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Einstein metrics and complex singularities

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Abstract

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 K X is trivial), and that if K X 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.

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Authors and Affiliations

  1. School of Mathematics, University of Edinburgh, King’s Buildings, Mayfield Road, EH9 3JZ, Edinburgh, UK

    David M.J. Calderbank & Michael A. Singer

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  1. David M.J. Calderbank
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  2. Michael A. Singer
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Correspondence to David M.J. Calderbank or Michael A. Singer.

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Calderbank, D., Singer, M. Einstein metrics and complex singularities. Invent. math. 156, 405–443 (2004). https://doi.org/10.1007/s00222-003-0344-1

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