Quasi-ALE Metrics with Holonomy SU(m) and Sp(m) |
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Authors: | Dominic Joyce |
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Institution: | (1) Lincoln College, Turl Street, Oxford, OX1 3DR, England |
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Abstract: | Let G be a finite subgroup of U(m),and X a resolution of
m
/G. We define aspecial class of Kähler metrics g on Xcalled Quasi Asymptotically Locally Euclidean (QALE) metrics. Thesesatisfy a complicated asymptotic condition, implying that gis asymptotic to the Euclidean metric on
m
/G away fromits singular set. When
m
/Ghas an isolated singularity,QALE metrics are just ALE metrics. Our main result is an existencetheorem for Ricci-flat QALE Kähler metrics: if G is afinite subgroup of SU(m) and X a crepant resolution of
m
/G, then there is a unique Ricci-flat QALE Kähler metric on X in each Kähler 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. |
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Keywords: | asymptotically locally Euclidean Calabi conjecture Ricci-flat |
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