Asymptotically Locally Euclidean metrics with holonomy SU(m)
Abstract
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 Ricciflat 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 Ricciflat 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 8manifolds with exceptional holonomy. They can also be used to describe the CalabiYau metrics on resolutions of a CalabiYau orbifold. The paper has a sequel, "QuasiALE metrics with holonomy SU(m) and Sp(m)", math.AG/9905043, which studies Kahler metrics on resolutions of nonisolated singularities C^m/G.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 1999
 arXiv:
 arXiv:math/9905041
 Bibcode:
 1999math......5041J
 Keywords:

 Algebraic Geometry;
 Differential Geometry
 EPrint:
 23 pages, LaTeX, uses packages amstex and amssymb