Martin Orr's Blog

Deligne's theorem on absolute Hodge classes

Posted by Martin Orr on Wednesday, 17 December 2014 at 19:00

Today I will outline the proof of Deligne's theorem that Hodge classes on an abelian variety are absolute Hodge. The proof goes through three steps of reducing to increasingly special types of abelian varieties, until finally one reaches a case where it is easy to finish off. This post has ended up longer than usual, but I don't think it is worth splitting into two.

A key ingredient is Deligne's Principle B, which is used for two of the three reduction steps. Principle B says that if we have a family of varieties \mathcal{A} \to S and a flat section of the relative de Rham cohomology bundle \mathcal{H}_{dR}^{n}(\mathcal{A}/S), such that the section specialises to an absolute Hodge class at one point of S, then in fact it is absolute Hodge everywhere. This means that, if we have a method for constructing suitable families of abelian varieties and sections of their relative de Rham cohomology, then we only have to prove that Hodge classes are absolute Hodge at one point of each relevant family. We use Shimura varieties to construct these families of abelian varieties on which to apply Principle B.

The outline of the proof looks like this:

  1. Reduce to Hodge classes on abelian varieties of CM type (using Principle B)
  2. Reduce to a special type of Hodge classes, called Weil classes, on a special type of abelian variety, called abelian varieties of split Weil type (using linear algebra)
  3. Reduce to Hodge classes on abelian varieties which are isogenous to a power of an elliptic curve (using Principle B)
  4. Observe that it is easy to prove Deligne's theorem (and indeed the Hodge conjecture) for abelian varieties which are isogenous to a power of an elliptic curve

no comments Tags abelian-varieties, alg-geom, hodge, maths, shimura-varieties