On a smooth complex connected quasi-projective variety, a rational Betti class is Hodge if it has an integral structure and lies as a de Rham class in the right levels of the Hodge and the weight filtrations. Let $f: X\to S$ be a projective morphism between smooth quasi-projective manifolds. Deligne’s Fix Part Theorem asserts that a Hodge class on a fibre $X_s$ has finite orbit under the fundamental group of $S$ based at $s$. if and only if it extends to a Hodge class on $X$ after base change to a finite étale cover of S, if $S$ is an Artin neighborhood. Replacing above the Hodge class with a rational polarizable variation of Hodge structure which admits an integral structure, Deligne’s Fix Part Theorem has a perfect non-abelian analog. The proof involves among other ingredients Deligne’s finiteness theorem, which itself echoes on the Hodge side Faltings’ celebrated finiteness theorem. The result should have arithmetic versions. This is joint work with Moritz Kerz

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