Integrality of the Betti moduli space
If in a given rank $r$, there is an irreducible complex local system with torsion determinant and quasi-unipotent monodromies at infinity on a smooth quasi-projective variety, then for every prime number $\ell$, there is an absolutely irreducible $\ell$-adic local system of the same rank, with the same determinant and monodromies at infinity, up to semi-simplification. A finitely presented group is said to be weakly integral with respect to a torsion character and a rank $r$ if once there is an irreducible rank $r$ complex linear representation, then for any $\ell$, there is an absolutely irreducible one of rank $r$ and determinant this given character which is defined over $\bar{ \mathbb{Z}}_\ell$. We prove that this property is a new obstruction for a finitely presented group to be the fundamental group of a smooth qusi-projective complex variety. The proofs rely on the arithmetic Langlands program via the existence of Deligne's companions (L. Lafforgue, Drinfeld) and the geometric Langlands program via de Jong's conjecture (Gaitsgory for $\ell \ge 3$). We also define weakly arithmetic complex local systems and show they are Zariski dense in the Betti moduli. Finally we show that our method gives an arithmetic proof of Corlette-T. Mochizuki theorem, proved using tame pure imaginary harmonic metrics, after which the pull-back by a morphism between two smooth complex algebraic varieties of a semi-simple complex local system is semi-simple. v2: a mistake pointed out by the kind referee in the proof of Theorem 7.3 is corrected. Final version: appears in Transactions AMS