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Let $X$ be a (not necessarily smooth or proper) variety over a number field $k$. Suppose we are given

  1. A subquotient $V_{dR}$ of the algebraic de Rham cohomology $H_{dR}^i(X)$ (defined in the non-smooth case via a thickening, as in e.g. Hartshorne's algebraic de Rham cohomology paper),

  2. For each embedding $\iota: k\to \mathbb{C}$, a subquotient $V_\iota$ of $$H^i(X_{\mathbb{C}}^{an}, \mathbb{Q})$$ respecting the $\mathbb{Q}$-mixed Hodge structure, and

  3. For some fixed prime $\ell$ and algebraic closure $\bar k$ of $k$, a subquotient $V_\ell$ of $$H^i(X_{\bar k, \text{ét}}, \mathbb{Q}_\ell)$$ respecting the action of $\text{Gal}(\bar k/k)$, such that

  4. For each $\iota$, the corresponding comparison isomorphism $$H^i_{dR}(X)\otimes_k \mathbb{C} \to H^i(X_{\mathbb{C}}^{an}, \mathbb{Q})\otimes \mathbb{C}$$ sends $V_{dR}$ to $V_\iota$, and

  5. For each $\iota$, the corresponding comparison isomorphism $$H^i(X_{\mathbb{C}}^{an}, \mathbb{Q})\otimes \mathbb{Q}_\ell \to H^i(X_{\bar k, \text{ét}}, \mathbb{Q}_\ell)$$ sends $V_\iota$ to $V_\ell$, and

  6. The $V_\iota$ is of mixed Tate type (i.e. all $h^{p,q}$'s vanish for $p\not=q$). Or equivalently $V_\ell$ is an iterated extension of powers of the cyclotomic character.

(If you'd like, you can think of $V$ as the realization of a mixed Tate motive over $k$ under any of the various formalisms for mixed Tate motives over number fields -- I think that a priori the conditions above are slightly weaker but expected to be equivalent.)

I'd like to know -- is the analogue of the Mumford-Tate conjecture known to be true for such $V$? Explicitly, one expects that the Lie algebra of the image of $$\text{Gal}(\bar k/k)\to GL(V_\ell)$$ is the extension of scalars to $\mathbb{Q}_\ell$ of the Lie algebra of a $\mathbb{Q}$-group, given as the Tannaka dual of the subcategory of mixed Tate Hodge structures over $k$, generated by the Hodge realization $(V_{dR}, V_{\iota_1}, V_{\iota_2}, \cdots)$.

My feeling is that one can probably extract this from the literature by comparing ranks of various Ext groups (in one's favorite category of mixed Tate motives and in the $\ell$-adic and Hodge settings) but I am hoping it is written explicitly somewhere.

EDIT: Fixed some errors in the formulation of the Hodge realization.

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    $\begingroup$ Hi Daniel. I'm not sure it is written down anywhere, so it would be nice if you did. $\endgroup$ – Donu Arapura Dec 30 '20 at 18:18
  • $\begingroup$ @DonuArapura It seems like I might have to, since I have an application in mind! $\endgroup$ – Daniel Litt Dec 30 '20 at 18:44

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