Hypercomplex Structures Associated To Quaternionic Manifolds

ArticleinDifferential Geometry and its Applications 9(3):273-292 · December 1998with 29 Reads 
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Abstract
If M is a quaternionic manifold and P is an S 1 -instanton over M , then Joyce constructed a hypercomplex manifold we call P(M ) over M . These hypercomplex manifolds admit a U(2)-action of a special type permuting the complex structures. We show that up to double covers, all such hypercomplex manifolds arise in this way. Examples, including that of a hypercomplex structure on SU(3), show the necessity of including double covers of P(M ). 1. Introduction In [13], it was shown that over any quaternionic Kahler manifold there is a hyperK ahler manifold with an isometric action of SO(3) rotating the complex structures. The hyperKahler manifolds which arise this way were characterised by the properties of this action and it was shown how to recover the original quaternionic Kahler manifold via a moment map construction. The purpose of this article is to study similar constructions and results for quaternionic and hypercomplex structures---geometries that do not involve Riemannian metr...

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