Math 2.0

Functions of a complex variable Definition

Abelian differential

Builds on Riemann surfaceMeromorphic functionDifferential formResidue

Definition

Let XX be a Riemann surface. An Abelian differential on XX is a meromorphic 11-form on XX. That is, to each chart zz of XX it assigns a meromorphic function ff on the chart domain, written f(z) dzf(z)\,dz. On the overlap of two charts zz and ww with forms f(z) dzf(z)\,dz and g(w) dwg(w)\,dw, these must satisfy f(z)=g(w(z)) dwdz.f(z) = g(w(z))\, \frac{dw}{dz}. The differential is classified as follows.

  1. It is of the first kind if every ff is holomorphic.
  2. It is of the second kind if every residue of every ff 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/ΛX = \C / \Lambda for a lattice Λ\Lambda, the form dzdz is well defined, since the coordinate changes are translations w=z+λw = z + \lambda with dw/dz=1dw/dz = 1. It is of the first kind and has no zeros.

Non-example: on the Riemann sphere, e1/z dze^{1/z}\,dz is not an Abelian differential, because e1/ze^{1/z} has an essential singularity at 00 and so is not meromorphic there. Also dzdz itself on the sphere has a pole of order 22 at ∞\infty: in the chart w=1/zw = 1/z it becomes −w−2 dw-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 11-form only.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Differential form, concept, no evidence levelDifferential formconceptMeromorphic function, concept, no evidence levelMeromorphic functionconceptResidue, concept, no evidence levelResidueconceptRiemann surface, concept, no evidence levelRiemann surfaceconceptAbelian differential, definition, no evidence levelAbelian differentialdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Differential form concept
  2. Meromorphic function concept
  3. Residue concept
  4. Riemann surface concept
  5. Abelian differential definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Differential_of_the_first_kind
About this page
TagV6ZC875F8R
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Kinddefinition
WritingW0 stub
MSC30F30, 14H55
DefinesAbelian differential
UsesRiemann surface, Meromorphic function, Differential form, Residue
Changed2 November 2026

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