Let $X$ be a compact Kähler manifold. Let $F$ be a coherent $O_X$-module. Is there always an open cover $\{U_i\}_i$ for $X$ such that for every $i$:
1.$X\setminus U_i$ is an analytic subset of $X$;
2.there is $n_i\ge 1$ and a surjective morphism $O_{U_i}^{n_i}\to F|_{U_i}$?
When $X$ is a projective manifold, it is a consequence of Serre's GAGA. But I don't know the general case.