Complex Analytic and Differential Geometry 2025, 02 - Complex Spaces

Define the comorphism of F at x as F_x*: O_{B, F(x)} ∋ g → g ○ F ∈ O_{A, x} A complex space X is a locally compact Hausdorff space, countable at infinity with a sheaf O of continuous functions on X, so that each λ there is a homomorphism F_λ: U_λ → A_λ for U_λ an open covering of X and A_λ ⊂ Ω_λ ⊂ ℂ^n_λ so that F_λ* is an isomorphism of sheaves of rings. O is then the structure sheaf of X. For every complex space X, X_reg is dense and open in X and is a disjoint union of connected complex manifolds X'_α with closure X_α. (X_α) is a locally finite family of analytic subsets of X. The sets X_α are the global irreducible components of X. Analytic subsets A, B in X with closure C = ⋓(A_λ ⊄ B). The family (A_λ) the intersection ⋓A_λ = A ⊂ X is analytic. The intersection is stationary on all compact subsets. An irreducible complex space X has a constant holomorphic function f on X defining an open map f: X → ℂ. If X is a compact irreducible analtyic space, all holomorphic f are constant. If all global holomorphic f in O(X) separating points in X, then compact analytic subsets A of X are finite. A coherent O_X-module S has a support Supp[S] = {x∈X; S_x≠0} which is an analytic subset of X.

A complex space X of pure dim p and an analytic A ⊂ X with codim_x A ≥ 2, then all holomorphic f on X\A is bounded near A. If X is irreducible and f holomorphic, non-zero, then f^-1(0) is empty or of pure dimension dim[X - 1]. A is a local complete intersection in X, if all points of A have neighborhoods Ω with A∩Ω = {x∈Ω: f_i(x) = 0 ∀ i ≤ p} with p = codim[A] functions f_i. A complex manifold with dim_ℂM = n, (A, x) an analytic germ of pure dim n - 1 and irreducible components A_j, 1 ≤ j ≤ N, then the ideal of (A, x) is a principal ideal (g) where g is a product of irreducible germs, and for all holomorphic functions f with f^-1(0)⊂(A, x) there is a unique decomposition f = Π_i ug_i^m_i with invertible germs u and order of vanishing of f at a point m. Mermorphic functions f on X, the sets P_f, Z_f and indterminacy set P_f ∩ Z_f are analytic subsets. For non-zero f on a manifold M with dim n, they of pure dim n - 1, and the indeterminacy set is of pure dim n - 2.