Absolute Hodge and $\ell$-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 $\sigma \in \text{Aut}(\mathbb{C}/K)$ 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 ${\mathbf{H}}_Z$ fixes $v$. Using relationships between ${\mathbf{H}}_Z$ and ${\mathbf{H}}_{Z_{\sigma }}$ 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 ${\mathbf{H}}_Z$. We then use our criterion to establish new cases of the absolute Hodge conjecture.

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