Holomorphic geometric structures on Oeljeklaus-Toma manifolds

We prove that any holomorphic geometric structure of affine type on an Oeljeklaus- Toma manifold is locally homogeneous. For locally conformal Kähler Oeljeklaus-Toma manifolds we prove that all holomorphic geometric structures, and also all holomorphic Cartan geometries, on them are locally homogeneous.