On exceptional rigid local systems

TL;DR

This paper proves new cases of Simpson's rigidity conjecture, showing that certain rigid local systems are motivic, and constructs motives over a punctured sphere leading to non-rigid $G_2$-local systems in $ ext{GL}_7$.

Contribution

It advances the understanding of Simpson's rigidity conjecture by providing new instances and constructing motives that produce non-rigid local systems in a specific group.

Findings

Proved new instances of Simpson's rigidity conjecture.

Constructed motives over the fourfold punctured sphere.

Identified $G_2$-rigid local systems that are not rigid in $ ext{GL}_7$.

Abstract

We prove new instances of Simpson's rigidity conjecture which states that quasi-unipotent rigid local systems should be motivic. We construct new relative motives over the fourfold punctured Riemann sphere which give rise to $G_2$-rigid local systems which are not rigid in the group $\GL_7.$