Deligne's theorem on absolute Hodge classes
Posted by Martin Orr on Wednesday, 17 December 2014 at 19:00
Today I will outline the proof of Deligne's theorem that Hodge classes on an abelian variety are absolute Hodge. The proof goes through three steps of reducing to increasingly special types of abelian varieties, until finally one reaches a case where it is easy to finish off. This post has ended up longer than usual, but I don't think it is worth splitting into two.
A key ingredient is Deligne's Principle B, which is used for two of the three reduction steps.
Principle B says that if we have a family of varieties and a flat section of the relative de Rham cohomology bundle , such that the section specialises to an absolute Hodge class at one point of , then in fact it is absolute Hodge everywhere.
This means that, if we have a method for constructing suitable families of abelian varieties and sections of their relative de Rham cohomology, then we only have to prove that Hodge classes are absolute Hodge at one point of each relevant family.
We use Shimura varieties to construct these families of abelian varieties on which to apply Principle B.
The outline of the proof looks like this:
 Reduce to Hodge classes on abelian varieties of CM type (using Principle B)
 Reduce to a special type of Hodge classes, called Weil classes, on a special type of abelian variety, called abelian varieties of split Weil type (using linear algebra)
 Reduce to Hodge classes on abelian varieties which are isogenous to a power of an elliptic curve (using Principle B)
 Observe that it is easy to prove Deligne's theorem (and indeed the Hodge conjecture) for abelian varieties which are isogenous to a power of an elliptic curve
Deligne's Principle B
Before stating Principle B, we will need the concept of the relative de Rham cohomology bundle for a family of abelian varieties .
Of course there is a general definition of the relative de Rham cohomology bundle for any scheme, but we will take a shortcut for families of abelian varieties.
Just as we defined the de Rham cohomology of an abelian variety to be exterior powers of the cotangent space of the universal vector extension, we can do the same in families.
Recall that we can construct the universal vector extension of the family, and its relative tangent bundle
is a vector bundle on
.
We define
where the dual and exterior power are taken in the category of vector bundles on .
The fibre of this vector bundle at each point is the de Rham cohomology
.
Now we can state Principle B. There are a variety of ways of stating this with slightly different input classes . It is not necessarily easy to prove directly the equivalence of these different statements, but if you know how to prove one version of Principle B then you probably prove them all.
Theorem. Let be a family of abelian varieties over , with connected base . Let be a global section of
. If
 is flat with respect to the GaussManin connection (don't worry what this means; I might explain it next time); and
 there is a point such that
is an absolute Hodge class on
,
then
is an absolute Hodge class on
for every .
Constructing families of abelian varieties: Shimura varieties
I am not going to give an explanation of the theory of Shimura varieties here (one day I want to write a book on the subject...) All I will say is that, if you take a suitable algebraic group over , you can form a Shimura variety which is the ``moduli space of abelian varieties whose MumfordTate group is contained in .'' Properly speaking, in order to fully specify the moduli problem, you need to add some extra data like a polarisation and a level structure and an action of on the Hodge structures associated with the abelian varieties but I shall pass over those details here. The Shimura variety is an algebraic variety over .
If you choose the level structure suitably, then is a fine moduli space.
This means that there is a family of abelian varieties such that, for each , the abelian variety which corresponds to (via the moduli interpretation of ) is in fact the fibre , plus the relevant extra data matches up.
As I mentioned above, the data when we set up a Shimura variety as a moduli space of abelian varieties include an action of on the cohomology groups of the fibres .
By the construction of Shimura varieties, the monodromy action of
on these cohomology groups factors through .
This implies that any invariant cohomology class
can be extended to a global section of the relative singular cohomology local system
(actually this is only valid on a connected component of , rather than on all of , but I shall ignore that).
In order to apply Principle B (in the version stated above), we need to know that these sections of are sections of the relative de Rham cohomology bundle
.
The comparison theorem between singular cohomology and de Rham cohomology implies that they are holomorphic sections of the relative de Rham cohomology bundle, but we need to know that they are algebraic sections.
So far as I can see, Deligne never mentions this issue about holomorphic vs algebraic sections but I might easily have missed it.
We can deduce it from his Theorem of the Fixed Part (also known as the Global Invariant Cycle Theorem in Hodge theory).
Alternatively, I think it might be possible to prove it "by hand" using the construction of Shimura varieties as quasiprojective varieties.
The fact we have constructed these sections from sections of implies that they are flat with respect to the GaussManin connection.
Abelian varieties of split Weil type
The notions of Weil classes and abelian varieties of Weil type were first introduced in order to construct examples of abelian varieties for which a certain simple strategy for proving some cases of the Hodge conjecture does not work. However, Deligne also found them very useful in proving his theorem on absolute Hodge classes, which is a positive statement towards the Hodge conjecture.
Fix a CM field and an even positive integer .
Suppose we have a complex abelian variety of dimension equipped with a homomorphism .
This induces a structure of dimensional vector space on , and hence the MumfordTate group of is a subgroup of
.
The complex homology group splits into eigenspaces for the action of , one for each embedding , which we will call
.
Define
and observe that
for all .
We say that is of Weil type (relative to ) if for all .
Equivalently, the MumfordTate group of is contained in the group
for some Hermitian form on with signature
with respect to every embedding .
We say that is of split Weil type if the Hermitian form is split i.e. equivalent to the standard Hermitian form
(this is a condition on a polarisation of ).
If is an abelian variety of Weil type, a cohomology class in is said to be a Weil class if it is invariant under the action of
.
Because of what we said about the MumfordTate group of abelian varieties of Weil type, Weil classes are always Hodge classes.
The usefulness of Weil classes is due to the following theorem of AndrĂ©, proved by linear algebra manipulations of CM Hodge structures:
Theorem. For every complex abelian variety of CM type, there exist complex abelian varieties
of split Weil type and homomorphisms
such that every Hodge class on is a linear combination of pullbacks of Weil classes on the
.
Putting it together
We can now carry out the steps of the proof of Deligne's theorem mentioned in the introduction as follows:

We start with an arbitrary abelian variety . Let and form the universal family
of abelian varieties whose MumfordTate group is contained in (plus suitable extra data). Then there is certainly a point such that
. Since every Hodge class of is fixed by , it extends (as discussed above) to a global section of
. By the theory of Shimura varieties, there is a point such that has complex multiplication. If we can prove that is an absolute Hodge class, then we can use Principle B to deduce that is absolute Hodge.

We now have a Hodge class on an abelian variety of CM type. Using AndrĂ©'s theorem and the fact that a pullback of an absolute Hodge class is absolute Hodge, we reduce to proving that Weil classes on abelian varieties of split Weil type are absolute Hodge.

Let
and again form the universal family of abelian varieties with MumfordTate group contained in . ( is an example of a Shimura variety of type PEL. It parameterises abelian varieties with the following structure: Polarisation:  Endomorphisms: containing  and a Level structure.) Note that will not necessarily stabilise every Hodge class on , but by definition it does stabilise Weil classes. Hence a Weil class on can be extended to a section of
. One can construct a point whose corresponding abelian variety is isogenous to a power of a nonCM elliptic curve. By Principle B it suffices to show that is an absolute Hodge class.

One can describe very explicitly all the Hodge classes on an abelian variety isogenous to a power of an elliptic curve  in particular they are all in the span of exterior powers of Hodge classes of weight 2. Hence one can easily prove that they are all absolute Hodge.