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Let $X$ be a compact complex manifold and $\mathcal{F}\to X$ a sheaf.

Is there a regularity criteria (or a condition) for $\mathcal{F}$ that determines whether we there exists a closed subvariety $i:Y\hookrightarrow X$ and a locally free sheaf $\mathcal{G}$ such that $\mathcal{F}=i_*\mathcal{G}$?

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    $\begingroup$ The natural choice for $Y$ will be the scheme-theoretic support of $\mathcal F$. Do you have something in mind for the $Y = X$ case? $\endgroup$
    – red_trumpet
    Commented May 26, 2022 at 16:53

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