RATIONAL CURVES AND AMPLENESS PROPERTIES OF THE TANGENT BUNDLE OF ALGEBRAIC VARIETIES

FRÉDÉRIC CAMPANA THOMAS PETERNELL

Abstract. The purpose of this paper is to translate positivity properties of the tangent bundle (and the anti-canonical bundle) of an algebraic manifold into existence and movability properties of rational curves and to investigate the impact on the global geometry of the manifold X. Among the results we prove are these:

If X is a projective manifold, and E ⊂ T_{X} is an ample locally free sheaf with n − 2 ≥ rkE ≥ n, then X ≃ ℙ_{n}.

Let X be a projective manifold. If X is rationally connected, then there exists a free T_{X}-ample family of (rational) curves. If X admits a free T_{X}-ample family of curves, then X is rationally generated.