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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 \pi \colon \mathcal{A} \to S be a family of abelian varieties over \mathbb{C}, with connected base S. Let v be a global section of R^{2p} \pi_* \mathbb{Q}.

If there is a point s \in S(\mathbb{C}) such that v_{s,dR} is an absolute Hodge class on \mathcal{A}_s, then for every t \in S(\mathbb{C}), v_{t,dR} is an absolute Hodge class on \mathcal{A}_t.

Notation in the theorem

In the above theorem, R^{2p} \pi_* \mathbb{Q} denotes the locally constant sheaf on S obtained by pushing forward the constant sheaf \mathbb{Q} on \mathcal{A} (in the category of holomorphic sheaves of \mathbb{Q}-vector spaces). This is a sheaf in which each fibre (R^{2p} \pi_* \mathbb{Q})_s (for s \in S(\mathbb{C})) is canonically isomorphic to the singular cohomology group H^{2p}(\mathcal{A}_s, \mathbb{Q}).

The notation v_{s,dR} means: first evaluate the section v at s, to get v_s \in H^{2p}(\mathcal{A}_s, \mathbb{Q}). Then v_{s,dR} is the preimage of v_s under the comparison isomorphism  H^{2p}_{dR}(\mathcal{A}_s/\mathbb{C}) \to H^{2p}(\mathcal{A}_s, \mathbb{C}). This is the isomorphism which would have been called \mathrm{id}^* in my post on absolute Hodge classes, because we are using identity embedding \mathbb{C} \hookrightarrow \mathbb{C} to turn the complex algebraic variety \mathcal{A}_s on the left into the complex manifold \mathcal{A}_s 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 \mathrm{id}^* and the comparison isomorphisms \sigma^* for other embeddings \sigma \colon \mathbb{C} \to \mathbb{C}, so I prefer to add the subscript "dR" in v_{s,dR}.

v_{t,dR} should be interpreted similarly, as the element of H^{2p}_{dR}(\mathcal{A}_t/\mathbb{C}) which the comparison isomorphism \mathrm{id}^* maps to v_t.

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 \pi \colon X \to S be a smooth projective morphism of smooth algebraic varieties over \mathbb{C}. Let \bar{X} be a smooth compactification of X. Then the natural map  \alpha \colon H^n(\bar{X}, \mathbb{Q}) \to \Gamma(S, R^n \pi_* \mathbb{Q}) is surjective.

I think the reason for the name of this theorem is that one can think of the space of global sections \Gamma(S, R^n \pi_* \mathbb{Q}) as being the part of the family of Hodge structures H^n(\mathcal{A}_s, \mathbb{Q}) which remains fixed when we vary s \in S(\mathbb{C}). 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 \Gamma(S, R^n \pi_* \mathbb{Q}) , a priori only a \mathbb{Q}-vector space, is in fact canonically a \mathbb{Q}-Hodge structure.

The proof of the theorem of the fixed part has two parts: degeneracy of the Leray spectral sequence implies that H^n(X, \mathbb{Q}) \to \Gamma(S, R^n \pi_* \mathbb{Q}) is surjective, and one can use mixed Hodge structures to show that H^n(\bar{X}, \mathbb{Q}) \to H^n(X, \mathbb{Q}) is surjective. Note that the reason why it is important to have a compactification \bar{X} in the theorem, and not just be content with the surjectivity of H^n(X, \mathbb{Q}) \to \Gamma(S, R^n \pi_* \mathbb{Q}), is because H^n(\bar{X}, \mathbb{Q}) is a pure \mathbb{Q}-Hodge structure, while H^n(X, \mathbb{Q}) is only a mixed \mathbb{Q}-Hodge structure.

For each s \in S(\mathbb{C}), there is an injection  \beta_s \colon \Gamma(S, R^n \pi_* \mathbb{Q}) \to H^n(X_s, \mathbb{Q}). (It is an injection because the local system R^n \pi_* \mathbb{Q} is locally constant.) The composition  \beta_s \circ \alpha \colon H^n(\bar{X}, \mathbb{Q}) \to H^n(X_s, \mathbb{Q}) is the morphism on cohomology induced by the inclusion of varieties X_s \hookrightarrow \bar{X}, so it is a morphism of Hodge structures. In particular, \ker(\beta_s \circ \alpha) is a sub-Hodge structure of H^n(\bar{X}, \mathbb{Q}). The fact that \beta_s is an injection implies that \ker(\beta_s \circ \alpha) = \ker(\alpha) (observe that this implies that \ker(\beta_s \circ \alpha) is independent of s).

The theorem of the fixed part implies that  \Gamma(S, R^n \pi_* \mathbb{Q}) \cong H^n(\bar{X}, \mathbb{Q}) / \ker(\alpha) as \mathbb{Q}-vector spaces, and the fact that \ker(\alpha) is a sub-Hodge structure of H^n(\bar{X}, \mathbb{Q}) implies that we can give \Gamma(S, R^n \pi_* \mathbb{Q}) the quotient Hodge structure.

Blasius's proof of Principle B

Blasius's proof is based upon the following commutative diagram, for each \sigma \colon \mathbb{C} \hookrightarrow \mathbb{C} and for each t \in S(\mathbb{C}):

 \usepackage[matrix,arrow]{xy}
\xymatrix{
   H^{2p}_{dR}(\bar{\mathcal{A}}/\mathbb{C}) \otimes_\sigma \mathbb{C}   \ar[rr]^{r_t^{dR}}   \ar[d]^{\sigma^*}
 &
 & H^{2p}_{dR}(\mathcal{A}_t/\mathbb{C}) \otimes_\sigma \mathbb{C}       \ar[d]^{\sigma^*}
\\ H^{2p}(\bar{\mathcal{A}}^\sigma, \mathbb{C})              \ar[r]^{\alpha^\sigma}
 & \Gamma(S^\sigma, R^{2p} \pi^\sigma_* \mathbb{C})          \ar[r]^{\beta_t^\sigma}
 & H^{2p}(\mathcal{A}_t^\sigma, \mathbb{C})
}

We begin by applying the theorem of the fixed part to our section v \in \Gamma(S, R^{2p} \pi_* \mathbb{Q}), getting a class w \in H^{2p}(\bar{\mathcal{A}}, \mathbb{C}) such that \alpha(w) = v. Let w_{dR} be the corresponding de Rham class in H^{2p}_{dR}(\bar{\mathcal{A}}/\mathbb{C}). Commutativity of the above diagram for \sigma = \mathrm{id}, together with the fact that the right vertical arrow is an isomorphism, implies that  r_t^{dR}(w_{dR}) = v_{t,dR} for all t \in S(\mathbb{C}).

We can then construct a section  u_\sigma = \alpha^\sigma(\sigma^*(w_{dR} \otimes 1)) \in \Gamma(S^\sigma, R^{2p} \pi^\sigma_* \mathbb{C}) for each \sigma. Commutativity of the diagram implies that  \beta_t^\sigma(u_\sigma) = \sigma^*(v_{t,dR}) for all t and \sigma.

Now it is fairly easy to finish off. First consider what happens at the point s, where we are given that v_{s,dR} is an absolute Hodge class. In other words, \sigma^*(v_{s,dR}) is a Hodge class on \mathcal{A}_s^\sigma for every \sigma. The fact that \beta_s^\sigma is an injection of Hodge structures allows us to deduce that u_\sigma is a Hodge class in \Gamma(S^\sigma, R^{2p} \pi^\sigma_* \mathbb{Q}) (with respect to the Hodge structure constructed in the last section).

But since \beta_t^\sigma is also a morphism of Hodge structures for every \sigma and t, this implies that \beta_t^\sigma(u_\sigma) = \sigma^*(v_{t,dR}) is always a Hodge class, which is just what we need.

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

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  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. ...

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