On the universal regular homomorphism in codimension 2
[Sur l’homomorphisme régulier universel en codimension 2]
Annales de l'Institut Fourier, Tome 71 (2021) no. 2, pp. 843-848.

On signale une lacune dans la preuve de l’existence d’un homomorphisme régulier universel pour les cycles de codimension 2 sur une variété projective lisse par Murre, et on donne deux arguments différents pour combler cette lacune.

We point out a gap in Murre’s proof of the existence of a universal regular homomorphism for codimension 2 cycles on a smooth projective variety, and offer two arguments to fill this gap.

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DOI : 10.5802/aif.3408
Classification : 14C25, 14K30
Keywords: Algebraic cycles, abelian varieties
Mot clés : Cycles algébriques, variétés abéliennes
Kahn, Bruno 1

1 IMJ-PRG Case 247 4 place Jussieu 75252 Paris Cedex 05 France
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Kahn, Bruno. On the universal regular homomorphism in codimension $2$. Annales de l'Institut Fourier, Tome 71 (2021) no. 2, pp. 843-848. doi : 10.5802/aif.3408. http://www.numdam.org/articles/10.5802/aif.3408/

[1] Achter, Jeffrey D.; Casalaina-Martin, Sebastian; Vial, Charles On descending cohomology geometrically, Compos. Math., Volume 153 (2017) no. 7, pp. 1446-1478 | DOI | MR | Zbl

[2] Beauville, Arnaud Quelques remarques sur la transformation de Fourier dans l’anneau de Chow d’une variété abélienne, Algebraic geometry (Tokyo/Kyoto, 1982) (Lecture Notes in Mathematics), Volume 1016, Springer, 1982, pp. 238-260 | DOI | Zbl

[3] Benoist, Olivier; Wittenberg, Olivier The Clemens-Griffiths method over non-closed fields, Algebr. Geom., Volume 7 (2020) no. 6, pp. 696-721 | DOI | MR | Zbl

[4] Bloch, Spencer Some elementary theorems about algebraic cycles on Abelian varieties, Invent. Math., Volume 37 (1976) no. 3, pp. 215-228 | DOI | MR | Zbl

[5] Bloch, Spencer; Esnault, Hélène The coniveau filtration and non-divisibility for algebraic cycles, Math. Ann., Volume 304 (1996) no. 2, pp. 303-314 | DOI | MR | Zbl

[6] Clemens, C. Herbert; Griffiths, Philip Augustus The intermediate Jacobian of the cubic threefold, Ann. Math., Volume 95 (1972), pp. 281-356 | DOI | MR | Zbl

[7] Colliot-Thélène, Jean-Louis; Pirutka, Alena Troisième groupe de cohomologie non ramifiée d’un solide cubique sur un corps de fonctions d’une variable, Épijournal de Géom. Algébr., EPIGA, Volume 2 (2018), 13, 13 pages | Zbl

[8] Eilenberg, Samuel; Mac Lane, Saunders On the groups H(Π,n), II; methods of computation, Ann. Math., Volume 60 (1954), pp. 49-139 | DOI | MR

[9] Lichtenbaum, Stephen New results on weight-two motivic cohomology, The Grothendieck Festschrift, Vol. III (Progress in Mathematics), Volume 88, Birkhäuser, 1990, pp. 35-55 | DOI | MR | Zbl

[10] Milne, James S. Jacobian varieties, Arithmetic Geometry (Cornell, G.; H., Silverman J., eds.), Springer, 1998, pp. 167-212 | DOI

[11] Murre, Jacob P. Applications of algebraic K-theory to the theory of algebraic cycles, Algebraic geometry, Proc. Conf., Sitges (Barcelona)/Spain 1983 (Lecture Notes in Mathematics), Volume 1124, Springer, 1985, pp. 216-261 | DOI | MR | Zbl

[12] Nori, Madhav V. Cycles on the generic abelian threefold, Proc. Indian Acad. Sci., Math. Sci., Volume 99 (1989) no. 3, pp. 191-196 | DOI | MR | Zbl

[13] Roǐtman, A. A. The torsion of the group of 0-cycles modulo rational equivalence, Ann. Math., Volume 111 (1980) no. 3, pp. 553-569 | DOI | MR | Zbl

[14] Rosenschon, Andreas; Srinivas, Vasudevan The Griffiths group of the generic abelian 3-fold, Cycles, motives and Shimura varieties. Proceedings of the international colloquium, Mumbai, India, January 3–12, 2008 (Tata Institute of Fundamental Research Studies in Mathematics), Volume 21, Narosa Publishing House / Tata Institute of Fundamental Research (2010), pp. 449-467 | MR | Zbl

[15] Schoen, Chad On certain exterior product maps of Chow groups, Math. Res. Lett., Volume 7 (2000) no. 2-3, pp. 177-194 | DOI | MR | Zbl

[16] Totaro, Burt Complex varieties with infinite Chow groups modulo 2, Ann. Math., Volume 183 (2016) no. 1, pp. 363-375 | DOI | MR | Zbl

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