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

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.