The construction of ALE spaces as hyper-Kähler quotients.

*(English)*Zbl 0671.53045Asymptotically locally Euclidean 4-manifolds were introduced over ten years ago by relativists as a gravitational analogue of the instantons of gauge theory. Such a manifold is a Riemannian manifold with the asymptotic geometry of the quotient of \(R^ 4\) by a finite group \(\Gamma\). The positive action theorem showed that there were no ALE solutions to Einstein’s equations with \(\Gamma =0\) (the most natural analogue of instantons) other than flat space, but G. W. Gibbons and S. W. Hawking [Phys. Lett. B 78, 430–432 (1978)] produced self- dual examples where \(\Gamma\) is cyclic which were geometrically non- trivial.

In this very beautiful paper, the author constructs self-dual solutions for all finite subgroups of \(\mathrm{SU}(2)\subset \mathrm{SO}(4)\). As is well-known, these groups correspond not only to regular solids in \(R^ 3\), but to rational double points of algebraic surfaces and Dynkin diagrams of type A, D, E. These features all play a role in the author’s construction. The metrics are produced by the hyper-Kähler quotient construction [N. J. Hitchin, A. Karlhede, U. Lindström and M. Roček Commun. Math. Phys. 108, 535–589 (1987; Zbl 0612.53043)] which starts with an action of a Lie group on a quaternionic vector space. Here the vector space and group are canonically constructed from the regular representation of the finite group \(\Gamma\) and the complex 2-dimensional representation of \(\Gamma\) \(\subset SU(2)\). In working out the spaces involved, the McKay correspondence is used. This procedure produces 4- dimensional hyper-Kähler manifolds.

The next step the author takes is to identify these with the minimal resolutions of the corresponding quotient singularities \(C^ 2/\Gamma\), a process which involves the different complex structures which a hyper- Kähler metric possesses. Finally, the ALE property is established. One of the most attractive features of the construction is the explicit way in which it produces the well-known simultaneous resolution property of rational double points in a natural (differential-) geometric context, studied earlier by E. Brieskorn [Math. Ann. 166, 76-102 (1966; Zbl 0145.094)].

In this very beautiful paper, the author constructs self-dual solutions for all finite subgroups of \(\mathrm{SU}(2)\subset \mathrm{SO}(4)\). As is well-known, these groups correspond not only to regular solids in \(R^ 3\), but to rational double points of algebraic surfaces and Dynkin diagrams of type A, D, E. These features all play a role in the author’s construction. The metrics are produced by the hyper-Kähler quotient construction [N. J. Hitchin, A. Karlhede, U. Lindström and M. Roček Commun. Math. Phys. 108, 535–589 (1987; Zbl 0612.53043)] which starts with an action of a Lie group on a quaternionic vector space. Here the vector space and group are canonically constructed from the regular representation of the finite group \(\Gamma\) and the complex 2-dimensional representation of \(\Gamma\) \(\subset SU(2)\). In working out the spaces involved, the McKay correspondence is used. This procedure produces 4- dimensional hyper-Kähler manifolds.

The next step the author takes is to identify these with the minimal resolutions of the corresponding quotient singularities \(C^ 2/\Gamma\), a process which involves the different complex structures which a hyper- Kähler metric possesses. Finally, the ALE property is established. One of the most attractive features of the construction is the explicit way in which it produces the well-known simultaneous resolution property of rational double points in a natural (differential-) geometric context, studied earlier by E. Brieskorn [Math. Ann. 166, 76-102 (1966; Zbl 0145.094)].

Reviewer: N.Hitchin

##### MSC:

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |

14J17 | Singularities of surfaces or higher-dimensional varieties |