# Absolute Hodge and $\ell$-adic Monodromy

@article{Urbanik2020AbsoluteHA, title={Absolute Hodge and \$\ell\$-adic Monodromy}, author={David Urbanik}, journal={arXiv: Algebraic Geometry}, year={2020} }

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 $\sigma \in \textrm{Aut}(\mathbb{C}/K)$ 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_…

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