ALE Ricci-flat Kähler metrics and deformations of quotient surface singularities (Q658364)

The article under review deals with quotient orbifolds of the form \(\mathbb{C}^2 / H\) with \(H\subset U(2)\) a finite subgroup acting freely on \(\mathbb{C}^{2} \setminus \{0\}\). The author proves that for any \(\mathbb{Q}\)-Gorenstein smoothings of \(N_{0}:=\mathbb{C}^2 / H\) a nearby fiber admits ALE Ricci-flat Kähler metrics in any Kähler class. Moreover an explicit classification of the ALE Ricci-flat Kähler surfaces is given, so generalizing Kronheimer's results on hyper-Kähler \(4\)-manifolds.