Paper detail

Rational curves and ampleness properties of the tangent bundle of algebraic varieties

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: \quad If $X$ is a projective manifold, and ${\cal E} \subset T_X$ is an ample locally free sheaf with $n-2\ge rk {\cal E}\ge n$, then $X \simeq \EP_n$. \quad 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.