Dual abelian variety

Dual Abelian Variety

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.


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

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: TA<sup>v</sup> so that L is isomorphic to the pullback of P along the morphism 1<sub>A</sub>×f: A×TA×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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