Dual abelian variety

Dual Abelian Variety

Dual abelian variety

to get instant updates about 'Dual Abelian Variety' on your MyPage. Meet other similar minded people. Its Free!

X 

All Updates


Description:
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 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...
Read More

No feeds found

All
wait Posting your question. Please wait!...


No updates available.
No messages found
Tell your friends >
about this page
 Create a new Page
for companies, colleges, celebrities or anything you like.Get updates on MyPage.
Create a new Page
 Find your friends
  Find friends on MyPage from