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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.

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

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. These are defined to be those classes on which the action of the Galois group is given by multiplying by the appropriate power of the cyclotomic character.

The notion of Tate class depends on the base field of our variety (because changing the base field changes the Galois group). They are mainly interesting in the case in which the base field is finitely generated. We will also define potentially Tate classes, which depend less strongly on the base field (they are unchanged by finite extensions).

We 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. We will also mention some other conjectures which are implied by or equivalent to the Tate conjecture or a slight strengthening of it.

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

Hodge classes on abelian varieties

Posted by Martin Orr on Monday, 25 August 2014 at 18:50

In this post I will define Hodge classes and state the Hodge conjecture. I will restrict my attention to the case of abelian varieties and say the minimum amount necessary to be able to discuss the relationships between the Hodge, Tate and Mumford-Tate conjectures and absolute Hodge classes in subsequent posts. There are many excellent accounts of this material already written, which may give greater detail and generality.

Hodge classes are cohomology classes on a complex variety A which are in the intersection of the singular cohomology H^n(A, \mathbb{Q}) and the middle component H^{n/2,n/2}(A) of the Hodge decomposition  H^n(A, \mathbb{Q}) \otimes_\mathbb{Q} \mathbb{C} = \bigoplus_{\substack{p,q\geq 0 \\ p+q=n}} H^{p,q}(A). They can also be defined as rational cohomology classes which are eigenvectors for the Mumford-Tate group. The Hodge conjecture predicts that these classes are precisely the \mathbb{Q}-span of cohomology classes coming from algebraic subvarieties of A.

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

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