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  "body": "\n## Summary\n\nAn abelian variety is a complete [[algebraic variety|HV6YBS0YDH]] that is also an [[algebraic group|GH81SYAPPN]], with group operations given by morphisms.\nIts group law is automatically commutative, and the variety is automatically projective.\nAbelian varieties of dimension one are the [[elliptic curves|427N6S6QQW]], and the [[Jacobian variety|ZQ84M9D9TC]] of a curve is a basic example in higher dimension.\nThey are central in arithmetic geometry, for instance in the [[Mordell–Weil theorem|MXRCPAH09H]].\n\n## Explanations\n\n### L1 human\n\nAn abelian variety is a shape cut out by polynomial equations whose points can be added like numbers.\nOver the complex numbers it looks like a doughnut in several complex dimensions, a [[complex torus|Y7MA6KBE9M]].\nThe simplest example is an elliptic curve such as $y^2 = x^3 - x$, together with its point at infinity.\nTwo points on it are added by drawing the line through them and taking the third intersection point, reflected.\nBecause the shape is closed up with no missing points, the group law is very rigid, and this rigidity makes the theory rich.\n",
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