Complex Analytic and Differential Geometry 2025, 04 - Complex Analytic Schemes
A ringed space (X, RX) consists of a topological space X and the structure sheaf RX, so that F: (X, RX) → (Y, RY) is a pair (f, F*) where f: X → Y is a continuous map and F*: f-1RY → RX, F*x: (RY)f(x) → (RX)x is a homomorphism of sheaves of rings on F. Such an F: (A, OΩ/J↑A) → (A', OΩ'/J'↑A') is analytic if for all x in A there is a nighborhood W in Ω and a holomorphic Φ: Wx → Ω' that is equal to f on ↑A∩Wx with comorphism induced by Φ*. A complex analytic scheme is a ringed space (X, OX) over a separable Hausdorff X with open coverings (Uλ) of X and isomorphic Gλ: (Uλ, OX↑Uλ) → (Aλ, OΩλ/Jλ↑Aλ) with Aλ is the 0-set on an open subset Ωλ with holomorphic, isomorphic transition morphisms equipped with the respective structure sheaves.
The set of nilpotent elements in an analytic scheme is the sheaf of ideals NX = {u∈OX: uk = 0} for some natural k. Locally, OX↑Aλ = (OΩλ/Jλ)↑Aλ The scheme (X, OX) is reduced if NX = 0. The associated ringed space is the reduced scheme. For an analytic scheme, a sheaf S of OX-modules is said to be coherent if X is a manifold, and S is locally finitely generated over OX, and for all open sets U in X and sections in S(U), the relation sheaf is locally finitely generated.
For Analytic schemes (X, OX), (Y, OY) an analytic F: X → Y is a "modification" if F is proper and ∃ a nowhere dense closed analytic B ⊂ Y with isomorphic F: X \ F-1(B) → Y \ B. If F is a modification, then the comorphism F*: f*OY &rarr OX induces an isomorphism F*: f*MY → MX for the sheaves of meromorphic functions on X and Y. All holomorphic functions v in the complement of B are meromorphic on X, if B is weakly holomorphic on Xred. A meromorphic map F is a scheme morphism from X\A to Y, defined in B with the analytic closure of the graph of F in X × Y.