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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Posted by Martin Orr on
Thursday, 17 April 2014 at 18:12

A theme of my posts on abelian varieties has been ad hoc constructions of objects which are equivalent to the (co)homology of abelian varieties together with their appropriate extra structures -- the period lattice for singular homology and the Hodge structure, the Tate module for -adic cohomology and its Galois representation. I want to do the same thing for de Rham cohomology. To prepare for this, I need to discuss vector extensions of abelian varieties -- that is extensions of abelian varieties by vector groups.

In this post I will define and classify extensions of an abelian variety by the additive group.
We will conclude that , the set of isomorphism classes of such extensions, forms a vector space isomorphic to the tangent space of the dual of .
Most of this was discovered by Rosenlicht in the 1950s.

Tags
abelian-varieties, alg-geom, maths
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Posted by Martin Orr on
Thursday, 03 January 2013 at 15:51

Today I will discuss the classification of endomorphism algebras of simple abelian varieties.
The endomorphism algebra of a non-simple abelian variety can easily be computed from the endomorphism algebras of its simple factors.
For a simple abelian variety, its endomorphism algebra is a division algebra of finite dimension over

.
(A division algebra is a not-necessarily-commutative algebra in which every non-zero element is invertible.)
As discussed last time, the endomorphism algebra also has a positive involution, the Rosati involution.
There may be many Rosati involutions, coming from different polarisations of the abelian variety, but all we care about today is the existence of a positive involution.
Division algebras with positive involutions were classified by Albert in the 1930s.

Tags
abelian-varieties, alg-geom, maths
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