Bi-algebraicity in the rank one Riemann-Hilbert correspondence via o-minimality

Abstract

For a smooth, projective, complex algebraic variety X, the Riemann–Hilbert correspondence establishes a complex analytic isomorphism between the ‘Betti moduli space’ of rank n local systems on Xan and the ‘de Rham moduli space’ of rank n vector bundles with flat connection on X. In the rank one case, C. Simpson precisely characterizes the subvarieties of these moduli spaces that are ‘bi-algebraic’ for this typically transcendental, analytic isomorphism. In this short note, we give a new proof of this characterization of Simpson, using methods from o-minimal geometry. We adapt the o-minimal proof to a p-adic setting, namely that of Mumford curves.