Chow groups of intersections of quadrics via homological projective duality and (Jacobians of) noncommutative motives

Marcello Bernardara (1) , Goncalo Tabuada (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 MIT - Massachusetts Institute of Technology

Making use of homological projective duality and the recent theory of (Jacobians of) noncommutative Chow motives, we compute the rational Chow groups of a complete intersection of either two quadrics or three odd-dimensional quadrics. We show moreover that the unique non-trivial algebraic Jacobians are the middle ones. As a first application, we describe the rational Chow motives of these complete intersections. As a second application, we prove that smooth fibrations in such complete intersections over small dimensional bases S verify Murre's conjecture (dim(S) less or equal to 1), Grothendieck's standard conjectures (dim(S) less of equal to 2), and Hodge's conjecture (dim(S) less or equal to 3).

Mots clés: intermediate Jacobians; Chow groups; noncommutative motives; semiorthogonal decompositions; intersection of quadrics

Domaines: Géométrie algébrique [math.AG]; Topologie algébrique [math.AT]; K-théorie et homologie [math.KT]; Anneaux et algèbres [math.RA]

Marcello Bernardara, Goncalo Tabuada. Chow groups of intersections of quadrics via homological projective duality and (Jacobians of) noncommutative motives. Izvestiya: Mathematics, 2016, ⟨10.1070/IM8409⟩. ⟨hal-00876512⟩