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).



Variety $X$ such that $TX$ is ample on any curve in $X$
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 $\mathbb P^1$.


