---
tag: V6ZC875F8R
kind: definition
name: Abelian differential
msc: [30F30, 14H55]
defines: P014C13K5H
uses: [5K0N8QMV6Q, 0ZZ7ZYDR8W, 0E3G0GRPZ6, 0PHN07KVQT]
sources:
  - url: https://en.wikipedia.org/wiki/Differential_of_the_first_kind
---

## Definition

Let $X$ be a [[Riemann surface|5K0N8QMV6Q]].
An *Abelian differential* on $X$ is a meromorphic $1$-form on $X$.
That is, to each chart $z$ of $X$ it assigns a [[meromorphic function|0ZZ7ZYDR8W]] $f$ on the chart domain, written $f(z)\,dz$.
On the overlap of two charts $z$ and $w$ with forms $f(z)\,dz$ and $g(w)\,dw$, these must satisfy
$$f(z) = g(w(z))\, \frac{dw}{dz}.$$
The differential is classified as follows.

1. It is of the *first kind* if every $f$ is holomorphic.
2. It is of the *second kind* if every [[residue|0PHN07KVQT]] of every $f$ is zero.
3. It is of the *third kind* in general, that is, with arbitrary poles and residues.

## Remarks

Example: on the torus $X = \C / \Lambda$ for a lattice $\Lambda$, the form $dz$ is well defined, since the coordinate changes are translations $w = z + \lambda$ with $dw/dz = 1$.
It is of the first kind and has no zeros.

Non-example: on the Riemann sphere, $e^{1/z}\,dz$ is not an Abelian differential, because $e^{1/z}$ has an essential singularity at $0$ and so is not meromorphic there.
Also $dz$ itself on the sphere has a pole of order $2$ at $\infty$: in the chart $w = 1/z$ it becomes $-w^{-2}\,dw$.
So it is an Abelian differential of the second kind but not of the first kind.

Some authors, notably in the study of translation surfaces, use "Abelian differential" for a holomorphic $1$-form only.