Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle

Let X be a compact connected Kahler 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 -> X. a complex torus T, and a holomorphic surjective submersion f :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 g-bundle over T given by f, where g is the group of all holomorphic automorphisms of a fiber, admits a flat holomorphic connection. (C) 2011 Elsevier B.V. All rights reserved.