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Izv. RAN. Ser. Mat., 2016, Volume 80, Issue 3, Pages 3–22 (Mi izv8409)  

This article is cited in 2 scientific papers (total in 2 papers)

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

M. Bernardaraabc, G. Tabuadad

a Université Paul Sabatier, Toulouse
b Université de Toulouse
c Institute de Mathématique de Toulouse
d Department of Mathematics, Massachusetts Institute of Technology

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

Keywords: quadrics, homological projective duality, Jacobians, non-commutative motives, non-commutative algebraic geometry.

Funding Agency Grant Number
National Science Foundation
G. Tabuada was partially supported by the National Science Foundation CAREER Award #1350472 and by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project grant UID/MAT/00297/2013 (Centro de Matemática e Aplicações).


DOI: https://doi.org/10.4213/im8409

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English version:
Izvestiya: Mathematics, 2016, 80:3, 463–480

Bibliographic databases:

UDC: 512.7
MSC: 14A22, 14C15, 14F05, 14J40, 14M10
Received: 14.05.2015

Citation: M. Bernardara, G. Tabuada, “Chow groups of intersections of quadrics via homological projective duality and (Jacobians of) non-commutative motives”, Izv. RAN. Ser. Mat., 80:3 (2016), 3–22; Izv. Math., 80:3 (2016), 463–480

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. M. Marcolli, G. Tabuada, “Feynman quadrics-motive of the massive sunset graph”, J. Number Theory, 195 (2019), 159–183  crossref  mathscinet  zmath  isi
    2. M. Ornaghi, L. Pertusi, “Voevodsky's conjecture for cubic fourfolds and gushel-mukai fourfolds via noncommutative K3 surfaces”, J. Noncommutative Geom., 13:2 (2019), 499–515  crossref  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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