Abelian variety

From Citizendium, the Citizens' Compendium

In algebraic geometry an Abelian variety over a field is a projective variety, together with a marked point and two algebraic maps: addition and inverse , such that these two maps, and the point satisfy the Abelian group axioms. One dimensional Abelian varieties are elliptic curves. Over the complex numbers Abelian varieties are a subset of the set of complex tori.