  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.
