## Vector extensions of abelian varieties

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.

### Definitions

Throughout this post, will be a field of characteristic zero (most of it works in positive characteristic as well, provided that you take care about separability in various places).

We define a *vector group* to be an algebraic group with an action of the multiplicative group on such that there is an isomorphism of algebraic groups between and some power of the additive group, under which the action of on corresponds to its natural action on .
(Because we are restricting to characteristic zero, there is only one possible action of on each , but in positive characteristic there is more than one.)

If is an abelian variety and is a vector group, then an *extension of by * is an algebraic group together with a short exact sequence
(in positive characteristic, we also require the morphisms in the short exact sequence to be separable).
Note that the short exact sequence is part of the data, so if we have an extension
and we replace by then we get a different extension.
Nevertheless, we will often write just as a shorthand for the extension.

The group is always commutative, even though this is not part of the definition. This follows from the fact that for any algebraic group , the quotient by the centre is an affine algebraic group.

Given any abelian variety and vector group , we can form the trivial extension , which is just the direct product of the groups with the obvious morphisms.

### Operations on vector extensions

Here are some methods of constructing new vector extensions from old.

First, the pushout. Given a vector extension
and a morphism of vector groups ,
there is a unique extension of by such that there exists a morphism of algebraic groups making the following diagram commute:
```
This extension is called the
```

*pushout* of by and denoted .

If we fix and , the set of isomorphism classes of extensions of by is denoted

We can define a -vector space structure on as follows:

Given two extensions and of by , we take their fibre product , which is an extension of by , then push out by the map This gives the sum .

To multiply an extension by a scalar , we simply push out by the multiplication-by- map . This is equivalent to replacing the morphism by .

### Principal bundles

In order to classify vector extensions, it is useful to consider more general objects called principal bundles.

Let be an algebraic group and a variety.

A *principal -bundle over * is an algebraic variety together with a morphism and an action of on such that there exists an open cover ```
of and isomorphisms
which commute with the projections down to and under which the restriction of to corresponds to the action of on given by
Note that this implies that restricts to a free and transitive action of on the -points of each fibre of .
A principal -bundle is also called a
```

*-torsor over *.

If is an extension of an abelian variety by a vector group , then letting act on by translations makes into a principal -bundle over . The key step in proving this is Rosenlicht's cross section theorem, which asserts that there is a rational map (defined over ) such that is the identity wherever it is defined.

The above paragraph works for any algebraic group and any solvable algebraic group but the converse which we will discuss later works only for abelian varieties and vector groups .

### Cohomological classification

It is a standard fact that isomorphism classes of principal -bundles on a variety are in canonical bijection with the sheaf cohomology group where is the sheaf . This is because, if we take isomorphisms as in the definition of a principal -bundle, then the functions are constant on the fibres of and form a Čech cocycle on with values in .

Combining this with the previous section, we get that there is a canonical map for any abelian variety . (The sheaf is the same as sheaf of regular functions .) One can show that this map is -linear.

This map is an isomorphism by the following proposition.

Proposition.Every principal -bundle over an abelian variety can uniquely be given the structure of an extension of by .

We observe that is canonically isomorphic to the tangent space of the dual abelian variety at the origin, by a nice argument using dual numbers (see Chapter III, Proposition 2.1 in Milne's notes on abelian varieties -- note that ).

Hence we get a canonical isomorphism of -vector spaces

We conclude with a sketch proof of the proposition.

Sketch proof of proposition.The extra ingredient which we must give to a principal -bundle in order to make it an extension of by is an algebraic group law , compatible with the action of and with the projection to .The uniqueness is easy: suppose that we had two group laws and on , and given consider the function defined by Because both group laws are compatible with the action of and the projection to , factors as and the morphism must be constant because is complete and is affine.

In order to prove existence, define a

translationof the principal -bundle to be a -morphism such that

- is constant (i.e. lifts a translation of ); and
- commutes with .
The key step is to prove that every translation of can be lifted to a translation of . We can deduce from this that there is a bijection between and the set of translations of ; since the latter forms a group, we get a group law on if we choose a point to be the identity element. This group law is compatible with and if we choose this point in (Rosenlicht's cross section theorem implies that there is a -point in ). This construction is functorial in the base ring, so in fact we get an algebraic group law.

Now we prove that every translation of can be lifted to a translation of . Let , and let be the corresponding translation. We get a new principal -bundle , and a translation of as defined above is the same as an isomorphism of principal -bundles Hence to prove the proposition, it suffices to prove that these bundles are isomorphic.

The translation induces a linear map

`If the bundle corresponds to the class , then corresponds to`

`. So we are reduced to showing that`

`is the identity.`

The map

`defines a group homomorphism Since is a finite-dimensional vector space, is an affine algebraic group. The homomorphism is functorial in the base ring, so it is a morphism of algebraic groups from an abelian variety to an affine group, and so it is trivial.`