Let be an abelian variety over an algebraically closed field . If , then corresponds to a *complex torus*: that is, can be expressed complex analytically as where is a complex vector space of dimension and is a lattice (i.e., a -free, discrete submodule of rank ). In this case, one can form the *dual abelian variety*

At least, as defined is a complex torus, but it turns out to admit the structure of an abelian variety.

The purpose of the next few posts is to describe an algebraic version of this duality: it turns out that can be constructed as a scheme, purely algebraically. I’d like to start with a couple of posts on Picard schemes. A useful reference here is this article of Kleiman.

**1. The Picard scheme analytically**

Let be a smooth projective variety over the complex numbers . The collection of line bundles is a very interesting invariant of . Usually, it splits into two pieces: the “topological” piece and the “analytic” piece. For instance, there is a first Chern class map

which picks out the topological type of a line bundle. (Topologically, line bundles on a space are classified by their first Chern class.) The admissible topological types are precisely the classes in which project to -classes in under the Hodge decomposition. (more…)