---
tag: Z4AMJS5KA6
kind: concept
name: Stein space
msc: ["32E10", 32C15]
specializes: [HXSZA44J5J]
sources:
  - url: https://en.wikipedia.org/wiki/Stein_manifold
---

## Summary

A Stein space is a [[complex analytic space|HXSZA44J5J]] with enough global holomorphic functions: they separate points and give compact holomorphically convex hulls of compact sets.
It extends the notion of Stein manifold to spaces with singularities, as affine schemes extend affine varieties.
Stein spaces are exactly the complex spaces on which every coherent analytic sheaf has vanishing higher cohomology.
Closed complex subspaces of $\C^N$ are the main examples.

## Explanations

### L1 human

A Stein space is a space defined locally by holomorphic equations, possibly with singular points, that carries many global holomorphic functions.
The cone $\set{(x, y, z) \in \C^3 : x^2 + y^2 = z^2}$ is an example: it has a singular point at the origin, and the coordinate functions already separate its points.
Stein spaces are the setting where the theorems of Cartan and Oka about holomorphic functions work in full generality.