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

Abstract

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 [20]. 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 [26], [27] to describe the rational Chow motives of these complete intersections and show that smooth fibrations into such complete intersections over bases of small dimension satisfy Murre's conjecture (when ), Grothendieck's standard conjecture of Lefschetz type (when ), and Hodge's conjecture (when ).