The Kuga-Satake Construction

Abstract:

Number theorists, algebraic geometers, and indeed all mathematicians love elliptic curves. They have been the subject of study for centuries and serve as a natural testing ground for many modern conjectures. Today we know a great deal about them. Modern advances have extended much of our knowledge from elliptic curves to their higher dimensional analogue, abelian varieties. For instance, these were some of the first varieties for which the now general Weil conjectures were known. The Tate conjecture (not fully general yet) was also first proven for abelian varieties.

These and other results have begun to be extended to yet more general varieties by application of the \emph{Kuga-Satake construction}. This takes as input a Hodge structure of a certain kind and outputs an abelian variety. When the Hodge structure arises from a nice variety, this can allow one to communicate results known for abelian varieties to this new nice variety. One case of particular interest is when the nice variety is a K3 surface. Deligne used this construction to prove the Weil conjectures for K3 surfaces (though his proof in general used different methods). Tankeev used the construction to prove the Tate conjecture for K3 surfaces over number fields. Work of many authors, culminating with papers of Madapusi-Pera, extended the construction to work over general fields and thus proved the Tate conjecture for K3 surfaces in general. There have been further applications since then.

These lectures will give an introduction to the Kuga-Satake construction. We will begin by reviewing complex abelian varieties and Hodge structures, defining K3 surfaces, and demonstrating the classical Kuga-Satake construction over $\mathbb{C}$. We will then discuss Shimura varieties and the work of Madapusi-Pera. There will be examples and applications throughout.

Schedule:

This is very tentative; we'll spend as long on each topic as suits the audience. If people would like to go slower or see more on a certain area, we can do that. Thus, depending on how things run, we may end up having more or less lectures total.

(02/09) Abelian varieties and Hodge structures

(09/09) K3 surfaces

(16/09) The classical Kuga-Satake construction

(23/09) Shimura Varieties and the extended Kuga-Satake