

Deformation stability of scalar-flat K"ahler ALE metrics.

Construction of a local moduli space for these metrics.

Dimension formula for moduli space in specific cases.

Abstract

We prove a Kuranishi-type theorem for deformations of complex structures on ALE K\"ahler surfaces. This is used to prove that for any scalar-flat K\"ahler ALE surface, all small deformations of complex structure also admit scalar-flat K\"ahler ALE metrics. A local moduli space of scalar-flat K\"ahler ALE metrics is then constructed, which is shown to be universal up to small diffeomorphisms (that is, diffeomorphisms which are close to the identity in a suitable sense). A formula for the dimension of the local moduli space is proved in the case of a scalar-flat K\"ahler ALE surface which deforms to a minimal resolution of $\mathbb{C}^2/\Gamma$, where $\Gamma$ is a finite subgroup of ${\rm{U}}(2)$ without complex reflections.