Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle

Abstract

Let X be a compact connected Kähler manifold such that the holomorphic tangent bundle T X is numerically effective. A theorem of [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.2000 Mathematics Subject Classification. 32M10, 14M17, 53C15.

Holomorphic Cartan geometries and rational curves

“…Biswas and Bruzzo [10] proved a special case of the main theorem above. Suppose that M is a compact connected Kähler manifold with numerically effective tangent bundle.…”