ALE Ricci-flat Kahler surfaces and weighted projective spaces

TL;DR

This paper demonstrates that explicit ALE Ricci-flat Kähler metrics, including those by Eguchi-Hanson and Kronheimer, can be constructed using Tian-Yau techniques through compactifications of certain surface singularities.

Contribution

It establishes a unifying framework connecting known ALE Ricci-flat Kähler metrics with Tian-Yau methods via compactifications of quotient surface singularities.

Findings

Explicit construction of ALE Ricci-flat Kähler metrics via Tian-Yau techniques

Identification of good compactifications of Q-Gorenstein deformations

Connection between classical metrics and modern algebraic geometry methods

Abstract

We show that the explicit ALE Ricci-flat Kahler metrics constructed by Eguchi-Hanson, Gibbons-Hawking, Hitchin and Kronheimer, and their free quotients are metrics obtained by Tian-Yau techniques. The proof relies on a construction of good compactifications of Q-Gorenstein deformations of quotient surface singularities as log del Pezzo surfaces with only cyclic quotient singularities at infinity.