Abstract
Using quaternionic Feix–Kaledin construction, we provide a local classification of quaternion-Kähler metrics with a rotating S^1-symmetry with the fixed point set submanifold S of maximal possible dimension. For any real-analytic Kähler manifold S equipped with a line bundle with a real-analytic unitary connection with curvature proportional to the Kähler form, we explicitly construct a holomorphic contact distribution on the twistor space obtained by the quaternionic Feix–Kaledin construction from these data. Conversely, we show that quaternion-Kähler metrics with a rotating S^1-symmetry induce on the fixed point set of maximal dimension a Kähler metric together with a unitary connection on a holomorphic line bundle with curvature proportional to the Kähler form and the two constructions are inverse to each other. Moreover, we study the case when S is compact, showing that in this case the quaternion-Kähler geometry is determined by the Kähler metric on the fixed point set (of maximal possible dimension) and by the contact line bundle along the corresponding submanifold on the twistor space. Finally, we relate the results to the c-map construction showing that the family of quaternion-Kähler manifolds obtained from a fixed Kähler metric on S by varying the line bundle and the hyperkähler manifold obtained by hyperkähler Feix–Kaledin construction from S are related by hyperkähler/quaternion-Kähler correspondence.
Highlights
A classification of hyperkähler metrics with rotating S1-symmetry near the fixed point submanifold S of maximal dimension was provided by Feix [12] and, independently, by Kaledin [17]
The c-projective Armstrong cone over S carries a natural Kähler metric if and only if S is Kähler–Einstein which could suggest that this condition is necessary. We show that this is not true: starting from any Kähler manifold with a holomorphic line bundle on S with a real-analytic unitary connection such that its curvature is proportional to the Kähler form, we explicitly construct a holomorphic contact distribution on the twistor space
In [9], it is shown that the twistor space of any quaternionic manifold with such an S1-action can be obtained locally in this way: on the twistor space Z, the fixed point set corresponds to a submanifold of complex dimension n with two components on which the corresponding C∗-action has constant single weight equal to 1 and one can deduce that Z is of the form as in the quaternionic Feix–Kaledin construction
Summary
A classification of hyperkähler metrics with rotating S1-symmetry near the fixed point submanifold S of maximal dimension was provided by Feix [12] and, independently, by Kaledin [17] They have shown that locally such metrics are uniquely determined by the real-analytic Kähler metric on S, and for any such metric the corresponding hyperkähler structure can be constructed on a neighbourhood of the zero section of the cotangent bundle of S. 5, we discuss the case when S is compact, proving in particular that the quaternion-Kähler structure on a neighbourhood of S is fully encoded in the Kähler metric on S and the contact line bundle on the twistor space restricted to the submanifold corresponding to S.
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