Complex Analytic and Differential Geometry 2025, 01 - Complex Analytic Sets

A complex analytic set can be defined locally by finitely many holomorphic equations. They generally have singular points. A complex analytic manifold M and A ⊂ M an analytic subset, then A is closed and ∀ x0 ∈ A ∃ some neighborhood U of x0 with holomorphic functions {g}n in O(U) with A∩U = {z∈U: g1(z) = ... = gN(z) = 0}. The holomorphisms g are "local equations of A in U". Obviously finite unions/intersections of analytic sets are themselves analytic. If M is connected, either A = M or A has no interior point by analytic continuation. Germs of a set at x∈M build an equivalence class in the power set of M for A~B if there is an open neighborhood V of x with A∪V = B∪V. Ideals in the ring OM,x are subsets of Ideals of the germ at x of zero variety. For germs of analytic sets (A, x) = (V(JA,x), x)

(A, x) is irreducible if it has no decomposition into germs of analytic sets with x. In turn it's irreducible iff JA,x is a prime ideal of OM,x. Decreasing sequences of (A, x) are stationary. (A, x) has some finite decomposition

(A, x) = ⋓1≤k≤N(Ak, x)

where the germs are irreducible and pairwise disjoint. It's unique up to notation order.

There is an integer d, a basis {e} of ℂn and coordinates (z) as follows: Jd = {0} and ∀k = d + 1, ..., n there is a Weierstrass polynomial Pk∈ Jk so that

Sets z', z'' and Δ'∈ℂd, Δ''∈ℂn-d centered on 0 with radius r', r'' respectively, then (A, 0) is contained in a cone |z''| ≤ C|z'|, where C is constant and the restriction π: A∩(Δ'×Δ'')→Δ' of the projection map ℂn→ℂd, (z', z'')→z' is proper if r'' is small enough and r'≤r''/C

For the discriminant δ(z') in Od of the Weierstrass polynomial Wu(z'; T), all g of the integral MA over Od, then δg∈Od[u]. For every germ f in On there is a unique polynomial R in Od[T], degT R ≤ q - 1, such that δ(z;)mf(z) = R(z'; u(z'')) (mod G)

where G is an ideal in J. If f ∈ J, then R = 0 and δmJ ⊂ G. If J is a prime ideal, and A = V(J), then the coordinates (z', z'') then On/J is a finite integral extension of Od. The degree of the extension q and the discriminant δ(z') of the irreducible polynomial of a primitive element. If the polydisk of sufficiently small radii, their projection is a ramified covering with q sheets, with locus contained in S = {z' ∈ Δ'; δ(z') = 0}. If J is prime in On and A = V(J), then JA, 0 = J.

For every ideal J ⊂ On: JV(J), 0 is the radiacal of J.

An analytic set A in M and point x in A a regular point of A if A ∩ Ω is a ℂ-analytic submanifold of Ω as a neighborhood of x. Otherwise x is singular. If (A, x) is irreducible, then there are arbitrarily small neighborhoods Ω of x, such that its intersection with the open subset Areg is dense and connected in A ∩ Ω. The dimension of an irreducible germ of analytic set (A, x) is defined by dim(A, x) = dim(Areg, x). Germs of analytic sets (B, x) in (A, x) for (A, x) irreducible and unequal (B, x), has dim(B, x) < dim(A, x) and B ∩ Ω with an empty interior in A ∩ Ω for all sufficiently small neighborhoods Ω of x.

For analytic sets A in a complex manifold M, the sheaf of ideals JA is a coherent analytic sheaf. Asing is an analytic subset of A. (A, x) with dimension d, then the maximal ideal of functions m, vanishing at x must have at least d generators. It has exactly d generators, iff x is a regular point. If A is of pure dimension d and B an analytic subset of codimension of at least p in A, then the ideal JB, x is generated by at least p generators in B, and at most p+1 generators in Breg∩Ssing.