Generalized Kuga-Satake theory and rigid local systems, I: the middle convolution

TL;DR

This paper presents a novel approach to constructing examples of generalized Kuga-Satake theory by leveraging Simpson's conjecture and rigid local systems, resulting in the first non-trivial examples outside traditional motive categories.

Contribution

It introduces a new method to produce generalized Kuga-Satake examples using rigid local systems and Simpson's conjecture, expanding the scope beyond abelian varieties.

Findings

Produced the first non-trivial examples outside abelian motives

Utilized work of Bogner and Reiter in a low-dimensional case

Established a strategy for generating generalized Kuga-Satake examples

Abstract

This note explains an approach to producing examples of 'generalized Kuga-Satake theory' based on establishing special cases of Simpson's conjecture that rigid local systems are motivic. This strategy is then carried out, using work of Bogner and Reiter, in a low-dimensional example, producing the first non-trivial examples of generalized Kuga-Satake theory outside of the Tannakian category of motives generated by abelian varieties. For a more robust collection of examples, see the sequel "Generalized Kuga-Satake theory and rigid local systems, II: rigid Hecke eigensheaves."