Martin Orr's Blog

Absolute Hodge classes in l-adic cohomology

Posted by Martin Orr on Friday, 26 June 2015 at 11:30

We can define absolute Hodge classes in \ell-adic cohomology in the same way as absolute Hodge classes in de Rham cohomology. We can then prove Deligne's theorem, that Hodge classes on an abelian variety are absolute Hodge, for \ell-adic cohomology. Because it is easy to prove that absolute Hodge classes in \ell-adic cohomology are potentially Tate classes, this implies half of the Mumford-Tate conjecture.

In particular, it implies that if A is an abelian variety over a number field, then a finite index subgroup of the image of the \ell-adic Galois representation on T_\ell A is contained in the \mathbb{Q}_\ell-points of the Mumford-Tate group of A. This is the goal I have been working towards for some time on this blog.

Deligne's definition of absolute Hodge classes considered \ell-adic cohomology (for all \ell) and de Rham cohomology simultaneously. The accounts I read of this theory focussed on the de Rham side, leading me to believe that the de Rham part was essential and the \ell-adic part an optional extra. This is why I wrote the past few posts about de Rham cohomology and am now adding the \ell-adic version on at the end, even though I am more interested in the \ell-adic version. Now that I understand what is going on, I realise that I could have used only \ell-adic cohomology from the beginning. One day I might write up a neater account which uses \ell-adic cohomology only.

Before the main part of this post, talking about absolute Hodge classes in \ell-adic cohomology, I need to talk about Tate twists in \ell-adic cohomology. These are more significant than Tate twists in singular cohomology because they change the Galois representations involved. This resulted in some mistakes in my previous posts on Tate classes, which I think I have now fixed.

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Tate twists in singular and de Rham cohomology

Posted by Martin Orr on Friday, 19 June 2015 at 19:30

Tate twists in singular cohomology are a device for dealing with factors of 2 \pi i which come up whenever we compare singular and de Rham cohomology of complex projective varieties. In this post I will explain the problem, including calculating the 2 \pi i in the case of \mathbb{P}^1, and define Tate twists to solve it.

In the case of singular cohomology, Tate twists are largely a matter of normalising things conveniently. Without them, we could just write out factors of 2 \pi i everywhere. On the other hand, there is also a notion of Tate twists for \ell-adic cohomology, which cannot be omitted so easily, and which I will discuss in a subsequent post.

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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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