Let $X$ be a nonsingular complex algebraic variety whose tangent bundle is $T_X$. I use Lazarsfeld's book for the definition of a big vector bundle. A line bundle $L$ is big if it has Itaka dimension dim$X$. A vector bundle $E$ is big if the line bundle $O_{P(E)}(1)$ is big on the projective bundle $P(E)$ of one dimensional quotients of $E$.

Questions:

- Does there exist a Fano manifold $X$ whose $T_X$ isn't big?
- Is it true that $T_X$ is big for any nonsingular toric variety $X$?
- How about for (partial) flag varieties or more generally homogeneous spaces G/P?