Martin's Blog

Hodge classes on abelian varieties

Posted by Martin Orr on Monday, 25 August 2014 at 18:50

In this post I will define Hodge classes and state the Hodge conjecture. I will restrict my attention to the case of abelian varieties and say the minimum amount necessary to be able to discuss the relationships between the Hodge, Tate and Mumford-Tate conjectures and absolute Hodge classes in subsequent posts. There are many excellent accounts of this material already written, which may give greater detail and generality.

Hodge classes are cohomology classes on a complex variety A which are in the intersection of the singular cohomology H^n(A, \mathbb{Q}) and the middle component H^{n/2,n/2}(A) of the Hodge decomposition  H^n(A, \mathbb{Q}) \otimes_\mathbb{Q} \mathbb{C} = \bigoplus_{\substack{p,q\geq 0 \\ p+q=n}} H^{p,q}(A). They can also be defined as rational cohomology classes which are eigenvectors for the Mumford-Tate group. The Hodge conjecture predicts that these classes are precisely the \mathbb{Q}-span of cohomology classes coming from algebraic subvarieties of A.

Note: Hodge classes are usually defined as living in the Tate twist H^n(A, \mathbb{Q})(n/2) rather than in the cohomology group H^n(A, \mathbb{Q}) itself. This is because the cycle class maps for singular and for de Rham cohomology differ by factors of 2 \pi i, as explained in my post on Tate twists. In this post, I shall only consider the singular cohomology normalisation of the cycle class map and hence omit Tate twists.

Hodge structures and H^1 of an abelian variety

So far in this blog, I have talked a lot about the homology group H_1 of an abelian variety and its Hodge structure. I also briefly mentioned the cohomology group H^2 when talking about Hodge symplectic forms. Now I need to talk about the cohomology groups H^n in general.

Let's start with H^1(A, \mathbb{Z}), which is the dual of H_1(A, \mathbb{Z}). Recall that H_1(A, \mathbb{Z}) is a free \mathbb{Z}-module of rank 2g. The Hodge decomposition  H_1(A, \mathbb{Z}) \otimes_\mathbb{Z} \mathbb{C}  =  H^{-1,0}(A) \oplus H^{0,-1}(A) induces a decomposition  H^1(A, \mathbb{Z}) \otimes_\mathbb{Z} \mathbb{C}  =  H^{1,0}(A) \oplus H^{0,1}(A) where  H^{1,0}(A) = \{ f \in (H_1(A, \mathbb{Z}) \otimes_\mathbb{Z} \mathbb{C})^\vee  \mid  f(H^{0,-1}(A)) = 0 \} \cong H^{-1,0}(A)^\vee and likewise for H^{0,1}(A).

We could call this type of structure an H^1 \mathbb{Z}-Hodge structure. (The precise definition is the n=1 case of the definition below.) It is also known as an effective \mathbb{Z}-Hodge structure of weight 1.

Hodge structures and higher cohomology of abelian varieties

We can generalise the above concept to an effective \mathbb{Z}-Hodge structure of weight n. This is defined to be a finite-rank \mathbb{Z}-module V_\mathbb{Z} together with a decomposition into complex vector spaces  V_\mathbb{Z} \otimes_\mathbb{Z} \mathbb{C} = V^{n,0} \oplus V^{n-1,1} \oplus \dotsc \oplus V^{0,n} satisfying  V^{p,q} = \overline{V^{q,p}} for all p and q. (The word effective means that we only have pieces V^{p,q} where p,q \geq 0. The words weight n mean that for every piece, p+q=n.)

The importance of this concept comes from the fact that for every smooth projective complex variety X (and more generally for every compact Kähler manifold, which includes all complex tori), the cohomology group H^n(X, \mathbb{Z}) comes with an effective \mathbb{Z}-Hodge structure of weight n. I will not discuss the general construction here, but I will discuss what these Hodge structures are in the case of abelian varieties.

The complex abelian variety A is homeomorphic to the product of 2g circles. Basic topology gives us that the cohomology groups are the exterior powers of H^1:  H^n(A, \mathbb{Z}) = \bigwedge^n H^1(A, \mathbb{Z}).

Given an effective Hodge structure V of weight 1, we can make its n-th exterior power into an effective Hodge structure of weight n by using the decomposition  \bigwedge^n (V^{1,0} \oplus V^{0,1})  =  \bigoplus_{\substack{p,q\geq 0 \\ p+q=n}}  \bigwedge^p V^{1,0} \otimes \bigwedge^q V^{0,1}. It turns out that  H^{p,q}(A) = \bigwedge^p H^{1,0}(A) \otimes \bigwedge^q H^{0,1}(A) is indeed the correct Hodge decomposition for H^n of an abelian variety.

Hodge classes

Let us consider the intersection  H^{p,q}(A) \cap H^n(A, \mathbb{Z}). We have already seen this in the case p=q=1, where we called elements of this intersection Hodge symplectic forms.

If p \neq q then this intersection is { 0 }. This is because each element of H^n(A, \mathbb{Z}) is fixed by complex conjugation, so  H^{p,q}(A) \cap H^n(A, \mathbb{Z})  \subset  H^{p,q}(A) \cap \overline{H^{p,q}(A)}. But H^{p,q}(A) \cap \overline{H^{p,q}}(A)(A) is zero whenever p \neq q because  \overline{H^{p,q}(A)} = H^{q,p}(A) and the Hodge decomposition is a direct sum.

On the other hand, when p=q=n/2, we have H^{p,p}(A) = \overline{H^{p,p}(A)} and so  H^{p,p}(A) \cap H^{2p}(A, \mathbb{Z}) may be non-zero. We call elements of  H^{p,p}(A) \cap H^{2p}(A, \mathbb{Z}) integral Hodge classes of weight 2p.

In an earlier post, we essentially defined a Hodge symplectic form to be an integral Hodge class of weight 2. Hence the existence of a polarisation (a Hodge symplectic form satisfying a positivity property) guarantees that an abelian variety possesses a non-zero integral Hodge class of weight 2. The group of integral Hodge classes of weight 2 is known as the Neron-Severi group and its rank, called the Picard number, is an interesting invariant of the abelian variety.

In these blog posts, it will be more convenient to work with  \operatorname{Hdg}^p(A) := H^{p,p}(A) \cap H^{2p}(A, \mathbb{Q}). We call elements of this group Hodge classes of weight 2p. (Sometimes people call these elements rational Hodge classes, and use the term Hodge class to mean what we call an integral Hodge class.)

Algebraic cycle classes and the Hodge conjecture

The Hodge conjecture is not part of the main things I intend to write about in this blog, but I feel that I should not talk about Hodge classes without mentioning it briefly. It also provides motivation for some of the things we do with Hodge classes, and indeed give one reason why we care about Hodge classes in the first place.

Let X be any smooth projective complex variety. If Z \subset X is a complex algebraic subvariety of codimension p, then we can use Poincaré duality to define a class cl(Z) in H^{2p}(X, \mathbb{Z}). It turns out that cl(Z) is always an integral Hodge class.

A class in H^{2p}(X, \mathbb{Q}) is called an algebraic cycle class if it is in the \mathbb{Q}-span (or sometimes the \mathbb{Z}-span) of those classes of the form cl(Z). The Hodge conjecture states that every Hodge class is an algebraic cycle class.

Hodge conjecture. The classes cl(Z) of algebraic subvarieties of X of codimension p span the \mathbb{Q}-vector space \operatorname{Hdg}^p(X).

Hodge classes and the Mumford-Tate group

We can give a characterisation of Hodge classes using the Mumford-Tate group. The key reason why this is important is that there is a sort of converse which I will discuss in a subsequent post, using Hodge classes to characterise the Mumford-Tate group.

Let us recall the definition of the Mumford-Tate group. Associated with the H_1 \mathbb{Q}-Hodge structure H_1(A, \mathbb{Q}), there is a group homomorphism  h \colon \mathbb{C}^\times \to \operatorname{GL}(H_1(A, \mathbb{R})) such that h(z) acts as multiplication by z on H^{-1,0}(A) and by \bar{z} on H^{0,-1}(A). The Mumford-Tate group of A is defined to be the smallest algebraic subgroup M \subset \operatorname{GL}(H_1(A, \mathbb{Q})) defined over \mathbb{Q} and such that M(\mathbb{R}) contains the image of h.

For each n, there is a standard representation of \operatorname{GL}(H_1(A, \mathbb{Q})) on \bigwedge^n H_1(A, \mathbb{Q})^\vee. We can compose these with h from the previous paragraph to get homomorphisms  h^n \colon \mathbb{C}^\times \to \operatorname{GL}(H^n(A, \mathbb{R})). If we look at the eigenspaces of these homomorphisms, we find that h^n(z) acts as multiplication by z^{-p} \bar{z}^{-q} on H^{p,q}(A).

Hence \operatorname{Hdg}^p(A) is the subspace of H^{2p}(A, \mathbb{Q}) on which h^{2p}(z) acts as (z \bar{z})^{-p} for all z \in \mathbb{C}^\times.

We would like to state this in terms of the Mumford-Tate group instead of h, so we proceed as follows.

The image of h (and hence also the Mumford-Tate group) is contained in the group \operatorname{GSp}(H_1(A, \mathbb{Q}), \psi) consisting of elements which preserve the symplectic form \psi associated with a chosen polarisation (up to scalars). There is a character \chi \colon \operatorname{GSp}(H_1(A, \mathbb{Q}), \psi) \to \mathbb{Q}^\times defined by  \psi(gv, gw) = \chi(g) \psi(v, w).

The fact that \psi is a Hodge symplectic form implies that \chi(h(z)) = z \bar{z}. So g \in h(\mathbb{C}^\times) acts on \operatorname{Hdg}^p(A) as multiplication by \chi(g)^{-p}.

For each Hodge class v of weight 2p, consider the algebraic group  G_v = \{ g \in \operatorname{GSp}(H_1(A, \mathbb{Q}), \psi)  \mid  g.v = \chi(g)v \}. This group is defined over \mathbb{Q}, and by the above its real points contain the image of h. Hence G_v contains the Mumford-Tate group of A, and thus all of \operatorname{MT}(A) acts on \operatorname{Hdg}^p(a) as multiplication by \chi^{-p}.

On the other hand, if we have some v \in H^{2p}(A, \mathbb{Q}) which is an eigenvector for all of \operatorname{MT}(A), then it must in particular be an eigenvector for h(\mathbb{C}^\times). We know that the eigenspaces of h(\mathbb{C}^\times) on H^{2p}(A, \mathbb{C}) are precisely the pieces of the Hodge decomposition, and the only such piece which can intersect H^{2p}(A, \mathbb{Q}) is H^{p,p}. Thus any eigenvector for \operatorname{MT}(A) must be a Hodge class.

We deduce that

Proposition. A class in H^{2p}(A, \mathbb{Q}) is a Hodge class if and only if it is an eigenvector for the action of \operatorname{MT}(A).

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

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