---
tag: T19P7J53A1
kind: concept
name: Holomorphic map between Riemann surfaces
msc: [30F10, 30F99]
specializes: [0P27W9RW0G]
sources:
  - url: https://en.wikipedia.org/wiki/Riemann_surface
---

## Summary

A holomorphic map between [[Riemann surfaces|5K0N8QMV6Q]] is a continuous map that is holomorphic when written in local coordinates on both sides.
These maps are the morphisms of the category of Riemann surfaces; the bijective ones are the biholomorphisms, which identify two Riemann surfaces.
Meromorphic functions on a Riemann surface are the same as holomorphic maps to the [[Riemann sphere|0CDMXGPSES]] that are not constantly $\infty$.

## Explanations

### L1 human

On a Riemann surface you cannot differentiate directly, but you can always look through a chart, which is a piece of the complex plane.
A map between two surfaces is holomorphic if every such view of it is an ordinary holomorphic function.
For instance $z \mapsto z^2$ on the Riemann sphere, with $\infty \mapsto \infty$, is holomorphic: near $\infty$ in the coordinate $w = 1/z$ it reads $w \mapsto w^2$.
Near any point where it is not constant, a holomorphic map looks like $z \mapsto z^k$ for some $k \ge 1$ in suitable coordinates.