Abstract
Let G be a finite subgroup of U(m),and X a resolution of ℂm/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 ℂm/G away fromits singular set. When ℂm/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 ℂm/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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References
Bando, S. and Kobayashi, R.: Ricci-Flat Kähler Metrics on Affine Algebraic Manifolds, Lecture Notes in Math. 1339, Springer-Verlag, New York, 1988, pp. 20-31.
Bando, S. and Kobayashi, R.: Ricci-flat Kähler metrics on affine algebraic manifolds. II, Math. Ann. 287 (1990), 175-180.
Beauville, A.: Variétés Kählériennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), 755-782.
Calabi, E.: On Kähler manifolds with vanishing canonical class, in: R. H. Fox, D. C. Spencer and A. W. Tucker (eds), Algebraic Geometry and Topology, a Symposium in Honour of S. Lefschetz, Princeton Univ. Press, Princeton, 1957, pp. 78-89
Eguchi, T. and Hanson, A. J.: Asymptotically flat solutions to Euclidean gravity, Phys. Lett. B 74 (1978), 249-251.
Gilbarg, D. and Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss. 224, Springer-Verlag, Berlin, 1977.
Joyce, D. D.: Compact 8-manifolds with holonomy Spin(7), Invent. Math. 123 (1996), 507-552.
Joyce, D. D.: Compact Riemannian 7-manifolds with holonomy G 2. I and II, J. Differential Geom. 43 (1996), 291-328 and 329-375.
Joyce, D. D.: Hypercomplex algebraic geometry, Quart. J. Math. Oxford 49 (1998), 129-162.
Joyce, D. D.: Compact Manifolds with Special Holonomy, Oxford Math. Monographs, Oxford University Press, Oxford, 2000.
Joyce, D.: Asymptotically locally Euclidean metrics with holonomy SU(m), Ann. Global Anal. Geom. 19, 2001, 55-73.
Kronheimer, P. B.: The construction of ALE spaces as hyperkähler quotients, J. Differential Geom. 29 (1989), 665-683.
Reid, M.: Young person's guide to canonical singularities, in: S. J. Bloch (ed.), Algebraic Geometry, Bowdoin 1985, Proc. Sympos. Pure Math. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 345-416.
Salamon, S. M.: Riemannian Geometry and Holonomy Groups, Pitman Res. Notes in Math. 201, Longman, Harlow, 1989.
Tian, G. and Yau, S.-T.: Complete Kähler manifolds with zero Ricci curvature. I, J. Amer.Math. Soc. 3 (1990), 579-609.
Tian, G. and Yau, S.-T.: Complete Kähler manifolds with zero Ricci curvature. II, Invent. Math. 106 (1991), 27-60.
Yau, S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equations. I, Comm. Pure Appl. Math. 31 (1978), 339-411.
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Joyce, D. Quasi-ALE Metrics with Holonomy SU(m) and Sp(m). Annals of Global Analysis and Geometry 19, 103–132 (2001). https://doi.org/10.1023/A:1010778214851
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DOI: https://doi.org/10.1023/A:1010778214851