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Let $X$ be a smooth projective complex variety and $t\in H^{2p}(X,\mathbf{Q})(p)$ a rational cohomology class.

Assume that there exist $$t_1,\ldots,t_N\in H^{2*}(X,\mathbf{Q})(*)$$ algebraic classes (i.e. in the image of the cycle map $cl : A^*(X)\to H^{2*}(X,\mathbf{Q})(*)$) and an algebraic subgroup $$G\subset\text{Aut}(H^*(X,\mathbf{Q}))\times\text{GL}(\mathbf{Q}(1))$$ (automorphisms as a graded $\mathbf{Q}$-algebra) fixing the $t_1,\ldots, t_N$ and largest with respect to this property, such that $G$ also fixes $t$.

Is $t$ algebraic?

From papers of Milne, one can deduce this is true under the variational Hodge conjecture. However, I wonder if this is already known unconditionally.

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  • $\begingroup$ What is the relation between $t$ and $G$ (and the $t_i$)? As stated, $t$ could just be any class and $G$ the trivial group... $\endgroup$
    – R. van Dobben de Bruyn
    Commented Feb 26, 2023 at 17:28
  • $\begingroup$ @R.vanDobbendeBruyn I've corrected and clarified. Thanks! $\endgroup$
    – user480741
    Commented Feb 26, 2023 at 17:41

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Essentially you are asking whether the category of homological motives generated by $X$ is tannakian, which is true if and only if homological equivalence coincides with numerical equivalence (Jannsen, Deligne). This coincidence is known for abelian varieties (Lieberman) but not much else, so I think the answer to your question is No in general.

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