HyperKahler and quaternionic Kahler manifolds with S^1-symmetries

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We study relations between quaternionic Riemannian manifolds admitting different types of symmetries.

• We show that any hyperKahler manifold admitting hyperKahler potential and triholomorphic action of S^1 can be constructed from another hyperKahler manifold (of lower dimention) with an action of S^1 which fixes one complex structure and rotates the other two and vice versa.
• We also study corresponding quaternionic Kahler manifolds equipped with a quaternionic Kahler action of the circle.
• In particular we show that any positive quaternionic Kahler manifold with S^1 symmetry admits a Kahler metric on an open everywhere dense subset.