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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 \mathcal{A} \to S and a flat section of the relative de Rham cohomology bundle \mathcal{H}_{dR}^{n}(\mathcal{A}/S), such that the section specialises to an absolute Hodge class at one point of S, 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:

  1. Reduce to Hodge classes on abelian varieties of CM type (using Principle B)
  2. 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)
  3. Reduce to Hodge classes on abelian varieties which are isogenous to a power of an elliptic curve (using Principle B)
  4. 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 \mathcal{H}_{dR}^{n}(\mathcal{A}/S) for a family of abelian varieties \mathcal{A} \to S. 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 \mathcal{E}_{\mathcal{A}} of the family, and its relative tangent bundle e^* T_{\mathcal{E}_{\mathcal{A}}/S} is a vector bundle on S. We define \mathcal{H}_{dR}^{n}(\mathcal{A}/S) = \bigwedge^n (e^* T_{\mathcal{E}_{\mathcal{A}}/S})^\vee where the dual and exterior power are taken in the category of vector bundles on S. The fibre of this vector bundle at each point s \in S(k) is the de Rham cohomology H_{dR}^n(\mathcal{A}_s/k) .

Now we can state Principle B. There are a variety of ways of stating this with slightly different input classes v. 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 \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 \mathcal{H}_{dR}^{2p}(\mathcal{A}/S). If

  1. v is flat with respect to the Gauss-Manin connection (don't worry what this means; I might explain it next time); and
  2. there is a point s \in S(\mathbb{C}) such that v_s is an absolute Hodge class on \mathcal{A}_s,

then v_t is an absolute Hodge class on \mathcal{A}_t for every t \in S(\mathbb{C}).

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 G over \mathbb{Q}, you can form a Shimura variety S which is the ``moduli space of abelian varieties whose Mumford-Tate group is contained in G.'' 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 G on the Hodge structures associated with the abelian varieties but I shall pass over those details here. The Shimura variety S is an algebraic variety over \mathbb{C}.

If you choose the level structure suitably, then S is a fine moduli space. This means that there is a family of abelian varieties \pi \colon \mathcal{A} \to S such that, for each s \in S(\mathbb{C}), the abelian variety which corresponds to s (via the moduli interpretation of S) is in fact the fibre \mathcal{A}_s, 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 G(\mathbb{Q}) on the cohomology groups of the fibres H^n(\mathcal{A}_s, \mathbb{Q}). By the construction of Shimura varieties, the monodromy action of \pi_1(S, s) on these cohomology groups factors through G^{\mathrm{der}}(\mathbb{Q}). This implies that any G^{\mathrm{der}}(\mathbb{Q})-invariant cohomology class v_s \in H^n(\mathcal{A}_s, \mathbb{Q}) can be extended to a global section v of the relative singular cohomology local system R^n \pi_* \mathbb{Q} (actually this is only valid on a connected component of S, rather than on all of S, but I shall ignore that).

In order to apply Principle B (in the version stated above), we need to know that these sections of R^n \pi_* \mathbb{Q} are sections of the relative de Rham cohomology bundle \mathcal{H}_{dR}^{n}(\mathcal{A}/S). 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 quasi-projective varieties. The fact we have constructed these sections from sections of R^n \pi_* \mathbb{Q} implies that they are flat with respect to the Gauss-Manin 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 E and an even positive integer d. Suppose we have a complex abelian variety A of dimension d[E:\mathbb{Q}]/2 equipped with a homomorphism E \to \operatorname{End} A \otimes \mathbb{Q}. This induces a structure of d-dimensional E-vector space on H_1(A, \mathbb{Q}), and hence the Mumford-Tate group of A is a subgroup of \operatorname{GL}_d(E).

The complex homology group H_1(A, \mathbb{C}) splits into eigenspaces for the action of E, one for each embedding \sigma \colon E \to \mathbb{C}, which we will call V_\sigma. Define r_\sigma = \dim (V_\sigma \cap H^{-1,0}(A)) and observe that r_\sigma + r_{\bar\sigma} = d for all \sigma.

We say that A is of Weil type (relative to E) if r_\sigma = d/2 for all \sigma \colon E \to \mathbb{C}. Equivalently, the Mumford-Tate group of A is contained in the group \mathbb{G}_{m,\mathbb{Q}}.\operatorname{SU}_d(E, \phi) = \{ g \in \operatorname{GL}_d(E) \mid \phi(gx, gy) = \phi(x, y) \text{ and } \det g \in \mathbb{Q}^\times \} for some E-Hermitian form \phi on E^d with signature (d/2, d/2) with respect to every embedding \sigma \colon E \to \mathbb{C}.

We say that A is of split Weil type if the E-Hermitian form \phi is split i.e. equivalent to the standard E-Hermitian form ((x_1, \dotsc, x_d), (y_1, \dotsc, y_d)) \mapsto \sum_{j=1}^d (-1)^d x_j \bar{y}_j (this is a condition on a polarisation of A).

If A is an abelian variety of Weil type, a cohomology class in H^n(A, \mathbb{Q}) is said to be a Weil class if it is invariant under the action of \operatorname{SU}_d(E, \phi). Because of what we said about the Mumford-Tate 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 André, proved by linear algebra manipulations of CM Hodge structures:

Theorem. For every complex abelian variety A of CM type, there exist complex abelian varieties B_1, \dotsc, B_n of split Weil type and homomorphisms A \to B_j such that every Hodge class on A is a linear combination of pullbacks of Weil classes on the B_j.

Putting it together

We can now carry out the steps of the proof of Deligne's theorem mentioned in the introduction as follows:

  1. We start with an arbitrary abelian variety A. Let G = MT(A) and form the universal family \pi \colon \mathcal{A} \to S of abelian varieties whose Mumford-Tate group is contained in G (plus suitable extra data). Then there is certainly a point s \in S(\mathbb{C}) such that \mathcal{A}_s \cong A. Since every Hodge class of A is fixed by G^{\mathrm{der}}(\mathbb{Q}), it extends (as discussed above) to a global section v of \mathcal{H}_{dR}^{n}(\mathcal{A}/S). By the theory of Shimura varieties, there is a point t \in S(\mathbb{C}) such that \mathcal{A}_t has complex multiplication. If we can prove that v_t is an absolute Hodge class, then we can use Principle B to deduce that v_s is absolute Hodge.

  2. We now have a Hodge class on an abelian variety of CM type. Using André'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.

  3. Let G = \mathbb{G}_{m,\mathbb{Q}}.\operatorname{SU}_d(E, \phi) and again form the universal family \mathcal{A} \to S of abelian varieties with Mumford-Tate group contained in G. (S is an example of a Shimura variety of type PEL. It parameterises abelian varieties with the following structure: Polarisation: \phi - Endomorphisms: containing E - and a Level structure.) Note that G^{\mathrm{der}}(\mathbb{Q}) will not necessarily stabilise every Hodge class on A, but by definition it does stabilise Weil classes. Hence a Weil class on A can be extended to a section v of \mathcal{H}_{dR}^{n}(\mathcal{A}/S). One can construct a point t \in S(\mathbb{C}) whose corresponding abelian variety is isogenous to a power of a non-CM elliptic curve. By Principle B it suffices to show that v_t is an absolute Hodge class.

  4. 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.

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

Trackbacks

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