An **Abelian variety** is a fundamental concept in algebraic geometry and is defined as a projective algebraic variety that has the structure of a group variety. More formally, an Abelian variety can be described as follows: 1. **Projective Variety**: It is a complex manifold that can be embedded in projective space \(\mathbb{P}^n\) for some integer \(n\). This means it can be described in terms of polynomial equations.
Articles by others on the same topic
An abelian variety over a field is a complete, geometrically connected group variety; equivalently it is a connected projective algebraic group in the usual smooth variety setting. Its group operation is necessarily commutative, a consequence of rigidity rather than an extra condition hidden in the name. An elliptic curve with a chosen origin is a one-dimensional abelian variety; products of elliptic curves give higher-dimensional examples. The additive and multiplicative algebraic groups are not complete and hence are not abelian varieties. Their geometry links line bundles, torsion elements and the Jacobian variety of a curve to questions in number theory.[ref]