QuasiALE 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 Ricciflat QALE Kahler metrics on X. We prove an existence result for Ricciflat 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 Ricciflat 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 8manifolds with exceptional holonomy. They can also be used to describe the CalabiYau metrics on resolutions of a CalabiYau orbifold.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 1999
 arXiv:
 arXiv:math/9905043
 Bibcode:
 1999math......5043J
 Keywords:

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