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If $X$ is a projective variety then Hilbert's syzygy theorem says that any coherent sheaf of $\mathcal O_X$ modules has a finite resolution by locally free modules.

By GAGA, I believe this should imply that over any smooth complex submanifold $X$ of $\mathbb CP^n$, a coherent sheaf of modules over the sheaf of holomorphic functions has a finite resolution by holomorphic vector bundles.

Assuming what I said is correct, is there a direct proof of this in the analytic setting? Also, is the analytic statement true for more general complex manifolds (e.g. Kahler)?

Note: this is a crosspost from MO.

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I don't know much about this topic but I think you mean finite resolutions. Infinite ones should always exist. Anyway, nice question. –  Marek Jan 8 at 15:35
    
@Marek you're right, thanks for the correction. I have edited it accordingly. –  Eric O. Korman Jan 8 at 15:39
    
Dear Eric, Since this has been answered in the comments on MO, you might want to say so here (to save duplication of effort; indeed, I was about to write something similar to the comment on MO, before thinking I should check there first). Regards, –  Matt E Jan 12 at 4:06

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