# Martin's Blog

## Deligne's Principle B

Posted by Martin Orr on Thursday, 22 January 2015 at 11:10

As I explained last time, one of the key steps in the the proof of Deligne's theorem on absolute Hodge classes is Principle B. This allows us to take an absolute Hodge classes on one fibre in a family of varieties, and deduce that certain classes are absolute Hodge on other fibres of the same family. In this post I shall explain a proof of Principle B due to Blasius, which I think is simpler than Deligne's original proof.

As I also mentioned last time, one can state Principle B in a number of slightly different forms. I have chosen to use the following version instead of the one I gave last time (where there was a lot hidden in the mention of the Gauss-Manin connection, which I am happy not to have to talk about). This version can be applied to Shimura varieties just as easily, or even more easily, than the previous one.

Theorem. Let be a family of abelian varieties over , with connected base . Let be a global section of .

If there is a point such that is an absolute Hodge class on , then for every , is an absolute Hodge class on .

### Notation in the theorem

In the above theorem, denotes the locally constant sheaf on obtained by pushing forward the constant sheaf on (in the category of holomorphic sheaves of -vector spaces). This is a sheaf in which each fibre (for ) is canonically isomorphic to the singular cohomology group .

The notation means: first evaluate the section at , to get . Then is the preimage of under the comparison isomorphism This is the isomorphism which would have been called in my post on absolute Hodge classes, because we are using identity embedding to turn the complex algebraic variety on the left into the complex manifold on the right. Often people will tacitly identify the two sides of this comparison isomorphism, but I find that when I do that I tend to get confused between and the comparison isomorphisms for other embeddings , so I prefer to add the subscript "dR" in .

should be interpreted similarly, as the element of which the comparison isomorphism maps to .

### The theorem of the fixed part

A key role in Blasius's proof of Principle B is played by the following theorem of Deligne:

Theorem of the fixed part. Let be a smooth projective morphism of smooth algebraic varieties over . Let be a smooth compactification of . Then the natural map is surjective.

I think the reason for the name of this theorem is that one can think of the space of global sections as being the part of the family of Hodge structures which remains fixed when we vary . Hodge theorists also call the theorem the Global Invariant Cycle Theorem. As we shall now explain below, the theorem has the consequence that this fixed part , a priori only a -vector space, is in fact canonically a -Hodge structure.

The proof of the theorem of the fixed part has two parts: degeneracy of the Leray spectral sequence implies that is surjective, and one can use mixed Hodge structures to show that is surjective. Note that the reason why it is important to have a compactification in the theorem, and not just be content with the surjectivity of , is because is a pure -Hodge structure, while is only a mixed -Hodge structure.

For each , there is an injection (It is an injection because the local system is locally constant.) The composition is the morphism on cohomology induced by the inclusion of varieties , so it is a morphism of Hodge structures. In particular, is a sub-Hodge structure of . The fact that is an injection implies that (observe that this implies that is independent of ).

The theorem of the fixed part implies that as -vector spaces, and the fact that is a sub-Hodge structure of implies that we can give the quotient Hodge structure.

### Blasius's proof of Principle B

Blasius's proof is based upon the following commutative diagram, for each and for each :

We begin by applying the theorem of the fixed part to our section ), getting a class such that . Let be the corresponding de Rham class in . Commutativity of the above diagram for , together with the fact that the right vertical arrow is an isomorphism, implies that for all .

We can then construct a section for each . Commutativity of the diagram implies that for all and .

Now it is fairly easy to finish off. First consider what happens at the point , where we are given that is an absolute Hodge class. In other words, is a Hodge class on for every . The fact that is an injection of Hodge structures allows us to deduce that is a Hodge class in (with respect to the Hodge structure constructed in the last section).

But since is also a morphism of Hodge structures for every and , this implies that is always a Hodge class, which is just what we need.

1. Absolute Hodge classes in l-adic cohomology From Martin's Blog

We can define absolute Hodge classes in l-adic cohomology in the same way as absolute Hodge classes in de Rham cohomology. We can then prove Deligne's theorem, that Hodge classes on an abelian variety are absolute Hodge, for l-adic cohomology. ...