Quaternion-Kahler manifolds near maximal fixed point sets of S1-symmetries

Using quaternionic Feix-Kaledin construction, we provide a local classification of quaternion-Kahler metrics with a rotating S-1-symmetry with the fixed point set submanifold S of maximal possible dimension. For any real-analytic Kahler manifold S equipped with a line bundle with a real-analytic unitary connection with curvature proportional to the Kahler 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-Kahler metrics with a rotating S-1-symmetry induce on the fixed point set of maximal dimension a Kahler metric together with a unitary connection on a holomorphic line bundle with curvature proportional to the Kahler 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-Kahler geometry is determined by the Kahler 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-Kahler manifolds obtained from a fixed Kahler metric on S by varying the line bundle and the hyperkahler manifold obtained by hyperkahler Feix-Kaledin construction from S are related by hyperkahler/quaternion-Kahler correspondence.