I’ve been trying to understand some complex analytic geometry as of late; here is an overview of Oka’s theorem.
Consider the space and the sheaf of holomorphic functions on it. One should think of this as the analog of complex affine space , with the Zariski topology, and with the sheaf of regular functions.
In algebraic geometry, if is an ideal, or if is a coherent sheaf of ideals, then we can define a closed subset of corresponding to the roots of the polynomials in . 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 is a coherent sheaf of ideals, then we define a ringed space ; this gives the associated affine variety. Here the “support” corresponds to taking the common zero locus of the functions in . In this way an affine variety is not just a subset of , but a locally ringed space.
Now we want to repeat this construction in the holomorphic category. If is a finitely generated ideal—that is, an ideal which is locally finitely generated—in the sheaf of holomorphic functions on , then we define the space cut out by to be . 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.
Now, there is a technical point here. Earlier, I ignored the distinction between “finitely generated” sheaves of ideals in and “coherent” sheaves of ideals in . 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 be a ringed space, and let be a sheaf of -modules. Then is a coherent sheaf if:
- is finitely generated. That is, to each point , there is a neighborhood of and a surjection of sheaves .
- has finitely generated “sheaves of relations.” That is, if is an open subset if is any morphism of -modules, then the kernel is finitely generated.
Coherence is an extremely useful finiteness condition. Let us suppose that the sheaf 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 of sheaves with the following properties:
- The property of a sheaf being in is “local” (in a manner easy to make precise).
- is closed under kernels and cokernels and direct sums.
- contains the constant sheaf .
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 of holomorphic functions on is coherent over itself.
We can unwind this in a concrete way. This states the sheaf is of “relation finite type:” in other words, if is an open subset containing a point and are holomorphic functions on , then there is a subset containing such that the kernel of the map defined by the ,
is generated by a finite number of -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 be a Hausdorff sheaf of integral domains on a space . Then is coherent as an -module if, for each section of on some open subset , is coherent
This is mostly a formal argument. After this, one reduces inductively to showing that the holomorphic sheaf on is coherent. To apply the lemma to the sheaf on , one uses the fact that is locally not that different from : in other words, a hypersurface is locally a branched cover of . 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 is useful. In particular, we can think of subspaces of an analytic space as being cut out by coherent sheaves of ideals.