---
tag: PN55VEF0J7
kind: concept
name: Normal complex space
msc: [32C20, 32C15]
specializes: [831N2Q4GCB]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_analytic_variety
---

## Summary

A normal complex space is a [[complex analytic space|HXSZA44J5J]] whose local rings are integrally closed domains.
Normal spaces are [[reduced|831N2Q4GCB]], locally irreducible, and their singular locus has codimension at least $2$.
On a normal space, the Riemann extension theorem holds: a function that is holomorphic off the singular locus and locally bounded is holomorphic.
Every reduced complex space has a normalization, so normal spaces are a standard first step in the study of singularities.

## Explanations

### L1 human

Singular points come in different degrees of badness.
At a normal point, the space does not cross itself and is not pinched like the cusp $z_2^2 = z_1^3$.
For example, the cone $\set{z_1 z_2 = z_3^2}$ in $\C^3$ is normal, even though its vertex is a singular point.
A normal curve is the same as a smooth curve, but in higher dimension normal spaces can have singularities.