2
$\begingroup$

Given an ample line bundle $L$ on a smooth projective variety of dimension $\geq 2$, let $C$ be the category of vector bundles that are direct sums of powers of $L$. Two related questions:

  1. Given a surjection in $C$ does the kernel have a filtration by line bundles? (a filtration that the successive quotients are line bundles)
  2. Given an injection in $C$ where the cokernel is a vector bundle, does the cokernel have a filtration by line bundles?

If both of these are not true is there any family of varieties of dimension $\geq 2$ that any of them are true?

$\endgroup$

1 Answer 1

7
$\begingroup$

Definitely not. For instance, consider the projective space $\mathbb{P}^n$ and the Euler sequence $$ 0 \to \Omega \to \mathcal{O}(-1)^{\oplus (n + 1)} \to \mathcal{O} \to 0. $$ Its second and third terms are in $C$ (if $L = \mathcal{O}(1)$), but the first term has no filtration by line bundles. Indeed, the first cohomology of any line bundle on $\mathbb{P}^n$ vanishes (when $n \ge 2$), hence any such filtration would split, but $\Omega$ is indecomposable.

Dualizing the sequence you can obtain an example with injection.

$\endgroup$

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.