I’ve been trying to understand some complex analytic geometry as of late; here is an overview of Oka’s theorem.

Consider the space ${\mathbb{C}^n}$ and the sheaf ${\mathcal{O}}$ of holomorphic functions on it. One should think of this as the analog of complex affine space ${\mathbb{C}^n}$, with the Zariski topology, and with the sheaf ${\mathcal{O}_{reg}}$ of regular functions.

In algebraic geometry, if ${I \subset \mathbb{C}[x_1, \dots, x_n]}$ is an ideal, or if ${\mathcal{I} \subset \mathcal{O}_{reg}}$ is a coherent sheaf of ideals, then we can define a closed subset of ${\mathbb{C}[x_1,\dots, x_n]}$ corresponding to the roots of the polynomials in ${I}$. This construction gives the notion of an affine variety, and by gluing these one gets general varieties.

More precisely, here is what an affine variety is. If ${\mathcal{I} \subset \mathcal{O}_{reg}}$ is a coherent sheaf of ideals, then we define a ringed space ${(\mathrm{supp} \mathcal{O}_{reg}/\mathcal{I}, \mathcal{O}_{reg}/\mathcal{I})}$; this gives the associated affine variety. Here the “support” corresponds to taking the common zero locus of the functions in ${\mathcal{I}}$. In this way an affine variety is not just a subset of ${\mathbb{C}^n}$, but a locally ringed space.

Now we want to repeat this construction in the holomorphic category. If ${\mathcal{I} \subset \mathcal{O}}$ is a finitely generated ideal—that is, an ideal which is locally finitely generated—in the sheaf of holomorphic functions on ${\mathbb{C}^n}$, then we define the space cut out by ${\mathcal{I}}$ to be ${(\mathrm{supp} \mathcal{O}/\mathcal{I}, \mathcal{O}_{reg}/\mathcal{I})}$. We can think of these as “affine analytic spaces.”

Definition 1 An analytic space is a locally ringed space which is locally isomorphic to an “affine analytic space.”

This is analogous to the definition of a scheme as a locally ringed space which is locally isomorphic to an affine scheme.

Coherence

Now, there is a technical point here. Earlier, I ignored the distinction between “finitely generated” sheaves of ideals in ${\mathcal{O}_{hol}}$ and “coherent” sheaves of ideals in ${\mathcal{O}}$. Namely, in the algebraic category—at least for noetherian schemes, e.g. varieties—there is a simple notion of “coherence:” a sheaf on an affine scheme is coherent if it comes from a finitely generated module over the ring.

Now this is not the most general definition of coherent.

Definition 2 (Serre) Let ${(X, \mathcal{O}_X)}$ be a ringed space, and let ${\mathcal{F}}$ be a sheaf of ${\mathcal{O}_X}$-modules. Then ${\mathcal{F}}$ is a coherent sheaf if:

1. ${\mathcal{F} }$ is finitely generated. That is, to each point ${x \in X}$, there is a neighborhood ${U \subset X}$ of ${x}$ and a surjection of sheaves ${\mathcal{O}_X^m \twoheadrightarrow \mathcal{F}|_U}$.
2. ${\mathcal{F}}$ has finitely generated “sheaves of relations.” That is, if ${U \subset X}$ is an open subset if ${\mathcal{O}_X|_U^m \rightarrow \mathcal{F}|_U}$ is any morphism of ${\mathcal{O}_X}$-modules, then the kernel is finitely generated.

Coherence is an extremely useful finiteness condition. Let us suppose that the sheaf ${\mathcal{O}_X}$ is coherent over itself: this is the interesting case. (We will see that it is verified in the cases that we care about.) Then the class of coherent sheaves is the smallest class ${\mathcal{C}}$ of sheaves with the following properties:

1. The property of a sheaf being in ${\mathcal{C} }$ is “local” (in a manner easy to make precise).
2. ${\mathcal{C}}$ is closed under kernels and cokernels and direct sums.
3. ${\mathcal{C}}$ contains the constant sheaf ${\mathcal{O}_X}$.

In practice, it is safe to think of coherence on an algebraic variety in the way I described earlier (which is what Hartshorne says, for instance). But it doesn’t make any sense for an analytic space. Coherence is a lot like the condition of being finitely presented as a module, though it is a bit more.

I stated above that an “affine analytic space” was one cut out by a finitely generated ideal in the sheaf of holomorphic functions. In fact, we can just say “coherent,” by the following result.

Theorem 3 (Oka) The sheaf ${\mathcal{O}_{hol}}$ of holomorphic functions on ${\mathbb{C}^n}$ is coherent over itself.

We can unwind this in a concrete way. This states the sheaf ${\mathcal{O}_{hol}}$ is of “relation finite type:” in other words, if ${U \subset \mathbb{C}^n}$ is an open subset containing a point ${x}$ and ${f_1, \dots, f_m}$ are holomorphic functions on ${U}$, then there is a subset ${V \subset U}$ containing ${x}$ such that the kernel of the map defined by the ${f_i}$,

$\displaystyle \mathcal{O}_{hol}^m|_V \rightarrow \mathcal{O}_{hol}|_V$

is generated by a finite number of ${V}$-sections.

This is a nontrivial fact, and took significant work to prove, especially since sheaves had not entered complex geometry when Oka discovered this result.

The essential idea is to prove:

Lemma 4 Let ${\mathcal{A}}$ be a Hausdorff sheaf of integral domains on a space ${X}$. Then ${\mathcal{A}}$ is coherent as an ${\mathcal{A}}$-module if, for each section ${s}$ of ${\mathcal{A}}$ on some open subset ${U \subset X}$, ${\mathcal{A}/s\mathcal{A}}$ is coherent

This is mostly a formal argument. After this, one reduces inductively to showing that the holomorphic sheaf on ${\mathbb{C}^{n-1}}$ is coherent. To apply the lemma to the sheaf on ${\mathbb{C}^n}$, one uses the fact that ${\mathcal{O}_{hol}/s \mathcal{O}_{hol}}$ is locally not that different from ${\mathbb{C}^{n-1}}$: in other words, a hypersurface is locally a branched cover of ${\mathbb{C}^{n-1}}$. Branched covers, which are finite maps, tend to preserve coherence, so one can reason inductively. (You can find this argument in the book by Grauert-Remmert, Coherent Analytic Sheaves.)

So, what does this mean? The basic formalism of Serre’s paper Faisceaux Algebriques Coherents can now be applied to the holomorphic category, and as a result the notion of “coherence” of ananalytic sheaf—a sheaf of modules over ${\mathcal{O}_{hol}}$ is useful. In particular, we can think of subspaces of an analytic space as being cut out by coherent sheaves of ideals.