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.

使用四元离子Feix–Kaledin构造，我们提供了具有旋转\（S ^ 1 \）-对称性且具有最大可能尺寸的不动点集子流形S的四元数-Kähler度量的局部分类。对于任何配备线束的实解析Kähler流形S，实线单一连接的曲率与Kähler形式成比例，我们根据这些数据通过四元Feix-Kaledin结构获得的扭转空间上明确构造了全纯接触分布。 。相反，我们证明了具有旋转\（S ^ 1 \）的四元数-Kähler度量-对称在最大维的不动点集上诱发Kähler度量，以及在全纯线束上的整体连接，其曲率与Kähler形式成比例，并且两个构造彼此相反。此外，我们研究了当S为紧时的情况，表明在这种情况下，四元数-Kähler几何形状由固定点集（最大可能尺寸）上的Kähler度量以及沿着线束上相应子流形的接触线束确定。扭曲空间。最后，我们将结果与c-map结构相关联，表明通过改变线束从S上的固定Kähler度量获取的四元数-Kähler流形的族和由hyperkählerFeix-Kaledin结构从S通过hyperkähler/quaternion-Kähler对应关系关联。