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.

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