---
tag: 831N2Q4GCB
kind: concept
name: Reduced complex space
msc: [32C15]
specializes: [HXSZA44J5J]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_analytic_variety
---

## Summary

A reduced complex space is a [[complex analytic space|HXSZA44J5J]] whose structure sheaf has no nonzero nilpotent germs.
On such a space, a function is determined by its values at points, so the structure sheaf is a sheaf of genuine complex-valued functions.
Reduced complex spaces are the same as complex analytic varieties with their sheaves of holomorphic functions.
Every complex space has a reduction, a reduced complex space with the same underlying topological space.

## Explanations

### L1 human

A complex space may carry extra infinitesimal information, such as the double point given by $z^2 = 0$, where $z$ is a nonzero function that vanishes at the only point.
A reduced complex space has none of this extra information.
Its functions are ordinary functions, and a function that is zero at every point is zero.
The cross $\set{z_1 z_2 = 0}$ in $\C^2$ is reduced, while the doubled line $\set{z_1^2 = 0}$ is not.