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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...

The Hodge filtration and universal vector extensions

Posted by Martin Orr on Friday, 13 June 2014 at 20:10

We will begin this post by looking at the isomorphism between the Hodge filtration  H^{0,-1}(A) \subset H_1(A, \mathbb{C}) of a complex abelian variety A and the natural filtration  T_0(A^\vee)^\vee \subset T_0(E_A) on the tangent space to the universal vector extension of A.

The significance of this isomorphism is that the Hodge filtration, as we defined it before, is constructed by transcendental methods, valid only over \mathbb{C}, but the universal vector extension is an object of algebraic geometry. So this gives us an analogue for the Hodge filtration for abelian varieties over any base field. Furthermore, in the usual way of algebraic geometry, the construction of the universal vector extension can be carried out uniformly in families of abelian varieties.

We will use the construction of the universal vector extension in families to show that “the Hodge filtration varies algebraically in families.” We will first have to explain what this statement means. We will also mention briefly why H^{-1,0}(A) does not vary algebraically.

A note on the general philosophy of this post: the usual construction of an algebraic-geometric object isomorphic to the Hodge filtration uses de Rham cohomology, which works for H^n of an arbitrary smooth projective variety. My aim in using universal vector extensions is to give an ad hoc construction of de Rham (co)homology, valid only for H_1 of an abelian variety, requiring less sophisticated technology than the general construction. This fits with previous discussion on this blog of the Hodge structure on H_1, constructed via the exponential map from the tangent space of A, and of the \ell-adic H_1, constructed as the Tate module.

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

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