Almost homogeneous torus actions of varieties with ample tangent bundle

The main purpose of the present paper is to prove: THEOREM 4.1. Let V be an n-dimensonal irreducible non-singular projective variety on which an n-dimensional algebraic torus (C*)* acts regularly and effectively. Assume that every irreducible non-singular hypersurface in V has ample normal bundle. Then V is isomorphic to In recent years, systematic studies of torus embeddings have been made by several authors, (cf. Demazure [1], Miyake-Oda [5], Mumford et al. [6]). Their method is, in its spirit, to translate geometric problems of varieties with almost homogeneous torus actions into purely combinatorial problems of the complexes constructed from the group actions. This very spirit is also the main motivation that has led us to re-examine the following conjecture of Hartshorne from a group theoretical point of view: