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 [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 $\dim(S)\leq 3$).
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