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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
- Type
- Research Article
- Information
- Copyright
- © 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence