The exceptional locus of a motivic local system

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 André 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 André's fullness theorem.