The generalized Hodge and Bloch conjectures are equivalent for general complete intersections

Voisin, Claire

doi:10.24033/asens.2193

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]

Voisin, Claire

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 3, pp. 449-475.

Résumé
Abstract

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 : 10.24033/asens.2193

Classification : 14C25, 14C30
Keywords: algebraic cycles, Bloch conjecture, generalized Hodge conjecture
Mot 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},
     pages = {449--475},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 46},
     number = {3},
     year = {2013},
     doi = {10.24033/asens.2193},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2193/}
}

TY  - JOUR
AU  - Voisin, Claire
TI  - The generalized Hodge and Bloch conjectures are equivalent for general complete intersections
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2013
SP  - 449
EP  - 475
VL  - 46
IS  - 3
PB  - Société mathématique de France
UR  - https://www.numdam.org/articles/10.24033/asens.2193/
DO  - 10.24033/asens.2193
LA  - en
ID  - ASENS_2013_4_46_3_449_0
ER  -

%0 Journal Article
%A Voisin, Claire
%T The generalized Hodge and Bloch conjectures are equivalent for general complete intersections
%J Annales scientifiques de l'École Normale Supérieure
%D 2013
%P 449-475
%V 46
%N 3
%I Société mathématique de France
%U https://www.numdam.org/articles/10.24033/asens.2193/
%R 10.24033/asens.2193
%G en
%F 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. https://www.numdam.org/articles/10.24033/asens.2193/

Bibliographie
Cité par

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

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

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

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

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

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

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

[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 | Zbl

[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. | EuDML | Numdam | MR | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

Cité par Sources :