Geodesic
Chow groups of intersections of quadrics via homological projective duality
Izvestiya. Mathematics , Tome 80 (2016) no. 3, pp. 463-480

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{IM2_2016_80_3_a0,
     author = {M. Bernardara and G. Tabuada},
     title = {Chow groups of intersections of quadrics via homological projective duality},
     journal = {Izvestiya. Mathematics },
     pages = {463--480},
     publisher = {mathdoc},
     volume = {80},
     number = {3},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2016_80_3_a0/}
}
TY  - JOUR
AU  - M. Bernardara
AU  - G. Tabuada
TI  - Chow groups of intersections of quadrics via homological projective duality
JO  - Izvestiya. Mathematics 
PY  - 2016
SP  - 463
EP  - 480
VL  - 80
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2016_80_3_a0/
LA  - en
ID  - IM2_2016_80_3_a0
ER  - 
%0 Journal Article
%A M. Bernardara
%A G. Tabuada
%T Chow groups of intersections of quadrics via homological projective duality
%J Izvestiya. Mathematics 
%D 2016
%P 463-480
%V 80
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2016_80_3_a0/
%G en
%F IM2_2016_80_3_a0
M. Bernardara; G. Tabuada. Chow groups of intersections of quadrics via homological projective duality. Izvestiya. Mathematics , Tome 80 (2016) no. 3, pp. 463-480. http://geodesic.mathdoc.fr/item/IM2_2016_80_3_a0/