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 1Ananalytic spaceis a locally ringed space which is locally isomorphic to an “affine analytic space.” (more…)