This small text was written for the AMS Notices. It is a survey of integrality properties of complex local systems, where I tried to single out one example which is not entirely explicit in the literature. The focus is on the obstruction it yields for a finitely presented group to be the topological fundamental group of a connected smooth quasi-projective complex variety. I thank Johan de Jong, Michael Groechenig, and Moritz Kerz.
The material in this expository note relies on joint work or discussions with them.
This version: I have tried several times to make the AMS bib file compatible with the arXiv requirement, it never worked..., today it seems to do!