HyperK�hler and quaternionic K�hler geometry

Abstract

 A quaternion-Hermitian manifold, of dimension at least 12, with closed fundamental 4-form is shown to be quaternionic Kahler. A similar result is proved for 8-manifolds. HyperKahler metrics are constructed on the fundamental quaternionic line bundle (with the zero-section removed) of a quaternionic Kahler manifold (indefinite if the scalar curvature is negative). This construction is compatible with the quaternionic Kahler and hyperKahier quotient constructions and allows quaternionic Kahler geometry to be subsumed into the theory of hyperKahler manifolds. It is shown that the hyperKahler metrics that arise admit a certain type of SU (2)- action, possess functions which are Kahler potentials for each of the complex structures simultaneously and determine quaternionic Kahler structures via a variant of the moment map construction. Quaternionic Kahler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperKahler metrics with circle actions on complex line bundles over Kahler-Einstein (complex) contact manifolds. Nilpotent orbits in a complex semi-simple Lie algebra, with the hyperKahler metrics defined by Kronheimer, are shown to give rise to quaternionic Kahler metrics and various examples of these metrics are identified. It is shown that any quaternionic Kahler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds. The twistor space structure of the projectivised nilpotent orbits is studied.