From the introduction: In 1979 in his famous solution of the Hartshorne-Frankel conjecture, S. Mori proved that a projective manifold \(X\) whose tangent bundle \(T_X\) is ample, must be the projective space \(\mathbb{P}_n\). Previously, Mori and Sumihiro showed that \(\mathbb{P}_n\) is the only manifold admitting a vector field vanishing along an ample divisor (and this is an Euler vector field if \(n\geq 2)\). This was generalised by J. Wahl to the extent that if \(T_X\otimes L^{-1}\) has a non-zero section with \(L\) an ample line bundle (not necessarily effective), then \(X\simeq\mathbb{P}_n\). – Although these two theorems look at first quite different in nature, they might be only special cases of a more general theorem.
Question. Let \(X\) be a projective manifold, \({\mathcal E}\) an ample locally free sheaf, \({\mathcal E}\subset T_X\). Is \(X\simeq \mathbb{P}_n\)?
We shall prove theorem 1: The question has a positive answer if \(\text{rk} {\mathcal E}\) is \(n-2\), \(n-1\) or \(n\).
A manifold is said to be rationally connected if two general points can be joined by a chain of rational curves. Examples are Fano manifolds.
Theorem 2. Let \(X\) be a projective manifold.
(1) If \(X\) is rationally connected, then there exists a free \(T_X\)-ample family of (rational) curves.
(2) If \(X\) admits a free \(T_X\)-ample family of curves, then \(X\) is rationally generated.