Chow–Künneth projectors for modular varieties
[Projecteurs de Chow–Künneth pour des variétés modulaires]
Comptes Rendus. Mathématique, Tome 335 (2002) no. 9, pp. 745-750.

Nous démontrons l'existence des projecteurs de Chow–Künneth pour certaines variétés, incluant les variétés de Kuga–Shimura des variétés modulaires de Hilbert. Les projecteurs de Chow–Künneth d'une variété lisse projective sont par définition des idempotents orthogonaux de l'anneau de Chow des auto-correspondances qui donnent la décomposition par les degrés de la cohomologie totale de la variété.

We show the existence of the Chow–Künneth projectors for certain varieties, including Kuga–Shimura varieties of Hilbert modular varieties. The Chow–Künneth projectors of a smooth projective variety are, by definition, mutually orthogonal idempotents of the Chow ring of self-correspondences which give decomposition of the total cohomology of the variety into degree pieces.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02506-2
Gordon, B.Brent 1 ; Hanamura, Masaki 2 ; Murre, Jacob P. 3

1 Department of Mathematics, University of Oklahoma, 601 Elm, Room 423, Norman, OK 73019, USA
2 Graduate School of Mathematics, Kyushu University, Fukuoka, 812 Japan
3 Department of Mathematics, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands
@article{CRMATH_2002__335_9_745_0,
     author = {Gordon, B.Brent and Hanamura, Masaki and Murre, Jacob P.},
     title = {Chow{\textendash}K\"unneth projectors for modular varieties},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {745--750},
     publisher = {Elsevier},
     volume = {335},
     number = {9},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02506-2},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/S1631-073X(02)02506-2/}
}
TY  - JOUR
AU  - Gordon, B.Brent
AU  - Hanamura, Masaki
AU  - Murre, Jacob P.
TI  - Chow–Künneth projectors for modular varieties
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 745
EP  - 750
VL  - 335
IS  - 9
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/S1631-073X(02)02506-2/
DO  - 10.1016/S1631-073X(02)02506-2
LA  - en
ID  - CRMATH_2002__335_9_745_0
ER  - 
%0 Journal Article
%A Gordon, B.Brent
%A Hanamura, Masaki
%A Murre, Jacob P.
%T Chow–Künneth projectors for modular varieties
%J Comptes Rendus. Mathématique
%D 2002
%P 745-750
%V 335
%N 9
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/S1631-073X(02)02506-2/
%R 10.1016/S1631-073X(02)02506-2
%G en
%F CRMATH_2002__335_9_745_0
Gordon, B.Brent; Hanamura, Masaki; Murre, Jacob P. Chow–Künneth projectors for modular varieties. Comptes Rendus. Mathématique, Tome 335 (2002) no. 9, pp. 745-750. doi : 10.1016/S1631-073X(02)02506-2. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02506-2/

[1] Ash, A.; Mumford, D.; Rapoport, M.; Tai, Y. Smooth Compactification of Locally Symmetric Varieties, Math. Sci. Press, 1975

[2] Beilinson, A.; Bernstein, J.; Deligne, P. Faisceaux Pervers, Analyse et topologie sur les espaces singuliers, Astérisque, Volume 100 (1982), pp. 7-171

[3] Borel, A. Intersection Cohomology, Prog. in Math., 50, Birkhäuser, 1984 (pp. 47–182)

[4] A. Corti, M. Hanamura, Motivic decomposition and intersection Chow groups I, Preprint

[5] Deninger, C.; Murre, J.P. Motivic decomposition of abelian schemes and the Fourier transform, J. Reine Angew. Math., Volume 422 (1991), pp. 201-219

[6] Fulton, W. Intersection Theory, Springer, 1984

[7] Goresky, M.; MacPherson, R. Intersection homology II, Inv. Math., Volume 71 (1983), pp. 77-129

[8] Gordon, B.; Murre, J.P. Chow motives of elliptic modular surfaces and threefolds, Report W 96-16, Math. Institut, Univ. of Leiden, 1996

[9] Gordon, B.; Murre, J.P. Chow motives of elliptic modular threefolds, J. Reine Angew. Math., Volume 514 (1999), pp. 145-164

[10] Grothendieck, A. Standard conjectures on algebraic cycles, Algebraic Geometry, Bombay Colloquium, Oxford, 1969, pp. 193-199

[11] S. Kleiman, Algebraic cycles and Weil conjectures, Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, pp. 359–386

[12] Murre, J.P. On the motive of an algebraic surface, J. Reine Angew. Math., Volume 409 (1990), pp. 190-204

[13] Murre, J.P. On a conjectural filtration on the Chow groups of an algebraic variety I and II, Indag. Math., Volume 2 (1993), pp. 177-188 (189–201)

[14] Namikawa, Y. Toroidal Compactification of Siegel Spaces, Lecture Notes in Math., 812, Springer-Verlag, 1980

[15] Saito, M. Mixed Hodge modules, Publ. RIMS, Kyoto Univ., Volume 26 (1990), pp. 221-333

[16] Scholl, A.J. Motives for modular forms, Invent. Math., Volume 100 (1990), pp. 419-430

Cité par Sources :