Local moduli of scalar-flat K\"ahler ALE surfaces

TL;DR

This paper surveys the construction of local moduli spaces for scalar-flat Kähler ALE metrics in complex dimension two and provides an explicit formula for their dimension based on singularity properties.

Contribution

It introduces a detailed survey of the moduli space construction and derives a formula for its dimension related to the minimal resolution of quotient singularities.

Findings

Explicit dimension formula for moduli space

Construction of local moduli space of scalar-flat Kähler ALE metrics

Application to minimal resolutions of quotient singularities

Abstract

In this article, we give a survey of our construction of a local moduli space of scalar-flat K\"ahler ALE metrics in complex dimension $2$. We also prove an explicit formula for the dimension of this moduli space on a scalar-flat K\"ahler ALE surface which deforms to the minimal resolution of $\mathbb{C}^2/\Gamma$, where $\Gamma$ is a finite subgroup of ${\rm{U}}(2)$ without complex reflections, in terms of the embedding dimension of the singularity.