Theta-vanishing implies finite monodromy on character varieties

Prove that if f: X → S is a smooth projective morphism of smooth complex varieties and the lifting of tangent vectors Θ_{X/S} vanishes identically on the Dolbeault moduli stack M_Dol(X/S, r), then the action of π1(S(C)^an, s) on the Betti moduli M_B(X_s, r)(C) factors through a finite group.

Background

The authors introduce a non-abelian analogue of the Hodge-theoretic Kodaira–Spencer map, the lifting of tangent vectors Θ{X/S}, and show that vanishing of Θ{X/S} has strong geometric consequences (e.g., compactness of orbits and metric preservation).

They conjecture a strong finiteness conclusion for the π1-action on character varieties under the vanishing of Θ_{X/S}, extending their main results and aligning with recent developments in non-abelian Hodge theory.

References

Conjecture Let f: X\to S be a smooth projective morphism of smooth complex varieties, and fix s\in S. If \Theta_{X/S}\equiv 0 on \mathscr{M}_{Dol}(X/S, r), then the action of \pi_1(S(\mathbb{C}){\an},s) on M_B(X_s, r)(\mathbb{C}) factors through a finite group.

— p-Curvature and Non-Abelian Cohomology  (2601.07933 - Lam et al., 12 Jan 2026) in Introduction, Subsection “Questions”, Conjecture 1 (labelled Conjecture \ref{conj:theta-vanishing})