Quasi-ALE Metrics with Holonomy SU(m) and Sp(m)

Abstract

Let G be a finite subgroup of U(m),and X a resolution of ℂm/G. We define aspecial class of Kahler metrics g on Xcalled Quasi Asymptotically Locally Euclidean (QALE) metrics. Thesesatisfy a complicated asymptotic condition, implying that gis asymptotic to the Euclidean metric on ℂm/G away fromits singular set. When ℂm/Ghas an isolated singularity,QALE metrics are just ALE metrics. Our main result is an existencetheorem for Ricci-flat QALE Kahler metrics: if G is afinite subgroup of SU(m) and X a crepant resolution of ℂm/G, then there is a unique Ricci-flat QALE Kahler metric on X in each Kahler class.This is proved using a version of the Calabi conjecture for QALEmanifolds. We also determine the holonomy group of the metrics in termsof G.