---
tag: R8GZAJ85YF
kind: definition
name: Complex analytic space
msc: [32C15]
defines: HXSZA44J5J
uses: [B7FPT8JTXR, TB6TM6R4MD, 0S1FJ8VEX9, 507C34W0ZT, N3PGYJJQ7W, EVPZ7ZG1HB, 0670T59HGE]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_analytic_variety
---

## Definition

A *local model* is built as follows.
Let $D \subseteq \C^n$ be open, let $\mathcal{O}_D$ be the [[sheaf|TB6TM6R4MD]] of [[holomorphic functions|507C34W0ZT]] on $D$, and let $f_1, \dots, f_k$ be holomorphic functions on $D$.
Let $\mathcal{I} \subseteq \mathcal{O}_D$ be the [[ideal sheaf|N3PGYJJQ7W]] generated by $f_1, \dots, f_k$, and let $Z = \set{z \in D : f_1(z) = \dots = f_k(z) = 0}$.
The local model is the ringed space $(Z, (\mathcal{O}_D / \mathcal{I})|_Z)$.

A *complex analytic space* is a pair $(X, \mathcal{O}_X)$ such that:

1. $X$ is a [[Hausdorff space|0S1FJ8VEX9]];
2. $\mathcal{O}_X$ is a sheaf of $\C$-algebras on $X$;
3. each point of $X$ has an open neighbourhood $U$ such that $(U, \mathcal{O}_X|_U)$ is isomorphic to a local model as a $\C$-ringed space.

## Remarks

Example: $\C^n$ with its sheaf of holomorphic functions is a complex space; take $D = \C^n$ and $k = 0$.
The double point is the local model with $D = \C$ and $f_1 = z^2$: the underlying space is $\set{0}$ and the ring of functions is $\C[z]/(z^2)$, which contains the nilpotent $z$.

Non-example: $\C$ with the sheaf of continuous complex functions satisfies clauses 1 and 2 but not 3.
The stalks of a local model are quotients of the [[ring of convergent power series|EVPZ7ZG1HB]], hence [[Noetherian|0670T59HGE]].
Let $\mathfrak{m}$ be the maximal ideal of the ring of germs of continuous functions at $0$.
Then $\mathfrak{m} = \mathfrak{m}^2$, since $f = \sqrt{\abs{f}} \cdot (f / \sqrt{\abs{f}})$.
By Nakayama's lemma $\mathfrak{m}$ is not finitely generated, so this ring is not Noetherian.

A complex space is a [[locally ringed space|B7FPT8JTXR]].
Morphisms of complex spaces, also called holomorphic maps, are morphisms of locally ringed spaces of $\C$-algebras.
Some authors drop the Hausdorff condition.