Martin Orr's Blog

Deligne's Principle B

Posted by Martin Orr on Thursday, 22 January 2015 at 11:10

As I explained last time, one of the key steps in the the proof of Deligne's theorem on absolute Hodge classes is Principle B. This allows us to take an absolute Hodge classes on one fibre in a family of varieties, and deduce that certain classes are absolute Hodge on other fibres of the same family. In this post I shall explain a proof of Principle B due to Blasius, which I think is simpler than Deligne's original proof.

As I also mentioned last time, one can state Principle B in a number of slightly different forms. I have chosen to use the following version instead of the one I gave last time (where there was a lot hidden in the mention of the Gauss-Manin connection, which I am happy not to have to talk about). This version can be applied to Shimura varieties just as easily, or even more easily, than the previous one.

Theorem. Let \pi \colon \mathcal{A} \to S be a family of abelian varieties over \mathbb{C}, with connected base S. Let v be a global section of R^{2p} \pi_* \mathbb{Q}.

If there is a point s \in S(\mathbb{C}) such that v_{s,dR} is an absolute Hodge class on \mathcal{A}_s, then for every t \in S(\mathbb{C}), v_{t,dR} is an absolute Hodge class on \mathcal{A}_t.

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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 Read more...

Absolute Hodge classes

Posted by Martin Orr on Thursday, 20 November 2014 at 18:55

Let A be an abelian variety over a field k of characteristic zero. For each embedding \sigma \colon k \hookrightarrow \mathbb{C}, we get a complex abelian variety A^\sigma by applying \sigma to the coefficients of equations defining A.

Whenever an object attached to A is defined algebraically, we will get closely related objects for each A^\sigma. On the other hand, whenever we use complex analysis to define an object attached to A^\sigma, we should expect to get completely unrelated things for different \sigma (if k = \mathbb{C} then most field embeddings k \hookrightarrow \mathbb{C} are horribly discontinuous so will mess up anything analytic).

Hodge classes provide a special case: the definition of Hodge classes on A^\sigma as H^{2p}(A^\sigma, \mathbb{Z}) \cap H^{p,p}(A^\sigma) is analytic so we expect no relation between Hodge classes on different A^\sigma. But the Hodge conjecture says that every Hodge class in H^{2p}(A^\sigma, \mathbb{C}) is an algebraic cycle class, and this implies the associated cohomology class in H^{2p}(A^{\sigma'}, \mathbb{C}) is also a Hodge class for every \sigma' \colon k \hookrightarrow \mathbb{C}. (We will explain in the post why there is a natural semilinear isomorphism H^{2p}(A^\sigma, \mathbb{C}) \to H^{2p}(A^{\sigma'}, \mathbb{C}).)

Deligne had the idea that we could pick this out as a partial step on the way to the Hodge conjecture: he defined an absolute Hodge class to be a cohomology class such that its associated class on A^\sigma is a Hodge class for every \sigma and proved that every Hodge class on an abelian variety is an absolute Hodge class. It turns out that this is sufficient to obtain some of the consequences which would follow from the Hodge conjecture. In this post we will explain the definition of absolute Hodge classes.

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Deligne's Principle A and the Mumford-Tate conjecture

Posted by Martin Orr on Wednesday, 10 September 2014 at 11:20

In this post I will fill in a missing detail from two weeks ago, where I mentioned that the Mumford-Tate group is determined by the Hodge classes. More precisely, I will show that an element g of \operatorname{GL}(H_1(A, \mathbb{Z})) is in the Mumford-Tate group if and only if every Hodge class on every Cartesian power A^r is an eigenvector of g. In the context of Deligne's theorem on absolute Hodge classes, this is known as Principle A.

We will also see that a version of this statement holds for the \ell-adic monodromy group and Tate classes. This implies a link between the Hodge, Tate and Mumford-Tate conjectures.

no comments Tags abelian-varieties, alg-geom, alg-groups, hodge, maths, number-theory Read more...

Tate classes

Posted by Martin Orr on Tuesday, 02 September 2014 at 19:30

In my last post I talked about Hodge classes on abelian varieties. Today I will talk about the analogue in \ell-adic cohomology, called Tate classes. Tate classes are defined to be classes in a Tate twist of the \ell-adic cohomology on which the absolute Galois group of the base field acts trivially.

The Tate classes on a variety change if we extend the base field (because this changes the Galois group). They are mainly interesting in the case in which the base field is finitely generated. In this post I will also define potentially Tate classes, which depend less strongly on the base field (they are unchanged by finite extensions).

I will state the Tate conjecture, the \ell-adic analogue of the Hodge conjecture, which says that if the base field is finitely generated, then the vector space of Tate classes is spanned by classes of algebraic cycles. I will also mention some other conjectures which are implied by or equivalent to the Tate conjecture or a slight strengthening of it.

Unlike in the case of Hodge classes, we cannot easily ignore the Tate twist in the definition of Tate classes. This post only contains brief remarks on Tate twists; there is a link to a later post with a more detailed discussion.

no comments Tags abelian-varieties, alg-geom, maths, number-theory Read more...

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