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Over a complex manifold, can every coherent sheaf be seen as a holomorphic vector bundle over an analytic subset?

Thanks in advance.

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    $\begingroup$ Not sure what the precise question is, but consider the ideal sheaf of the origin within any higher-dimensional affine space. $\endgroup$ – Keerthi Madapusi Pera Mar 13 at 15:53
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    $\begingroup$ I'm not sure what you mean by that either, but given a coherent sheaf $\mathcal E$ on a complex manifold $X$, there exists a non-empty Zariski open subset $U\subset X$ such that $\mathcal E|_U$ is locally free, hence is the sheaf of sections of a holomorphic vector bundle $E$ on $U$. $\endgroup$ – Henri Mar 13 at 17:38
  • $\begingroup$ Henri, do you have please a reference for that? I am quite a novice in the subject. $\endgroup$ – BinAcker Mar 14 at 10:54

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