Martin Orr's Blog

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...

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...

Universal vector extensions of abelian varieties

Posted by Martin Orr on Friday, 09 May 2014 at 18:25

In the last post we showed that extensions of an abelian variety A by the additive group \mathbb{G}_a are classified by a vector space \operatorname{Ext}^1(A, \mathbb{G}_a) which is canonically isomorphic to T_0(A^\vee). In this post I will show that there is a so-called universal vector extension of A, that is, a vector extension E_0 of A such that every vector extension of A can be obtained in a unique way as a pushout of E_0. The vector group part of E_0 is \operatorname{Ext}^1(A, \mathbb{G}_a)^\vee, and an ingredient which we require from last time is that this is finite-dimensional.

Even if you are not interested in vector extensions of abelian varieties for their own sake (and I am not), they still have interesting properties. For example, I will show that they provide examples of algebraic varieties over the complex numbers which are non-isomorphic as algebraic varieties but which are isomorphic as complex manifolds.

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