Complex Analytic and Differential Geometry 2025, 03 - Normal Spaces

A complex space X with holomorphic f on X_reg, so that all x in X_sing have a neighborhood V for which f is bounded on X_reg∩V. If h is holomorphic on V, such that h^-1(0) is nowhere dense in V. For all y in V, a function h is a universal denominator on V. The associated sheaf (O_{X, x}) is contained in the ring of meromorphic functions. If (X, x) is irreducible and the ring of germs of weakly holomorphic functions over neighborhoods of x (Õ_{X, x}) is the integral closure in its quotient field. Every germ admits a limit to f(z→x). A complex X is normal at a point x, if (X, x) is irreducible, and Õ_{X, x} = O_{X, x}. The set of normal points is X_norm. X is normal, if it is normal at all points. The non-normal set X_n-n is an analytic subset of X. X_norm is open in X. If x is a normal point, then (X_sing, x) has a codimension of at least 2 in (X, x). Complex curves are normal iff they're regular. A normalization (Y, π) is a normal complex space Y together with a holomorphic map π: Y→X so that π is proper and has finite fibers, and the set of singular points of X Σ, A = π^-1(Σ), Y \ A is dense in Y and π is an analytic isomorphism. A complex space has a normalization (Y, π). For locally irreducible complex space, the normalization π is homeomorphic.

Holomorphic and finite maps F: X→Y where X, Y are complex, then dim[X] ≤ dim[Y]. If F is also surjective, then dim[X] = dim[Y]. The fibre dimension is an upper semi-continuous function. The rank of F at x is defined as ρ_F(x) = dim[X, x] - dim[F^-1(F(x)), x] If F is holomorphic and Z an analytic subset of X, then maximal Rank of F on Z is less or equal than that of the maximal Rank of F. If Y is pure dimensional, and the max Rank of F is less than dim[Y], then F has empty interior in Y. For surjective F, its maximal rank is equal to dim[Y].

For some complex space X, an analytic subset A and Z ⊂ X \ A also analytic, where a non-negative integer p with dim[A] ≤ p, dim[Z, x] > p ∀ x ∈ Z, then the closure of Z is an analytic subset. If F: X→Y is proper holomorphic, then F(X) is an analytic subset of Y.