Absolute Hodge and ℓ-adic monodromy

@article{Urbanik2022AbsoluteHA,
  title={Absolute Hodge and ℓ-adic monodromy},
  author={David Urbanik},
  journal={Compositio Mathematica},
  year={2022},
  volume={158},
  pages={568 - 584}
}
  • D. Urbanik
  • Published 21 November 2020
  • Mathematics
  • Compositio Mathematica
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… 

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