Local systems with quasi-unipotent monodromy at infinity are dense

TL;DR

This paper proves that complex local systems with quasi-unipotent monodromy at infinity are densely distributed in their moduli space on normal complex varieties, with implications for the structure of such systems.

Contribution

It establishes the Zariski density of local systems with quasi-unipotent monodromy at infinity, extending understanding of their distribution and properties.

Findings

Local systems with quasi-unipotent monodromy are Zariski dense in their moduli.

Added consequences of Alexandr Petrov's theorem to the main results.

Discussed limitations of a related conjecture based on recent preprints.

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.