---
tag: R7W03TXRS9
kind: concept
name: Abelian variety
msc: [14K05, 14K20, 11G10]
specializes: [GH81SYAPPN]
sources:
  - url: https://en.wikipedia.org/wiki/Abelian_variety
same_as:
  wikidata: Q318751
  wikipedia: Abelian variety
---

## Summary

An abelian variety is a complete [[algebraic variety|HV6YBS0YDH]] that is also an [[algebraic group|GH81SYAPPN]], with group operations given by morphisms.
Its group law is automatically commutative, and the variety is automatically projective.
Abelian varieties of dimension one are the [[elliptic curves|427N6S6QQW]], and the [[Jacobian variety|ZQ84M9D9TC]] of a curve is a basic example in higher dimension.
They are central in arithmetic geometry, for instance in the [[Mordell–Weil theorem|MXRCPAH09H]].

## Explanations

### L1 human

An abelian variety is a shape cut out by polynomial equations whose points can be added like numbers.
Over the complex numbers it looks like a doughnut in several complex dimensions, a [[complex torus|Y7MA6KBE9M]].
The simplest example is an elliptic curve such as $y^2 = x^3 - x$, together with its point at infinity.
Two points on it are added by drawing the line through them and taking the third intersection point, reflected.
Because the shape is closed up with no missing points, the group law is very rigid, and this rigidity makes the theory rich.