MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $E$.


  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?
share|cite|improve this question
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
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
I agree with J.C Ottem. This seems to have gotten two downvotes now. Is this just because the definition of big is missing? – user5117 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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.