Locally conformal Kähler structures in quaternionic geometry

Abstract

We consider compact locally conformal quaternion Kähler manifolds M M . This structure defines on M M a canonical foliation, which we assume to have compact leaves. We prove that the local quaternion Kähler metrics are Ricci-flat and allow us to project M M over a quaternion Kähler orbifold N N with fibers conformally flat 4-dimensional real Hopf manifolds. This fibration was known for the subclass of locally conformal hyperkähler manifolds; in this case we make some observations on the fibers’ structure and obtain restrictions on the Betti numbers. In the homogeneous case N N is shown to be a manifold and this allows a classification. Examples of locally conformal quaternion Kähler manifolds (some with a global complex structure, some locally conformal hyperkähler) are the Hopf manifolds quotients of H n − { 0 } \mathbb H^n-\{0\} by the diagonal action of appropriately chosen discrete subgroups of C O + ( 4 ) CO^+(4) .