ALE Ricci-flat Kahler metrics and deformations of quotient surface singularities

TL;DR

This paper proves that Q-Gorenstein smoothings of certain quotient surface singularities admit ALE Ricci-flat Kähler metrics, classifies these surfaces explicitly, and constructs new ALF Ricci-flat Kähler examples with specific geometric properties.

Contribution

It extends Kronheimer's classification to a broader class of singularities and constructs new ALF Ricci-flat Kähler manifolds with cyclic fundamental groups.

Findings

Nearby fibers admit ALE Ricci-flat Kähler metrics in any Kähler class.

Explicit classification of ALE Ricci-flat Kähler surfaces related to quotient singularities.

Construction of new ALF Ricci-flat Kähler manifolds with cubic volume growth.

Abstract

Let N_0 = C^2/H be an isolated quotient singularity with H in U (2) a finite subgroup. We show that for any Q-Gorenstein smoothings of N_0 a nearby fiber admits ALE Ricci-flat Kahler metrics in any Kahler class. Moreover, we generalize Kronheimer's results on hyperkahler 4-manifolds, by giving an explicit classification of the ALE Ricci-flat Kahler surfaces. We construct ALF Ricci-flat Kahler metrics on the above non-simply connected manifolds. These provide new examples of ALF Ricci-flat Kahler 4-manifolds, with cubic volume growth and cyclic fundamental group at infinity.