On the classification of ALE K\"ahler manifolds

TL;DR

This paper studies ALE Kähler manifolds, showing their complex structures relate to resolutions of quotient singularities, leading to finiteness results for their diffeomorphism types with fixed asymptotic groups.

Contribution

It establishes a link between ALE Kähler manifolds and resolutions of singularities, proving finiteness of diffeomorphism types for minimal ALE Kähler surfaces with a given group at infinity.

Findings

Complex structure as a resolution of a quotient singularity

Finiteness of diffeomorphism types with fixed group at infinity

Existence of only finitely many minimal ALE Kähler surfaces for each asymptotic group

Abstract

The underlying complex structure of an ALE K\"ahler manifold is exhibited as a resolution of a deformation of an isolated quotient singularity. As a consequence, there exist only finitely many diffeomorphism types of minimal ALE K\"ahler surfaces with a given group at infinity.