This is a very naive question.

Let $X$ be a complex manifold. Let $\mathcal{O}_X$ be the structure sheaf of $X$, a sheaf of rings whose sections over opens $U\subset X$ are just the holomorphic functions $U\rightarrow\mathbb{C}$.

A sheaf of $\mathcal{O}_X$-modules $F$ is coherent if:

- It is locally finitely generated: Ie, there is an open cover $\{U_i\}$ of $X$ such that $F|_{U_i}$ admits surjections $\mathcal{O}_{U_i}\rightarrow F|_{U_i}$ for each $i$.
- For any open $V\subset X$, and any morphism $f : \mathcal{O}_V^s\rightarrow F|_V$, $\text{Ker}(f)$ is a locally finitely generated sheaf on $V$ (ie, satisfies condition 1).

Then, Oka's coherence theorem states that the structure sheaf $\mathcal{O}_X$ is coherent.

When $X$ is a scheme, this statement is almost tautological. What is it about the setting of complex manifolds that makes this theorem deep?

something. $\endgroup$