Complex manifolds with ample tangent bundles ∗

Let M be a close complex manifold and T M its holomorphic tangent bundle. We prove that if the global holomorphic sections of tangent bundle generate each fibre, then M is a complex homogeneous manifold. It implies that every irreducible close Kähler manifold with ample tangent bundle is isomorphic to the projective space. This provides an alternative proof of Hartshore's conjecture in algebraic geometry in characteristic zero and Frankel's conjecture in Kähler geometry. These two conjectures were completely proved by Mori, Siu and Yau. Our proof is quite different from theirs and depends on the complex version of Chow-Rashevskii theorem in Carnot-Caratheodory spaces. The method of this paper gives new proof on the famous Poincare conjecture in dimension three and a complete solution on smooth Poincare conjecture in dimension four.