Abelian variety

In algebraic geometry an Abelian variety $A$ over a field $K$ is a projective variety, together with a marked point $0$ and two algebraic maps: addition $A^{2}\to A$ and inverse $A\to A$, such that these two maps, and the point $0$ satisfy the Abelian group axioms. One dimensional Abelian varieties are elliptic curves. Over the complex numbers Abelian varieties are as subset of the set of complex tori.