HyperKähler and quaternionic Kähler manifolds with S1 -symmetries

Abstract

We study relations between quaternionic Riemannian manifolds admitting different types of symmetries. We show that any hyperKähler manifold admitting hyperKähler potential and triholomorphic action of S 1 can be constructed from another hyperKähler manifold (of lower dimension) with an action of S 1 that fixes one complex structure and rotates the other two and vice versa. We also study the corresponding quaternionic Kähler manifolds equipped with a quaternionic Kähler action of the circle. In particular we show that any positive quaternionic Kähler manifolds with S 1 -symmetry admits a Kähler metric on an open everywhere dense subset.