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
- 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
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...
Read More