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Let $i:Z\hookrightarrow X$ be a subvariety of a compact Kahler manifold. Assume that $Z$ can be realize as the zero locus of a section $s$ of a holomorphic vector bundle $E\to X$ of rank $r$. The Koszul complex $$\Lambda^rE^*\to \Lambda^{r-1}E^*\to ...\to E^*\to \mathcal{O}(X)\to i_*\mathcal{O}(Z)\to 0$$ gives a resolution of $i_*\mathcal{O}(Z)$ by vector bundles.

Let now $F\to Z$ be a holomorphic vector bundle on $Z$. How can we modify the complex above to obtain a resolution of $i_*F$ ?

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    $\begingroup$ Apply the projection formula. $\endgroup$
    – Z. M
    Commented May 30, 2022 at 7:59
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    $\begingroup$ @Z.M I guess that in your analogy $\mathcal{F}$ is $\mathcal{O}(Z)$ for me, but what would be $\mathcal{E}$ ? I would be grateful if you could develop a little $\endgroup$
    – BinAcker
    Commented May 30, 2022 at 8:47
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    $\begingroup$ If $F$ extends to a holomorphic vector bundle $G$ on $X$, then you want to tensor each term with $G$ to obtain a resolution of $i_* F=( i_* \mathcal O(Z)) \otimes G$. If not, then I don't think Z. M's idea will help. $\endgroup$
    – Will Sawin
    Commented May 30, 2022 at 10:30
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    $\begingroup$ @WillSawin I indeed did not assume that $F$ comes from a restriction of a bundle on $X$ $\endgroup$
    – BinAcker
    Commented May 30, 2022 at 11:04
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    $\begingroup$ Probably there is no way to do it by modifying the Koszul complex and you need to use a different method (e.g. twisting by ample line bundles and taking global sections) $\endgroup$
    – Will Sawin
    Commented May 30, 2022 at 11:27

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