Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle

Let X be a compact connected Kähler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly et al. (1994) [11] says that there is a finite unramified Galois covering M→XM→X, a complex torus T  , and a holomorphic surjective submersion f:M→Tf:M→T, such that the fibers of f are Fano manifolds with numerically effective tangent bundle. A conjecture of Campana and Peternell says that the fibers of f are rational and homogeneous. Assume that X admits a holomorphic Cartan geometry. We prove that the fibers of f   are rational homogeneous varieties. We also prove that the holomorphic principal GG-bundle over T given by f  , where GG is the group of all holomorphic automorphisms of a fiber, admits a flat holomorphic connection.