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

of

is in the Mumford-Tate group if and only if every Hodge class on every Cartesian power is an eigenvector of .
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 -adic monodromy group and Tate classes.
This implies a link between the Hodge, Tate and Mumford-Tate conjectures.

Tags
abelian-varieties, alg-geom, alg-groups, hodge, maths, number-theory
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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 -adic cohomology, called Tate classes.
Tate classes are defined to be classes in a Tate twist of the

-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 -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.

Tags
abelian-varieties, alg-geom, maths, number-theory
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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

which are in the intersection of the singular cohomology

and the middle component

of the Hodge decomposition

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

-span of cohomology classes coming from algebraic subvarieties of

.

Tags
abelian-varieties, alg-geom, hodge, maths
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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

of a complex abelian variety

and the natural filtration

on the tangent space to the universal vector extension of

.

The significance of this isomorphism is that the Hodge filtration, as we defined it before, is constructed by transcendental methods, valid only over

,
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

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

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

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

, constructed via the exponential map from the tangent space of

, and of the

-adic

, constructed as the Tate module.

Tags
abelian-varieties, alg-geom, hodge, maths
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