Quaternion-Kähler manifolds near maximal fixed points sets of -symmetries
arXiv:1904.08474
Abstract
Using quaternionic Feix--Kaledin construction we provide a local classification of quaternion-Kähler metrics with a rotating -symmetry with the fixed point set submanifold of maximal possible dimension. For any Kähler manifold 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 -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 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 by varying the line bundle and the hyperkähler manifold obtained by hyperkähler Feix--Kaledin construction form are related by hyperkähler/quaternion-Kähler correspondence.
16 pages