In mathematics, a dual abelian variety can be defined from an abelian variety A, defined over a field k. A 1-dimensional abelian variety is an elliptic curve, and every elliptic curve is isomorphic to its dual, but this fails for higher-dimensional abelian varieties, so the concept of dual becomes more interesting in higher dimensions.

Let A be an abelian variety over a field k. We define ${\displaystyle \operatorname {Pic} ^{0}(A)\subset \operatorname {Pic} (A)}$ to be the subgroup of the Picard group consisting of line bundles L such that ${\displaystyle m^{*}L\cong p^{*}L\otimes q^{*}L}$, where ${\displaystyle m,p,q}$ are the multiplication and projection maps ${\displaystyle A\times _{k}A\to A}$ respectively. An element of ${\displaystyle \operatorname {Pic} ^{0}(A)}$ is called a degree 0 line bundle on A.

To A one then associates a dual abelian variety A^v (over the same field), which is the solution to the following moduli problem. A family of degree 0 line bundles parametrized by a k-variety T is defined to be a line bundle L on A×T such that

Then there is a variety A^v and a line bundle ${\displaystyle P\to A\times A^{\vee }}$, called the Poincaré bundle, which is a family of degree 0 line bundles parametrized by A^v in the sense of the above definition. Moreover, this family is universal, that is, to any family L parametrized by T is associated a unique morphism f: T → A^v so that L is isomorphic to the pullback of P along the morphism 1_A×f: A×T → A×A^v. Applying this to the case when T is a point, we see that the points of A^v correspond to line bundles of degree 0 on A, so there is a natural group operation on A^v given by tensor product of line bundles, which makes it into an abelian variety.

In the language of representable functors one can state the above result as follows. The contravariant functor, which associates to each k-variety T the set of families of degree 0 line bundles parametrised by T and to each k-morphism f: T → T' the mapping induced by the pullback with f, is representable. The universal element representing this functor is the pair (A^v, P).

This association is a duality in the sense that there is a natural isomorphism between the double dual A^vv and A (defined via the Poincaré bundle) and that it is contravariant functorial, i.e. it associates to all morphisms f: A → B dual morphisms f^v: B^v → A^v in a compatible way. The n-torsion of an abelian variety and the n-torsion of its dual are dual to each other when n is coprime to the characteristic of the base. In general - for all n - the n-torsion group schemes of dual abelian varieties are Cartier duals of each other. This generalizes the Weil pairing for elliptic curves.

The theory was first put into a good form when K was the field of complex numbers. In that case there is a general form of duality between the Albanese variety of a complete variety V, and its Picard variety; this was realised, for definitions in terms of complex tori, as soon as André Weil had given a general definition of Albanese variety. For an abelian variety A, the Albanese variety is A itself, so the dual should be Pic⁰(A), the connected component of the identity element of what in contemporary terminology is the Picard scheme.

For the case of the Jacobian variety J of a compact Riemann surface C, the choice of a principal polarization of J gives rise to an identification of J with its own Picard variety. This in a sense is just a consequence of Abel's theorem. For general abelian varieties, still over the complex numbers, A is in the same isogeny class as its dual. An explicit isogeny can be constructed by use of an invertible sheaf L on A (i.e. in this case a holomorphic line bundle), when the subgroup