Quaternion-K\"ahler manifolds near maximal fixed points sets of $S^1$-symmetries

Abstract

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