---
tag: HMCD0JBFST
kind: concept
name: Stein manifold
msc: [32Q28, "32E10"]
specializes: [JSGWAN2RJV, Z4AMJS5KA6]
sources:
  - url: https://en.wikipedia.org/wiki/Stein_manifold
same_as:
  wikidata: Q931187
  wikipedia: Stein manifold
---

## Summary

A Stein manifold is a complex manifold with enough global holomorphic functions: they separate points and give compact holomorphically convex hulls of compact sets.
Stein manifolds are the complex analogue of affine varieties, and they are exactly the closed complex submanifolds of some $\C^N$.
On them the main problems of several complex variables, such as the Cousin problems, are governed by topology, through Cartan's theorems A and B.

## Explanations

### L1 human

A compact complex manifold has no nonconstant global holomorphic functions, so global function theory there is empty.
Stein manifolds are at the opposite end: global holomorphic functions are plentiful, as on $\C^n$ itself.
Examples are $\C^n$, every domain of holomorphy in $\C^n$, every open Riemann surface, and every closed complex submanifold of $\C^n$, such as the curve $\set{(z, w) \in \C^2 : w^2 = z^3 - z}$.
They are the natural spaces on which to do holomorphic function theory in several variables.