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.” (more…)