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I hear sometimes that there is many analogies between the Hodge conjecture and the Tate conjecture. If we take a look at the statements of this two conjectures, we have the followings :

  • The Tate conjecture :

Let $k$ be a field and let $X$ be a smooth geometrically irreducible projective variety over $k$ of dimension $d$.

We denote by $\overline{X} = X \times_k \overline{k}$ the base change of $X$ to the algebraic closure $\overline{k}$.

The Galois group $G = Gal ( \overline{k} / k )$ then acts on $\overline{X}$ via the second factor.

Let $Z^r ( \overline{X} )$ be the free abelian group generated by the irreducible closed subvarieties of $\overline{X}$ of codimension $r$ ( $1 \leq r \leq d$ ). An element of $ Z^r ( \overline{X} ) $ is called an algebraic cycle of codimension $ r $ on $\overline{X}$.

Let $\ell $ be a prime different from $p = \mathrm{car} (k) \geq 0$. There is a cycle map ( of $G$ - modules ) :

$$ c^r \ : \ Z^r ( \overline{X} ) \otimes \mathbb{Q}_{ \ell } \to H^{2r} ( \overline{X} , \mathbb{Q}_{ \ell } ( r ) )^G $$

which associates to every algebraic cycle an $ \ell $ - adic etale cohomology class.

Suppose $k$ is finitely generated over its prime field.

Then, the Tate conjecture says that the map $c^r$ is surjective.

  • The Hodge conjecture :

For each integer $ p \in \mathbb{N} $, let $ H^{p,p} (X) $ denotes the subspace of $ H^{2p} ( X ,\mathbb{C} ) $ of type $ (p,p) $.

The group of rational $ (p,p) $- cycles : $ H^{p,p} (X , \mathbb{Q} ) = H^{2p} ( X , \mathbb{Q} ) \cap H^{p,p} (X) $ is called the group of rational Hodge classes of type $ (p,p) $.

An $ r $ -cycle of an algebraic variety $ X $ is a formal finite linear combination $ \displaystyle \sum_{ i \in [1,h] } m_i Z_i $ of closed irreducible subvarieties $ Z $ of dimension $ r $ with integer coefficients $ m_i $.

The group of $ r $ -cycles is denoted by $ \mathcal{Z}_r (X) $.

On a compact complex algebraic manifold, the class of closed irreducible subvarieties of codimension $ p $ extends into a linear morphism :

$$ \mathrm{cl}_{ \mathbb{Q} } \ : \ \mathcal{Z}_{p} (X) \otimes \mathbb{Q} \to H^{p,p} (X, \mathbb{Q} ) $$ defined by : $ \mathrm{cl}_{ \mathbb{Q} } \big( \sum_{ i \in [1,h] } m_i Z_i \big) = \sum_{ i \in [1,h] } m_i \eta_{Z_{i}} \ , \ \forall m_i \in \mathbb{Q} $.

The elements of the image of $ \mathrm{cl}_{ \mathbb{Q} } $ are called rational algebraic Hodge classes of type $ (p,p) $.

The Hodge conjecture says :

On a non-singular complex projective variety, any rational Hodge class of type $ (p,p) $ is algebraic, i.e : in the image of $ \mathrm{cl}_{ \mathbb{Q} } $.

  • Questions :

If we look at the statements of the two conjectures above, and look for similarities between them, which group $G$ is it such that: $H^{2k} (X, \mathbb{Q})^G = H^{2p } (X, \mathbb {Q}) \cap H ^ {p, p} (X)$ to bring the statement of Hodge's conjecture closer to the statement of Tate's conjecture?

Thanks in advance for your help.

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You can find similarities between both conjectures by observing that the $\ell$-adic cohomologies appearing in Tate conjecture are objects of the category of $\mathbb{Q}_{\ell}[G]$-modules (may be continuous), where $G$ is the Galois group here, and $$H^{2r} ( \overline{X} , \mathbb{Q}_{ \ell } ( r ) )^G\cong \operatorname{Hom}_{\mathbb{Q}_{\ell}[G]}(\mathbb{Q}_{\ell}, H^{2r} ( \overline{X} , \mathbb{Q}_{ \ell } ( r )) ).$$ Now, for the Hodge Conjecture, you need to find an appropriate category to play the role of $\mathbb{Q}_{\ell}[G]$-modules, and this is the Hodge Mixed Structures (I abbreviate MHS). This is a Tannakian category, and it has an object called $\mathbb{Q}[0]$, which plays the analogous role of $\mathbb{Q}_{\ell}$ in the $\ell$-adic setting. We have that the cohomology $H^{2r} ( X , \mathbb{Q} ) $ together with the Hodge Filtration when tensoring by $\mathbb{C}$ and with the "trivial" weight filtration of weight $2r$ (since $X$ is smooth and projective) is a Hodge Mixed Structure (so $H^{2r} ( X , \mathbb{Q} )(r) $ has weight $0$) , and that $$ H^{p,p} (X , \mathbb{Q} ) = H^{2p} ( X , \mathbb{Q} ) \cap H^{p,p} (X) \cong \operatorname{Hom}_{MHS}(\mathbb{Q}[0], H^{2r} ( X , \mathbb{Q}) ( r ) ).$$

What it is more interesting, we also get an interpretation of the (Griffiths) intermediate Jacobian (in fact the group of points and tensor with $\mathbb{Q}$) as $$\operatorname{Ext}^1_{MHS}(\mathbb{Q}[0], H^{2r-1} ( X , \mathbb{Q}) ( r ) ).$$ and a unified construction of the cycle map and the Abel-Jacobi map in both settings (where the $\ell$-adic intermediate Jacobian is essentially $H^1(G,H^{2r-1} ( \overline{X} , \mathbb{Q}_{ \ell } ( r ))$ ).

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