Locally conformal Kähler structures in quaternionic geometry

Abstract:

We consider compact locally conformal quaternion Kähler manifolds $M$. This structure defines on $M$ 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 $M$ over a quaternion Kähler orbifold $N$ 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 $N$ 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 $\mathbb H^n-\{0\}$ by the diagonal action of appropriately chosen discrete subgroups of $CO^+(4)$.