Quaternion-Kähler manifolds near maximal fixed points sets of S¹-symmetries

الأصل · EN

Using quaternionic Feix--Kaledin construction we provide a local classification of quaternion-Kähler metrics with a rotating S¹-symmetry with the fixed point set submanifold S of maximal possible dimension. For any Kähler manifold S equipped with a line bundle with a unitary connection of 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 this data. Conversely, we show that quaternion-Kähler metrics with a rotating S¹-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. 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 form S are related by hyperkähler/quaternion-Kähler correspondence.