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  "body": "\n## Summary\n\nThe dual of an [[abelian variety|R7W03TXRS9]] $A$ is an abelian variety $A^\\vee$ of the same dimension.\nIts points are the line bundles on $A$ that are algebraically equivalent to zero.\nIt is the identity component of the [[Picard group|3Z9DCWYXQZ]] of $A$, made into a variety.\nDuality is an involution: the dual of $A^\\vee$ is $A$ again.\nPolarizations of $A$ are certain maps $A \\to A^\\vee$, and the Weil pairing comes from duality.\n\n## Explanations\n\n### L1 human\n\nOn an abelian variety one can look at all line bundles, the algebraic analogues of twisted strips over the space.\nThose that can be deformed continuously to the trivial one form a group that is itself an abelian variety.\nThis new abelian variety is the dual.\nFor an elliptic curve $E$ with origin $O$, the point $P$ corresponds to the line bundle of the divisor $P - O$, so $E$ is its own dual.\nIn higher dimension $A$ and $A^\\vee$ are isogenous but need not be isomorphic.\n",
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