The exceptional locus of a motivic local system

Abstract

To every Nori motivic local system over a smooth, connected complex algebraic variety, we associate an exceptional locus controlling the variation in the complexity of its stalks; the definition is given explicitly in terms of motivic Galois groups and Artin motives. We prove 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 it is defined over any algebraically closed subfield over which the motivic local system admits a model, and stable under Galois conjugation when the latter descends to a smaller subfield. This extends and strengthens previous results by André in the pure case. We obtain a similar description for the splitting locus of the motivic weight filtration.

42 pages; v2: removed section 5, some typos fixed, some references added