Let be an abelian variety over the algebraically closed field . In the previous post, we studied the Picard scheme , or rather its connected component at the identity. The main result was that was itself an abelian variety (in particular, smooth) of the same dimension as , which parametrizes precisely the *translation-invariant* line bundles on .

We also saw how to construct isogenies between and . Given an ample line bundle on , the map

is an isogeny. Such maps were in fact the basic tool in proving the above result.

The goal of this post is to show that the contravariant functor

from abelian varieties over to abelian varieties over , is a well-behaved duality theory. In particular, any abelian variety is canonically isomorphic to its bidual. (This explains why the double Picard functor on a general variety is the universal abelian variety generated by that variety, the so-called Albanese variety.) In fact, we won’t quite finish the proof in this post, but we will finish the most important step: the computation of the cohomology of the universal line bundle on .

Motivated by this, we set the notation:

Definition 11We write for .

The main reference for this post is Mumford’s *Abelian varieties. (more…)*