The generalized Hodge and Bloch conjectures are equivalent for general complete intersections  [ Les conjectures de Hodge et de Bloch généralisées sont équivalentes pour les intersections complètes générales ]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 3, p. 449-475
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consulter l'article sur le site de la revue
Nous montrons la conjecture de Bloch pour les surfaces avec p g =0 obtenues comme lieux des zéros X σ d’une section σ d’un fibré vectoriel très ample sur une variété X à groupes de Chow « triviaux ». Nous obtenons un résultat similaire en présence d’une action d’un groupe fini, montrant que si un projecteur du groupe agit comme 0 sur les 2-formes holomorphes de X σ , il agit comme 0 sur les 0-cycles de degré 0 de X σ . En dimension supérieure, nous obtenons un résultat similaire mais conditionnel montrant que la conjecture de Hodge généralisée pour X σ générale entraîne la conjecture de Bloch généralisée pour tout X σ lisse, en supposant satisfaite la conjecture de Lefschetz standard (cette dernière hypothèse n’étant pas nécessaire en dimension 3).
We prove that Bloch’s conjecture is true for surfaces with p g =0 obtained as 0-sets X σ of a section σ of a very ample vector bundle on a variety X with “trivial” Chow groups. We get a similar result in presence of a finite group action, showing that if a projector of the group acts as 0 on holomorphic 2-forms of X σ , then it acts as 0 on 0-cycles of degree 0 of X σ . In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general X σ implies the generalized Bloch conjecture for any smooth X σ , assuming the Lefschetz standard conjecture (the last hypothesis is not needed in dimension 3).
DOI : https://doi.org/10.24033/asens.2193
Classification:  14C25,  14C30
Mots clés: cycles algébriques, conjecture de Bloch, conjecture de Hodge généralisée
@article{ASENS_2013_4_46_3_449_0,
     author = {Voisin, Claire},
     title = {The generalized Hodge and Bloch conjectures are equivalent for general complete intersections},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 46},
     number = {3},
     year = {2013},
     pages = {449-475},
     doi = {10.24033/asens.2193},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2013_4_46_3_449_0}
}
Voisin, Claire. The generalized Hodge and Bloch conjectures are equivalent for general complete intersections. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 3, pp. 449-475. doi : 10.24033/asens.2193. http://www.numdam.org/item/ASENS_2013_4_46_3_449_0/

[1] A. Albano & A. Collino, On the Griffiths group of the cubic sevenfold, Math. Ann. 299 (1994), 715-726. | MR 1286893

[2] S. Bloch, Lectures on algebraic cycles, second éd., New Mathematical Monographs 16, Cambridge Univ. Press, 2010. | MR 2723320

[3] S. Bloch & V. Srinivas, Remarks on correspondences and algebraic cycles, Amer. J. Math. 105 (1983), 1235-1253. | MR 714776

[4] F. Charles, Remarks on the Lefschetz standard conjecture and hyperkähler varieties, preprint 2010, to appear in Comm. Math. Helv. | MR 3048193

[5] J.-L. Colliot-Thélène & C. Voisin, Cohomologie non ramifiée et conjecture de Hodge entière, Duke Math. J. 161 (2012), 735-801. | MR 2904092

[6] P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Publ. Math. I.H.É.S. 35 (1968), 259-278. | MR 244265

[7] H. Esnault, M. Levine & E. Viehweg, Chow groups of projective varieties of very small degree, Duke Math. J. 87 (1997), 29-58. | MR 1440062

[8] W. Fulton & R. Macpherson, A compactification of configuration spaces, Ann. of Math. 139 (1994), 183-225. | MR 1259368

[9] M. Green & P. Griffiths, Hodge-theoretic invariants for algebraic cycles, Int. Math. Res. Not. 2003 (2003), 477-510. | MR 1951543

[10] A. Grothendieck, Hodge's general conjecture is false for trivial reasons, Topology 8 (1969), 299-303. | MR 252404 | Zbl 0177.49002

[11] M. Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), 941-1006. | MR 1839919 | Zbl 1009.14001

[12] S.-I. Kimura, Chow groups are finite dimensional, in some sense, Math. Ann. 331 (2005), 173-201. | MR 2107443 | Zbl 1067.14006

[13] S. L. Kleiman, Algebraic cycles and the Weil conjectures, in Dix exposés sur la cohomologie des schémas, North-Holland, 1968, 359-386. | MR 292838 | Zbl 0198.25902

[14] R. Laterveer, Algebraic varieties with small Chow groups, J. Math. Kyoto Univ. 38 (1998), 673-694. | MR 1669995 | Zbl 0961.14003

[15] M. Lehn & C. Sorger, Letter to the author, June 24th, 2011.

[16] J. D. Lewis, A generalization of Mumford's theorem. II, Illinois J. Math. 39 (1995), 288-304. | MR 1316539 | Zbl 0823.14002

[17] D. Mumford, Rational equivalence of 0-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), 195-204. | MR 249428 | Zbl 0184.46603

[18] J. P. Murre, On the motive of an algebraic surface, J. reine angew. Math. 409 (1990), 190-204. | MR 1061525 | Zbl 0698.14032

[19] M. V. Nori, Algebraic cycles and Hodge-theoretic connectivity, Invent. Math. 111 (1993), 349-373. | MR 1198814 | Zbl 0822.14008

[20] A. Otwinowska, Remarques sur les cycles de petite dimension de certaines intersections complètes, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 141-146. | MR 1710511 | Zbl 0961.14004

[21] A. Otwinowska, Remarques sur les groupes de Chow des hypersurfaces de petit degré, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 51-56. | MR 1703267 | Zbl 0981.14004

[22] K. H. Paranjape, Cohomological and cycle-theoretic connectivity, Ann. of Math. 139 (1994), 641-660. | MR 1283872 | Zbl 0828.14003

[23] C. Peters, Bloch-type conjectures and an example of a three-fold of general type, Commun. Contemp. Math. 12 (2010), 587-605. | MR 2678942 | Zbl 1200.14015

[24] A. A. Rojtman, The torsion of the group of 0-cycles modulo rational equivalence, Ann. of Math. 111 (1980), 553-569. | MR 577137 | Zbl 0504.14006

[25] S. Saito, Motives and filtrations on Chow groups, Invent. Math. 125 (1996), 149-196. | MR 1389964 | Zbl 0897.14001

[26] C. Schoen, On Hodge structures and nonrepresentability of Chow groups, Compositio Math. 88 (1993), 285-316. | Numdam | MR 1241952 | Zbl 0802.14004

[27] A. J. Sommese, Submanifolds of Abelian varieties, Math. Ann. 233 (1978), 229-256. | MR 466647 | Zbl 0381.14007

[28] T. Terasoma, Infinitesimal variation of Hodge structures and the weak global Torelli theorem for complete intersections, Ann. of Math. 132 (1990), 213-235. | MR 1070597 | Zbl 0732.14005

[29] C. Voisin, Sur les zéro-cycles de certaines hypersurfaces munies d'un automorphisme, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 19 (1992), 473-492. | Numdam | MR 1205880 | Zbl 0786.14006

[30] C. Voisin, Remarks on zero-cycles of self-products of varieties, in Moduli of vector bundles (Sanda, 1994; Kyoto, 1994), Lecture Notes in Pure and Appl. Math. 179, Dekker, 1996, 265-285. | MR 1397993 | Zbl 0912.14003

[31] C. Voisin, Sur les groupes de Chow de certaines hypersurfaces, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), 73-76. | MR 1390824 | Zbl 0867.14002

[32] C. Voisin, Hodge theory and complex algebraic geometry. I and II, Cambridge Studies in Advanced Math. 76 and 77, Cambridge Univ. Press, 2002, 2003. | MR 1967689 | Zbl 1005.14002

[33] C. Voisin, Coniveau 2 complete intersections and effective cones, Geom. Funct. Anal. 19 (2010), 1494-1513. | MR 2585582 | Zbl 1205.14009

[34] C. Voisin, Lectures on the Hodge and Grothendieck-Hodge conjectures, Rend. Semin. Mat. Univ. Politec. Torino 69 (2011), 149-198. | MR 2931228 | Zbl 1249.14003