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Let $\mathbb{V}$ be a motivic variation of Hodge structure on a $K$-variety$S$, let $\mathcal{H}$ be the associated $K$-algebraic Hodge bundle, and let$\sigma \in \textrm{Aut}(\mathbb{C}/K)$ be an automorphism. The absolute Hodgeconjecture predicts that given a Hodge vector $v \in \mathcal{H}_{\mathbb{C},s}$ above $s \in S(\mathbb{C})$ which lies inside $\mathbb{V}_{s}$, theconjugate vector $v_{\sigma} \in \mathcal{H}_{\mathbb{C}, s_{\sigma}}$ is Hodgeand lies inside $\mathbb{V}_{s_{\sigma}}$. We study this problem in thesituation where we have an algebraic subvariety $Z \subset S_{\mathbb{C}}$containing $s$ whose algebraic monodromy group $\mathbf{H}_Z$ fixes $v$. Usingrelationships between $\mathbf{H}_Z$ and $\mathbf{H}_{Z_{\sigma}}$ coming fromthe theories of complex and $\ell$-adic local systems, we establish a criterionthat implies the absolute Hodge conjecture for $v$ subject to a group-theoreticcondition on $\mathbf{H}_{Z}$. We then use our criterion to establish new casesof the absolute Hodge conjecture.

Absolute Hodge and $\ell$-adic Monodromy