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Variational motivic principle for local systems of geometric origin

Ascertain that, for a smooth proper morphism f: X→S of smooth complex schemes, if a local system V on X has V|_{X_s} of geometric origin for some s∈S(C), then V|_{X_t} is of geometric origin for all t∈S(C).

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Background

This conjecture is a variational form of the Fontaine–Mazur philosophy, predicting that geometric origin persists across fibers in families. It is presented as a consequence of a relative Fontaine–Mazur conjecture and underlies expected stability of motivic properties along isomonodromy leaves.

The authors discuss how, combined with their arithmetic/algebraicity conjectures, it predicts that integrality-based algebraicity forces motivic origin to persist along isomonodromic deformations.

References

Conjecture (A variational motivic conjecture). Let f: X\to S be a smooth proper morphism of (say) smooth \mathbb{C}-schemes, s\in S(\mathbb{C}), and \mathbb{V} is a local system on X with \mathbb{V}|{X_s} of geometric origin, then \mathbb{V}|{X_t} is of geometric origin for all t\in S(\mathbb{C}).

— Algebraicity and integrality of solutions to differential equations (2501.13175 - Lam et al., 22 Jan 2025) in Conjecture, Section “A motivic variational principle”