In algebraic geometry one has the following result:
Let $X$ be a noetherian scheme and $\mathcal{F}$ a coherent sheaf with support $Z \neq X$. Then $\mathcal{F}$ has a finite filtration $\mathcal{F} = \mathcal{F}_0 \supset \mathcal{F}_1 \supset \dots \supset \mathcal{F}_n=0$ with all the quotients $\mathcal{F}_j /\mathcal{F}_{j+1}$ being coherent sheaves of $\mathcal{O}_Z-$modules.
If we take now $X$ to be a compact complex manifold, then $Z$ will be a closed variety (because $\mathcal{F}$ is coherent. Does a similar statement remain true in this case?