An algebraic group that is also a complete algebraic variety. The group is necesarily an abelian group.

A smooth projective geometrically integral group variety over a field. Over the complex numbers abelian varieties are tori.

In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a complex torus that can be embedded into projective space. Also it is used for the generalization of this concept studied in algebraic geometry over fields more general than the complex numbers. One-dimensional abelian varieties are elliptic curves.