Integrality of components of the Betti moduli

Determine whether, for every smooth projective complex variety X and integer r ≥ 1, each irreducible component of the Betti moduli space M_B(X, r) contains a point defined over the ring of algebraic integers \overline{\mathbb{Z}}.

Background

This conjecture strengthens Simpson’s integrality conjecture by asking for algebraic integer points on every component, not just zero-dimensional ones. It is known in some cases (e.g., reduced zero-dimensional components and for rank r=2) and is pivotal for deducing full finiteness results for isomonodromy in the authors’ framework.

The conjecture provides the arithmetic input needed to upgrade compactness of orbits to global finiteness of the monodromy action across components.