A note on the Schur-finiteness of linear sections

TL;DR

This paper demonstrates that Schur-finiteness of mixed motives remains invariant under homological projective duality and applies this to show Schur-finiteness of motives of certain algebraic varieties, extending to Kimura-finiteness.

Contribution

It introduces the invariance of Schur-finiteness under homological projective duality using noncommutative motives and applies this to specific algebraic varieties.

Findings

Schur-finiteness is invariant under homological projective duality.

Mixed motives of certain linear sections are Schur-finite.

Applications extend to Kimura-finiteness.

Abstract

Making use of the recent theory of noncommutative motives, we prove that Schur-finiteness in the setting of Voevodsky's mixed motives is invariant under homological projective duality. As an application, we show that the mixed motives of smooth linear sections of certain (Lagrangian) Grassmannians, spinor varieties, and determinantal varieties, are Schur-finite. Finally, we upgrade our applications from Schur-finiteness to Kimura-finiteness.