The exceptional locus of a motivic local system

TL;DR

This paper introduces the concept of the exceptional locus for motivic local systems, demonstrating it is a countable union of algebraic subvarieties and exploring its properties and implications in motivic and Hodge theory.

Contribution

It provides a motivic analogue of a classical theorem, describing the structure of the exceptional locus and its stability properties, extending previous pure case results.

Findings

The exceptional locus is a countable union of closed algebraic subvarieties.

Maximal subvarieties are defined over certain subfields and are Galois-stable.

Results extend to the splitting locus of the motivic weight filtration and to 1-motivic local systems.

Abstract

Given a Nori motivic local system over a smooth, connected complex algebraic variety, we define its exceptional locus as a way to measure the variation in the motivic complexity of its stalks. The definition is given explicitly in terms of motivic Galois groups and Artin motives. Our main result is a motivic analogue of the Cattani--Deligne--Kaplan Theorem, asserting that the exceptional locus is a countable union of closed algebraic subvarieties. Moreover, we show that the maximal such subvarieties are defined over any algebraically closed subfield over which the ambient variety and the motivic local system admit models, and that they are stable under Galois conjugation when these models descend to a further subfield. This strengthens and extends previous results by Andr\'e in the pure case. We obtain a similar geometric description for the splitting locus of the motivic weight filtration. In the case of 1-motivic local systems, the above properties pass to the underlying variations of mixed Hodge structure via Andr\'e's fullness theorem.