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Let $X$ be a smooth complex projective variety such that the restriction of $TX$ on any curve $C$ in $X$ is ample. Is true in this case that $X$ is isomorphic to $\mathbb CP^n$?

I guess the above condition implies that that $TX$ is nef (i.e. $O(1)$ is nef on $\mathbb P(TX)$), but it is not clear for me that this condition implies that $TX$ is ample (which then implies that $X\cong \mathbb CP^n$ according a theorem of Mori).

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In this very famous paper:

Mori, Shigefumi, Projective manifolds with ample tangent bundles. Ann. of Math. (2) 110 (1979), no. 3, 593–606.

it is proven that over an algebraically closed field of characteristic 0 $\mathbb P^n(K)$ is the only manifold $X$ with ample tangent bundle.

In the introduction the author points out that the statement is true under the weaker conditions: 1) $-K_X$ is ample; 2) for any non constant map $\mathbb P^1\to X$ the pull back of $T_X$ is ample on $\mathbb P^1$.

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  • $\begingroup$ Is there a minus sign missing from condition 1? $\endgroup$
    – user5117
    Commented Sep 11, 2013 at 14:41
  • $\begingroup$ yes, there is a minus sign before the canonical. $\endgroup$
    – diverietti
    Commented Sep 11, 2013 at 15:02
  • $\begingroup$ Sign fixed! Thank you to both for pointing it out. $\endgroup$
    – rita
    Commented Sep 11, 2013 at 15:05

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