Dual Abelian Variety

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In mathematics, a **dual abelian variety** can be defined from an abelian variety *A*, defined over a field *K*.

## Definition

To an abelian variety *A* over a field *k*, one associates a **dual abelian variety** *A*<sup>v</sup> (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*<sup>v</sup> and a family of degree 0 line bundles *P*, the Poincaré bundle, parametrized by *A*<sup>v</sup> such that a family *L* on *T* is associated a unique morphism *f*: *T* → *A*<sup>v</sup> so that *L* is isomorphic to the pullback of *P* along the morphism 1<sub>A</sub>×*f*: *A*×*T* → *A*×*A*<sup>v</sup>. Applying this to the case when *T* is a point, we see that the points of *A*<sup>v</sup> correspond to line bundles of degree 0 on *A*, so there is a natural group operation on *A*<sup>v</sup> given by tensor product of line bundles, which makes it into an abelian variety.

This association is a duality in the sense that there is a natural isomorphism between the double dual*A*<sup>vv</sup> and *A* (defined via the Poincaré bundle) and that it is contravariant functorial, i.e. it...

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- for all <math>t in T</math>, the restriction of
*L*to*A*× is a degree 0 line bundle, - the restriction of
*L*to ×*T*is a trivial line bundle (here 0 is the identity of*A*).

Then there is a variety

This association is a duality in the sense that there is a natural isomorphism between the double dual

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