= Abelian variety
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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>.<Abelian Varieties by J. S. Milne (course notes)>[Chapter I, basic definitions and rigidity]{ref}
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