Absolute Hodge and $\mathrm{\ell }$-adic Monodromy

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 $\sigma \in \text{Aut}(\mathbb{C}\mathrm{/}K)$ be an automorphism. The absolute Hodge conjecture predicts that given a Hodge vector $v \in $\mathcal{H}${$\mathbb{C}$, s}$above$s \in S($\mathbb{C}$)$whichliesinside$$\mathbb{V}${s}$, the conjugate vector $v_{\sigma }\in {\mathcal{H}}_{\mathbb{C},s_{\sigma }}$ is Hodge and lies inside ${\mathbb{V}}_{s_{\sigma }}$. We study this problem in the situation where we have an algebraic subvariety $Z\subset S_{\mathbb{C}}$ containing $s$ whose algebraic monodromy group ${\mathbf{H}}_Z$ fixes $v$. Using relationships between ${\mathbf{H}}_Z$ and ${\mathbf{H}}_{Z_{\sigma }}$ coming from the theories of complex and $\mathrm{\ell }$-adic local systems, we establish a criterion that implies the absolute Hodge conjecture for $v$ subject to a group-theoretic condition on ${\mathbf{H}}_Z$. We then use our criterion to establish new cases of the absolute Hodge conjecture.