---
tag: HXSZA44J5J
kind: concept
name: Complex analytic space
msc: [32C15]
specializes: [B7FPT8JTXR]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_analytic_variety
---

## Summary

A complex analytic space, or complex space, is a [[locally ringed space|B7FPT8JTXR]] that is locally modelled on an analytic set.
The model is the common zero set of finitely many [[holomorphic functions|507C34W0ZT]] on an open subset of $\C^n$.
Its structure sheaf is the quotient of the sheaf of holomorphic functions by the ideal that these functions generate.
Complex manifolds and [[complex analytic varieties|GB7M582M5Y]] are complex spaces.
The structure sheaf may have nilpotent elements, which record multiplicities, as for the double point $z^2 = 0$.
Complex spaces are the analytic counterpart of schemes and the setting of the global theory of several complex variables.

## Explanations

### L1 human

A complex manifold looks locally like an open subset of $\C^n$.
A complex space may look locally like the solution set of holomorphic equations, which may have singular points.
An example is the curve $z_2^2 = z_1^3$ with its cusp at the origin.
The space remembers not only the points but also the equations, through the functions that live on it.
So the equation $z^2 = 0$ in $\C$ gives a single point that remembers it was counted twice.