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Advances in Geometry and Lie Algebras from Supergravity pp 447–551Cite as

(Hyper)Kähler Quotients, ALE-Manifolds and \(\mathbb {C}^n/\varGamma \) Singularities

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Abstract

In this last chapter we turn to the analysis of important developments in complex geometry which took place in the 1980–1990s, directly motivated by supersymmetry and supergravity and completely inconceivable outside such a framework. Notwithstanding their roots in the theoretical physics of the superworld, such developments constitute, by now, the basis of some of the most innovative and alive research directions of contemporary geometry.

Quando chel cubo con le cose appresso

Se agguaglia a qualche numero discreto

Trouan dui altri differenti in esso.

Niccoló Tartaglia

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Notes

  1. 1.

    The unitary induced representations of the \(\mathrm {Osp}(\mathscr {N}\mid 4)\) superalgebra in their double interpretation as gravitational multiplets or as multiplets of superconformal fields were discussed in [12] and have been systematically reviewed in Chap. 12 of the book [33].

  2. 2.

    This result was derived in private conversations of the author with Dimitry Markushevich.

  3. 3.

    A resolution of singularities \(X \rightarrow Y\) is crepant when the canonical bundle of X is the pullback of the canonical bundle of Y.

  4. 4.

    A variety is Gorenstein when the canonical divisor is a Cartier divisor, i.e., a divisor corresponding to a line bundle.

  5. 5.

    Following standard mathematical nomenclature, we call compatible connection on a holomorphic vector bundle, one whose (0, 1) part is the Cauchy Riemann operator of the bundle.

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  1. Pietra Marazzi, Alessandria, Italy

    Pietro Giuseppe Fré

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Fré, P.G. (2018). (Hyper)Kähler Quotients, ALE-Manifolds and \(\mathbb {C}^n/\varGamma \) Singularities. In: Advances in Geometry and Lie Algebras from Supergravity. Theoretical and Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-74491-9_8

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