# Abelian variety

In algebraic geometry an Abelian variety ${\displaystyle A}$ over a field ${\displaystyle K}$ is a projective variety, together with a marked point ${\displaystyle 0}$ and two algebraic maps: addition ${\displaystyle A^{2}\to A}$ and inverse ${\displaystyle A\to A}$, such that these two maps, and the point ${\displaystyle 0}$ 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.