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Sasakian Geometry$
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Charles Boyer and Krzysztof Galicki

Print publication date: 2007

Print ISBN-13: 9780198564959

Published to Oxford Scholarship Online: January 2008

DOI: 10.1093/acprof:oso/9780198564959.001.0001

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Quaternionic Kähler and Hyperkähler Manifolds

Quaternionic Kähler and Hyperkähler Manifolds

(p.421) Chapter 12 Quaternionic Kähler and Hyperkähler Manifolds
Sasakian Geometry

Charles P. Boyer

Krzysztof Galicki

Oxford University Press

This chapter gives an extensive overview of various quaternionic geometries. The main focus is on positive quaternionic Kähler manifolds (orbifolds) and on hyper Kähler manifolds (orbifolds). Various other quaternionic and hypercomplex geometries are introduced along the way. The hyper Kähler and quaternionic Kähler quotient construction is described. Other topics include the theory of toric hyper Kähler manifolds, the classification of positive toric self-dual and Einstein orbifolds, Hitchin's construction of SO(3)-invariant orbifold self-dual Einstein metrics on a 4-sphere, McKay's correspondence and Kronheimer's construction of ALE gravitational instantons.

Keywords:   quaternionic geometry, hypercomplex geometry, quaternionic Kähler orbifolds, hyper Kähler manifolds, twistor spaces, quaternionic Kähler reduction, hyper Kähler reduction, Hitchin orbifolds, Konishi orbibundle, self-dual Einstein orbifolds

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