Chow groups of intersections of quadrics via homological projective duality and (Jacobians of) non-commutative motives

Conjectures of Beilinson–Bloch type predict that the low-degree rational Chow groups of intersections of quadrics are one-dimensional. This conjecture was proved by Otwinowska in [1]. By making use of homological projective duality and the recent theory of (Jacobians of) non-commutative motives, we give an alternative proof of this conjecture in the case of a complete intersection of either two quadrics or three odd-dimensional quadrics. Moreover, we prove that in these cases the unique non-trivial algebraic Jacobian is the middle one. As an application, we make use of Vial's work [2], [3] to describe the rational Chow motives of these complete intersections and show that smooth fibrations into such complete intersections over bases $S$ of small dimension satisfy Murre's conjecture (when $\dim (S)\leq 1$), Grothendieck's standard conjecture of Lefschetz type (when $\dim (S)\leq 2$), and Hodge's conjecture (when $\dim(S)\leq 3$).