Quasi-ALE metrics with holonomy SU(m) and Sp(m)

Abstract

This is the sequel to "Asymptotically Locally Euclidean metrics with holonomy SU(m)", math.AG/9905041. Let G be a subgroup of U(m), and X a resolution of C^m/G. We define a special class of Kahler metrics g on X called Quasi Asymptotically Locally Euclidean (QALE) metrics. These satisfy a complicated asymptotic condition, implying that g is asymptotic to the Euclidean metric on C^m/G away from its singular set. When C^m/G has an isolated singularity, QALE metrics are just ALE metrics. Our main interest is in Ricci-flat QALE Kahler metrics on X. We prove an existence result for Ricci-flat QALE Kahler metrics: if G is a subgroup of SU(m) and X a crepant resolution of C^m/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 QALE manifolds. We also determine the holonomy group of the metrics in terms of G. These results will be applied in the author's book ("Compact manifolds with special holonomy", to be published by OUP, 2000) to construct new examples of compact 7- and 8-manifolds with exceptional holonomy. They can also be used to describe the Calabi-Yau metrics on resolutions of a Calabi-Yau orbifold.