Cohomologically rigid local systems and integrality

TL;DR

This paper proves that certain rigid local systems on complex varieties have integral monodromy, confirming a special case of Simpson's conjecture, using techniques involving $ ext{l}$-adic companions and weight control.

Contribution

It establishes the integrality of monodromy for a class of cohomologically rigid local systems, advancing understanding of their arithmetic properties.

Findings

Monodromy of irreducible cohomologically rigid local systems is integral.

The proof uses Drinfeld's theorem on $ ext{l}$-adic companions.

Results confirm a special case of Simpson's conjecture.

Abstract

We prove that the monodromy of an irreducible cohomologically complex rigid local system with finite determinant and quasi-unipotent local monodromies at infinity on a smooth quasiprojective complex variety $X$ is integral. This answers positively a special case of a conjecture by Carlos Simpson. On a smooth projective variety, the argument relies on Drinfeld's theorem on the existence of $\ell$-adic companions over a finite field. When the variety is quasiprojective, one has in addition to control the weights and the monodromy at infinity.