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

Abstract

Let $${N_0={\mathbb C}^2/H}$$ be an isolated quotient singularity with $${H\subset U(2)}$$ a finite subgroup. We show that for any $${\mathbb Q}$$ -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 (J Differ Geom 29(3):685–697, 1989), 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.