Generalized Kuga-Satake theory and rigid local systems, II: rigid Hecke eigensheaves

TL;DR

This paper constructs new examples of generalized Kuga-Satake theory using rigid Hecke eigensheaves, linking motives with exceptional Galois groups to intersection cohomology motives, and extends foundational results on motives of quasi-projective varieties.

Contribution

It provides the first robust examples of generalized Kuga-Satake theory outside abelian motives and establishes that intersection cohomology motives originate from motivated motives, extending prior work.

Findings

Constructed examples of generalized Kuga-Satake lifts outside abelian motives.

Proved that intersection cohomology of quasi-projective varieties arises from motivated motives.

Extended results on the motivic nature of intersection cohomology to a broad class of varieties.

Abstract

This paper uses rigid Hecke eigensheaves, building on Yun's work on the construction of motives with exceptional Galois groups, to produce the first robust examples of `generalized Kuga-Satake theory' outside the Tannakian category of motives generated by abelian varieties. To strengthen our description of the `motivic' nature of Kuga-Satake lifts, we digress to establish a result that should be of independent interest: for any quasi-projective variety over a (finitely-generated) characteristic zero field, the associated weight-graded of its intersection cohomology arises from a motivated motive in the sense of Andr\'{e}, and in particular from a classical homological motive if one assumes the Standard Conjectures. This extends work of de Cataldo and Migliorini.