---
tag: TM528Q96ZD
kind: concept
name: Proper holomorphic map
msc: [32H35, 32H02]
specializes: [15XEBGNGHC, 3MQTWKBBBT]
sources:
  - url: https://en.wikipedia.org/wiki/Proper_map
---

## Summary

A proper holomorphic map is a [[holomorphic map|3MQTWKBBBT]] between domains, or complex manifolds, that is also a [[proper map|15XEBGNGHC]]: preimages of compact sets are compact.
For bounded domains this means that the map sends points near the boundary of the source to points near the boundary of the target.
Proper holomorphic maps between domains of the same dimension are finite branched coverings, so they generalize [[biholomorphisms|FX92PP3HZR]].
In several variables they are rigid: by [[Alexander's theorem|T3VN45KV0D]], every proper holomorphic self-map of the ball in $\C^n$ with $n \ge 2$ is an automorphism.

## Explanations

### L1 human

A proper map cannot hide the boundary: as a point runs to the edge of the source domain, its image runs to the edge of the target.
The map $z \mapsto z^2$ on the unit disc is a typical example.
It wraps the disc twice onto itself, and points near the unit circle go to points near the unit circle.
The inclusion of a smaller disc into the unit disc is not proper, since points near the edge of the small disc land in the middle of the big one.