Local systems with quasi-unipotent monodromy at infinity are dense

Abstract: We show that complex local systems with quasi-unipotent monodromy at infinity over a normal complex variety are Zariski dense in their moduli. v2: we waited for feedback and added a consequence of Alexandr Petrov's theorem. 3: we tightened the last section. Final version: appears in Israel Journal of Mathematics. footnote added to Conjecture 1.1: Aaron Landesman and Daniel Litt just made available a preprint showing that there is a lower bound for the rank of geometric local systems with infinite mon-odromy on certain curves, and consequently the conjecture can not be true in this generality.