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

• Main Authors: Bernardara, M. (Author), Trigo Neri Tabuada, Goncalo Jorge (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
• Format: Article
• Language: English
• Published: Turpion, 2016-09-26T19:26:40Z.
• Subjects: Article
• Online Access: Get fulltext

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 S of small dimension satisfy Murre's conjecture (when dim(S) [less than or equal to] 1), Grothendieck's standard conjecture of Lefschetz type (when dim(S) [less than or equal to] 2), and Hodge's conjecture (when dim(S) [less than or equal to] 3).
National Science Foundation (U.S.) (CAREER Award #1350472)
Fundação para a Ciência e a Tecnologia (Portugal) (project grant UID/MAT/00297/2013 (Centro de Matemática e Aplicações))