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Let's assume we are working a smooth projective variety. Let $C$ be the category of vector bundles constructed by taking successive extensions of line bundles of the form $\mathcal{O}(n)$ for $n\in \mathbb{Z}$. I have several related questions:

  1. Which vector bundles on $X$ admit a finite resolution by vector bundles in $C$? There might be obstructions in chern classes. You can assume the vector bundle on $X$ has whatever the chern classes you want it to have. For example you can assume all of its chern classes vanish.

  2. Are there examples of varieties that this happens for all vector bundles?

  3. Is it true that for ACM (arithmetically cohen-macaulay) varieties (2) is true?

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    $\begingroup$ Plane curves are ACM, but 2 is not true. $\endgroup$
    – Mohan
    Commented Jan 26, 2021 at 20:22

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This holds true for projective spaces. Indeed, if $E$ is any vector bundle (even any coherent sheaf), the Beilinson spectral sequence for $E(n)$ with $n \gg 0$ gives the required resolution for $E(n)$. Twisting it back, one obtains a resolution for $E$.

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  • $\begingroup$ Thanks for the answer. The projective space is a special case of ACM variety, which I suspect it might be true for them. $\endgroup$
    – user127776
    Commented Jan 26, 2021 at 6:56
  • $\begingroup$ You could choose a different embedding for a projective space, and then it you would not be true any longer. $\endgroup$
    – Angelo
    Commented Jan 28, 2021 at 6:16

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