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Einstein metrics and complex singularities

Calderbank, D.M.J. and Singer, M.A. (2003) Einstein metrics and complex singularities. Inventiones Mathematicae, 156 (2). pp. 405-443. ISSN 1432-1297

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This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkähler gravitational instantons, but we focus on a different class of singularities. We show that any resolution X of an isolated cyclic quotient singularity admits a complete scalar-flat Kähler metric (which is hyperkähler if and only if KX is trivial), and that if KX is strictly nef, then X also admits a complete (non-Kähler) self-dual Einstein metric of negative scalar curvature. In particular, complete self-dual Einstein metrics are constructed on simply-connected non-compact 4-manifolds with arbitrary second Betti number. Deformations of these self-dual Einstein metrics are also constructed: they come in families parameterized, roughly speaking, by free functions of one real variable. All the metrics constructed here are toric (that is, the isometry group contains a 2-torus) and are essentially explicit. The key to the construction is the remarkable fact that toric self-dual Einstein metrics are given quite generally in terms of linear partial differential equations on the hyperbolic plane.

Item Type: Article
Institution: The University of York
Academic Units: The University of York > Mathematics (York)
Depositing User: York RAE Import
Date Deposited: 29 May 2009 11:41
Last Modified: 29 May 2009 11:41
Published Version: http://dx.doi.org/10.1007/s00222-003-0344-1
Status: Published
Publisher: Springer Verlag (Germany)
Refereed: Yes
Identification Number: 10.1007/s00222-003-0344-1
URI: http://eprints.whiterose.ac.uk/id/eprint/6456

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