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Absolute Hodge and ℓ-adic monodromy
Published online by Cambridge University Press: 13 May 2022
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.
MSC classification
Primary:
14D07: Variation of Hodge structures
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- Research Article
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- © 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
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