  Let G be a nontrivial finite subgroup of U(m) acting freely on C^m - 0. Then
C^m/G has an isolated quotient singularity at 0. Let X be a resolution of
C^m/G, and g a Kahler metric on X. We say that g is Asymptotically Locally
Euclidean (ALE) if it is asymptotic in a certain way to the Euclidean metric on
C^m/G.
  In this paper we study Ricci-flat ALE Kahler metrics on X. We show that if G
is a subgroup of SU(m) acting freely on C^m - 0, and X is a crepant resolution
of C^m/G, then there is a unique Ricci-flat ALE Kahler metric in each Kahler
class. This is proved using a version of the Calabi conjecture for ALE
manifolds. We also show the metrics have holonomy SU(m).
  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. The
paper has a sequel, "Quasi-ALE metrics with holonomy SU(m) and Sp(m)",
math.AG/9905043, which studies Kahler metrics on resolutions of non-isolated
singularities C^m/G.
