Deligne's absolute Hodge conjecture

Let $X$ be a smooth projective variety over $\mathbb C$ and $p\ge0$. Let $\mathbb A_f$ be the ring of finite adèles of $\mathbb Q$. Consider three cohomology theories:

• Betti cohomology $H^{2p}_B(X)=H^{2p}(X(\mathbb C),\mathbb Q)(p)$, with its Hodge structure of weight $0$;
• algebraic de Rham cohomology $H^{2p}_{\mathrm{dR}}(X/\mathbb C)$, with its Hodge filtration $F^\bullet$, Tate-twisted so that it has weight $0$;
• étale cohomology $H^{2p}_{\mathrm{\acute et}}(X,\mathbb A_f(p))$.

The comparison isomorphisms send a class $\alpha\in H^{2p}_B(X)$ to a pair

$$\alpha_{\mathbb A}=(\alpha_{\mathrm{dR}},\alpha_{\mathrm{\acute et}})\in H^{2p}_{\mathrm{dR}}(X/\mathbb C)\times H^{2p}_{\mathrm{\acute et}}(X,\mathbb A_f(p)).$$

A Hodge class is an $\alpha\in H^{2p}_B(X)$ with $\alpha_{\mathrm{dR}}\in F^0$, that is, a rational class of type $(p,p)$.

Let $\sigma\in\operatorname{Aut}(\mathbb C)$, which need not be continuous, and let $X^\sigma=X\times_{\mathbb C,\sigma}\mathbb C$. Base change along $\sigma$ gives a $\sigma$-linear isomorphism $H_{\mathrm{dR}}(X/\mathbb C)\to H_{\mathrm{dR}}(X^\sigma/\mathbb C)$ respecting Hodge filtrations. It also gives an isomorphism of étale cohomology, since $X^\sigma\cong X$ as abstract schemes. Write $\sigma(\alpha_{\mathbb A})$ for the image pair. In general it is unrelated to the Betti cohomology of the different complex manifold $X^\sigma(\mathbb C)$.

Definition (Deligne). A Hodge class $\alpha$ is absolute Hodge if for every $\sigma\in\operatorname{Aut}(\mathbb C)$ the pair $\sigma(\alpha_{\mathbb A})$ is the image $\beta_{\mathbb A}$ of a Hodge class $\beta\in H^{2p}_B(X^\sigma)$.

Conjecture (Deligne). For every smooth projective complex variety $X$ and every $p$, every Hodge class in $H^{2p}(X(\mathbb C),\mathbb Q)(p)$ is absolute Hodge.

Classes of algebraic cycles are absolute Hodge, because the cycle class of $Z^\sigma$ is $\sigma$ applied to that of $Z$. So the Hodge conjecture implies this conjecture. Applying the conjecture to powers $X^r$ covers Hodge classes in all tensor spaces $H^*(X)^{\otimes m}\otimes H^*(X)^{\vee\otimes n}(j)$. Here one uses that the Künneth components of the diagonal are absolute Hodge [1, §2].

Background

Deligne introduced absolute Hodge classes in his 1978–79 seminar, published in 1982 [1]. The goal was to build a category of motives over a field of characteristic $0$ that is unconditionally Tannakian and works for arithmetic applications. Absolute Hodge classes replace algebraic cycles, whose theory depends on the Hodge conjecture. They give an unconditional Tannakian category of motives over any field of characteristic $0$. The motives generated by abelian varieties in this category are the ones used in the theory of Shimura varieties. A Hodge class defined over a subfield $k\subseteq\mathbb C$ that is absolute Hodge has its étale realization acted on by $\operatorname{Gal}(\bar k/k)$ through a finite quotient. So the conjecture also connects Hodge classes with Tate classes.

The conjecture is a purely Hodge-theoretic and arithmetic statement. It does not assert that Hodge classes are algebraic. Voisin's results on Hodge loci show what it predicts geometrically. Under a natural nondegeneracy condition, the locus where a Hodge class stays Hodge is defined over $\overline{\mathbb Q}$, and its Galois conjugates are again Hodge loci [3].