7
$\begingroup$

Serre-Swan's theorem (see the MO discussion) says that any locally free sheaf over an affine variety is a direct summand of a free sheaf. However, this is not true on projective varieties. It is not hard to check that a non-trivial line bundle with non-zero global sections can not be a direct summand of a free sheaf. The reason is that, being a direct summand of a free sheaf implies that the dual line bundle also has non-zero global sections. But that implies the line bundle is trivial.

I am wondering if the same holds for vector bundles, i.e. if a vector bundle and its dual over a projective variety both have non-zero global sections, then the vector bundle is trivial.

Another I think related question is the following:

Is a direct summand of a direct sum of line bundles on a projective variety also a direct sum of line bundles?

Improve this question
$\endgroup$
0

1 Answer 1

Reset to default
4
$\begingroup$

1) The vector bundle $E=\mathcal O(1) \oplus \mathcal O(-1)$ over $\mathbb P^1$ has non trivial sections and so has its dual (which happens to be the same bundle $E$ ).
However $E$ is non trivial because each of its global sections has a zero.

2) Yes, a direct summand of a direct sum of line bundles on a complete variety is also a direct sum of line bundles.
This follows from Atiyah's general version of the Krull-Schmidt theorem .

Edit
Since Fei asks, let me remark that the trick in 1) works also for bundles which are not direct sums of line bundles:
If $T$ is the tangent bundle to $\mathbb P^2$, then $E=T \oplus T^* $ has non-trivial global sections , and so has $E^*=E$
However, since $T$ is indecomposable $E$ is not a sum of line bundles by Atiyah's result and is thus a fortiori not trivial.

Improve this answer
$\endgroup$
6
  • $\begingroup$ Thank you so much for your answer. For the first question, do you have a counterexample that $E$ is not a direct sum of line bundles? $\endgroup$
    – Fei YE
    Mar 25, 2012 at 11:07
  • $\begingroup$ Dear @Fei,yes: I have added such a counterexample in an Edit. $\endgroup$
    – Georges Elencwajg
    Mar 25, 2012 at 11:26
  • $\begingroup$ Dear George, Thank you so much for the example. I saw you used a fact $E^\ast=E$ for $E=T\oplus T^\ast$. Is there are general relation between $E$ and $E^\ast$. I saw somewhere that $E^\ast\otiems \det(E)=E$. Is this really true in general? Thanks again for you help! $\endgroup$
    – Fei YE
    Mar 25, 2012 at 12:14
  • 1
    $\begingroup$ Dear @Fei, I have just used that the dual of a direct sum is the direct sum of the duals. The formula you quote is already false for every non trivial line bundle $L$ since $L^* \otimes det(L)=L^* \otimes L$ is trivial. $\endgroup$
    – Georges Elencwajg
    Mar 25, 2012 at 12:24
  • 1
    $\begingroup$ Dear @Fei, yes, for rank $2$ it is correct: Hartshorne, Ex.II 5.16 $\endgroup$
    – Georges Elencwajg
    Mar 25, 2012 at 14:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.