Comptes Rendus
no. 5
Algebraic geometry
On the Chow ring of certain rational cohomology tori
[Sur l'anneau de Chow de certaines variétés dont la cohomologie rationnelle est celle d'un tore]

Présenté par : Claire Voisin

Zhi Jiang1, 2 ; Qizheng Yin3
1 Département de mathématiques, CNRS UMR 8628, Université Paris-Sud, Bâtiment 425, 91405 Orsay cedex, France
2 Shanghai Center for Mathematical Sciences, Fudan University, No. 220 Handan Road, Yangpu District, Shanghai 200433, China
3 Beijing International Center for Mathematical Research, Peking University, No. 5 Yiheyuan Road, Haidian District Beijing 100871, China
Comptes Rendus. Mathématique, Volume 355 (2017) no. 5, pp. 571-576.

Soit f:X→A un revêtement abélien d'une variété algébrique complexe à singularités-quotient dans une variété abélienne. Nous montrons que f⁎ induit un isomorphisme entre les anneaux de cohomologie rationnelle H•(A,Q) et H•(X,Q) si et seulement si f⁎ induit un isomorphisme entre les anneaux de Chow à coefficients rationnels CH•(A)Q et CH•(X)Q.

Let f:X→A be an abelian cover from a complex algebraic variety with quotient singularities to an abelian variety. We show that f⁎ induces an isomorphism between the rational cohomology rings H•(A,Q) and H•(X,Q) if and only if f⁎ induces an isomorphism between the Chow rings with rational coefficients CH•(A)Q and CH•(X)Q.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.04.006
Zhi Jiang 1, 2 ; Qizheng Yin 3

1 Département de mathématiques, CNRS UMR 8628, Université Paris-Sud, Bâtiment 425, 91405 Orsay cedex, France
2 Shanghai Center for Mathematical Sciences, Fudan University, No. 220 Handan Road, Yangpu District, Shanghai 200433, China
3 Beijing International Center for Mathematical Research, Peking University, No. 5 Yiheyuan Road, Haidian District Beijing 100871, China 
@article{CRMATH_2017__355_5_571_0,
     author = {Zhi Jiang and Qizheng Yin},
     title = {On the {Chow} ring of certain rational cohomology tori},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {571--576},
     publisher = {Elsevier},
     volume = {355},
     number = {5},
     year = {2017},
     doi = {10.1016/j.crma.2017.04.006},
     language = {en},
}
TY  - JOUR
AU  - Zhi Jiang
AU  - Qizheng Yin
TI  - On the Chow ring of certain rational cohomology tori
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 571
EP  - 576
VL  - 355
IS  - 5
PB  - Elsevier
DO  - 10.1016/j.crma.2017.04.006
LA  - en
ID  - CRMATH_2017__355_5_571_0
ER  - 
%0 Journal Article
%A Zhi Jiang
%A Qizheng Yin
%T On the Chow ring of certain rational cohomology tori
%J Comptes Rendus. Mathématique
%D 2017
%P 571-576
%V 355
%N 5
%I Elsevier
%R 10.1016/j.crma.2017.04.006
%G en
%F CRMATH_2017__355_5_571_0
Zhi Jiang; Qizheng Yin. On the Chow ring of certain rational cohomology tori. Comptes Rendus. Mathématique, Volume 355 (2017) no. 5, pp. 571-576. doi : 10.1016/j.crma.2017.04.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.04.006/

[1] A. Beauville Sur l'anneau de Chow d'une variété abélienne, Math. Ann., Volume 273 (1986) no. 4, pp. 647-651

[2] Ch. Birkenhake; H. Lange Complex Abelian Varieties, Grundlehren der Mathematischen Wissenschaften, vol. 302, Springer-Verlag, Berlin, 2004

[3] O. Debarre; Z. Jiang; M. Lahoz; W. Savin Rational cohomology tori, Geom. Topol., Volume 21 (2017) no. 2, pp. 1095-1130

[4] B. Fantechi; E. Mann; F. Nironi Smooth toric Deligne–Mumford stacks, J. Reine Angew. Math., Volume 648 (2010), pp. 201-244

[5] Y. Kawamata Characterization of abelian varieties, Compos. Math., Volume 43 (1981) no. 2, pp. 253-276

[6] R. Pardini Abelian covers of algebraic varieties, J. Reine Angew. Math., Volume 417 (1991), pp. 191-213

[7] A. Vistoli Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math., Volume 97 (1989) no. 3, pp. 613-670

Cité par Sources :

Commentaires - Politique