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.