Dual abelian variety

Summary

The dual of an abelian variety $A$ is an abelian variety $A^\vee$ of the same dimension. Its points are the line bundles on $A$ that are algebraically equivalent to zero. It is the identity component of the Picard group of $A$, made into a variety. Duality is an involution: the dual of $A^\vee$ is $A$ again. Polarizations of $A$ are certain maps $A \to A^\vee$, and the Weil pairing comes from duality.