HyperKahler and quaternionic Kahler manifolds with S^1-symmetries

TL;DR

This paper explores the relationships between hyperKahler and quaternionic Kahler manifolds with S^1-symmetries, providing constructions and properties that connect different types of these manifolds.

Contribution

It demonstrates how hyperKahler manifolds with specific symmetries can be constructed from lower-dimensional examples and analyzes the structure of quaternionic Kahler manifolds with circle actions.

Findings

HyperKahler manifolds with S^1-symmetry can be derived from lower-dimensional hyperKahler manifolds.

Positive quaternionic Kahler manifolds with S^1 symmetry admit a Kahler metric on a dense subset.

The study establishes relations between different symmetry types in quaternionic Riemannian manifolds.

Abstract

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.