---
tag: P014C13K5H
kind: concept
name: Abelian differential
msc: [30F30, 14H55]
specializes: [0E3G0GRPZ6]
sources:
  - url: https://en.wikipedia.org/wiki/Differential_of_the_first_kind
---

## Summary

An Abelian differential is a meromorphic $1$-form on a [[Riemann surface|5K0N8QMV6Q]], written locally as $f(z)\,dz$ with $f$ meromorphic.
The holomorphic ones are the differentials of the first kind.
Their integrals along paths give the abelian integrals, the periods, and the [[Abel–Jacobi map|HE8F7KBF42]].
On a compact Riemann surface of genus $g$, the holomorphic Abelian differentials form a complex vector space of dimension $g$.

## Explanations

### L1 human

A function on a Riemann surface has a value at each point, but a differential is the thing you integrate along a path.
In a local coordinate $z$ it looks like $f(z)\,dz$, and changing the coordinate changes $f$ by the derivative of the change.
On the torus $\C / \Lambda$, the form $dz$ is a holomorphic differential that never vanishes.
Integrating it along the two basic loops of the torus gives the two generators of the lattice $\Lambda$.
On the Riemann sphere there is no nonzero holomorphic differential, which matches genus $0$.