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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:

  1. Does there exist a Fano manifold $X$ whose $T_X$ isn't big?
  2. Is it true that $T_X$ is big for any nonsingular toric variety $X$?
  3. How about for (partial) flag varieties or more generally homogeneous spaces G/P?
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I think it would be a very good thing if you included the definition of "big" rather than expecting people to get hold of a particular book and look it up. –  Steven Landsburg Apr 18 '13 at 16:31
    
Just to clarify: I think that bigness of $T_X$ refers to the line bundle $O(1)$ being big on $\mathbb{P}(T_X)$. As usual, a line bundle $L^m$ is big if $\dim H^0(mL)$ grows asymtotically as $m^{\dim X}$ for $m$ large. –  J.C. Ottem Apr 18 '13 at 19:23
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Also, I feel that the downvote is a bit harsh. I think it makes sense to ask this question and it it interesting. –  J.C. Ottem Apr 18 '13 at 20:30
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I agree with J.C Ottem. This seems to have gotten two downvotes now. Is this just because the definition of big is missing? –  Artie Prendergast-Smith Apr 18 '13 at 20:37
    
Thanks J.C. Ottem for clarify the definition. This is the definition I use. Sorry for being lazy. –  jhsiao Apr 18 '13 at 23:40
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