Absolute Hodge and ℓ-adic monodromy

Abstract

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 $\sigma \in \mathrm {Aut}(\mathbb {C}/K)$ be an automorphism. The absolute Hodge conjecture predicts that given a Hodge vector $v \in \mathcal {H}_{\mathbb {C}, s}$ above $s \in S(\mathbb {C})$ which lies inside $\mathbb {V}_{s}$, 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 $\textbf {H}_{Z}$ fixes $v$. Using relationships between $\textbf {H}_{Z}$ and $\textbf {H}_{Z_{\sigma }}$ coming from the theories of complex and $\ell$-adic local systems, we establish a criterion that implies the absolute Hodge conjecture for $v$ subject to a group-theoretic condition on $\textbf {H}_{Z}$. We then use our criterion to establish new cases of the absolute Hodge conjecture.