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Let $X$ be a complex manifold and let $V\subset X$ be a subvariety. Let $F\rightarrow V$ be a holomorphic vector bundle over $V$ and let $\mathcal{S}=\Gamma(F)$ be the sheaf of holomorphic section of $F$. As $\mathcal{S}$ is coherent over $X$ it admits a resolution by holomorphic vector bundles $E^i$ over $X$.

Is there a canonical recipe for building these vector bundles $E^i$?

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    $\begingroup$ What do you mean by "canonical"? $\endgroup$
    – Will Sawin
    Commented Jun 11, 2020 at 14:18
  • $\begingroup$ @WillSawin I actually mean just a recipe with the fewest choices. $\endgroup$
    – BinAcker
    Commented Jun 11, 2020 at 14:21
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    $\begingroup$ Don’t you need an extra condition on $X$ in there like ‘algebraic’? $\endgroup$
    – Aaron Bergman
    Commented Jun 11, 2020 at 14:45
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    $\begingroup$ If $X$ is projective, one can use coproduces of twisted invertible sheaves $\mathcal{O}(n)$ but probably you already know this. $\endgroup$
    – Leo Alonso
    Commented Jun 11, 2020 at 14:52
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    $\begingroup$ Let me summarize the above comments, with some additional remarks: 1) If $X$ is just a complex manifold, then a resolution by bundles on $X$ need not exist 2) If $X$ is projective with fixed very ample bundle, and you know the Castelnuovo-Mumford regularity of $V$, then you can give a canonical resolution. But I don't see how you would do it without additional information. $\endgroup$
    – Donu Arapura
    Commented Jun 11, 2020 at 15:07

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