The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

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

Claire Voisin

Annales scientifiques de l'École Normale Supérieure (2013)

  • Volume: 46, Issue: 3, page 449-475
  • ISSN: 0012-9593

Abstract

top
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 ).

How to cite

top

Voisin, Claire. "The generalized Hodge and Bloch conjectures are equivalent for general complete intersections." Annales scientifiques de l'École Normale Supérieure 46.3 (2013): 449-475. <http://eudml.org/doc/272244>.

@article{Voisin2013,
abstract = {We prove that Bloch’s conjecture is true for surfaces with $p_g=0$ obtained as $0$-sets $X_\sigma $ of a section $\sigma $ 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_\sigma $, then it acts as $0$ on $0$-cycles of degree $0$ of $X_\sigma $. In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general $X_\sigma $ implies the generalized Bloch conjecture for any smooth $X_\sigma $, assuming the Lefschetz standard conjecture (the last hypothesis is not needed in dimension $3$).},
author = {Voisin, Claire},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {algebraic cycles; Bloch conjecture; generalized Hodge conjecture},
language = {eng},
number = {3},
pages = {449-475},
publisher = {Société mathématique de France},
title = {The generalized Hodge and Bloch conjectures are equivalent for general complete intersections},
url = {http://eudml.org/doc/272244},
volume = {46},
year = {2013},
}

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
PB - Société mathématique de France
VL - 46
IS - 3
SP - 449
EP - 475
AB - We prove that Bloch’s conjecture is true for surfaces with $p_g=0$ obtained as $0$-sets $X_\sigma $ of a section $\sigma $ 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_\sigma $, then it acts as $0$ on $0$-cycles of degree $0$ of $X_\sigma $. In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general $X_\sigma $ implies the generalized Bloch conjecture for any smooth $X_\sigma $, assuming the Lefschetz standard conjecture (the last hypothesis is not needed in dimension $3$).
LA - eng
KW - algebraic cycles; Bloch conjecture; generalized Hodge conjecture
UR - http://eudml.org/doc/272244
ER -

References

top
  1. [1] A. Albano & A. Collino, On the Griffiths group of the cubic sevenfold, Math. Ann.299 (1994), 715–726. Zbl0803.14022MR1286893
  2. [2] S. Bloch, Lectures on algebraic cycles, second éd., New Mathematical Monographs 16, Cambridge Univ. Press, 2010. Zbl1201.14006MR2723320
  3. [3] S. Bloch & V. Srinivas, Remarks on correspondences and algebraic cycles, Amer. J. Math.105 (1983), 1235–1253. Zbl0525.14003MR714776
  4. [4] F. Charles, Remarks on the Lefschetz standard conjecture and hyperkähler varieties, preprint 2010, to appear in Comm. Math. Helv. MR3048193
  5. [5] J.-L. Colliot-Thélène & C. Voisin, Cohomologie non ramifiée et conjecture de Hodge entière, Duke Math. J.161 (2012), 735–801. Zbl1244.14010MR2904092
  6. [6] P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Publ. Math. I.H.É.S. 35 (1968), 259–278. Zbl0159.22501MR244265
  7. [7] H. Esnault, M. Levine & E. Viehweg, Chow groups of projective varieties of very small degree, Duke Math. J.87 (1997), 29–58. Zbl0916.14001MR1440062
  8. [8] W. Fulton & R. MacPherson, A compactification of configuration spaces, Ann. of Math.139 (1994), 183–225. Zbl0820.14037MR1259368
  9. [9] M. Green & P. Griffiths, Hodge-theoretic invariants for algebraic cycles, Int. Math. Res. Not.2003 (2003), 477–510. Zbl1049.14002MR1951543
  10. [10] A. Grothendieck, Hodge’s general conjecture is false for trivial reasons, Topology8 (1969), 299–303. Zbl0177.49002MR252404
  11. [11] M. Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc.14 (2001), 941–1006. Zbl1009.14001MR1839919
  12. [12] S.-I. Kimura, Chow groups are finite dimensional, in some sense, Math. Ann. 331 (2005), 173–201. Zbl1067.14006MR2107443
  13. [13] S. L. Kleiman, Algebraic cycles and the Weil conjectures, in Dix exposés sur la cohomologie des schémas, North-Holland, 1968, 359–386. Zbl0198.25902MR292838
  14. [14] R. Laterveer, Algebraic varieties with small Chow groups, J. Math. Kyoto Univ.38 (1998), 673–694. Zbl0961.14003MR1669995
  15. [15] M. Lehn & C. Sorger, Letter to the author, June 24th, 2011. 
  16. [16] J. D. Lewis, A generalization of Mumford’s theorem. II, Illinois J. Math. 39 (1995), 288–304. Zbl0823.14002MR1316539
  17. [17] D. Mumford, Rational equivalence of 0 -cycles on surfaces, J. Math. Kyoto Univ.9 (1968), 195–204. Zbl0184.46603MR249428
  18. [18] J. P. Murre, On the motive of an algebraic surface, J. reine angew. Math. 409 (1990), 190–204. Zbl0698.14032MR1061525
  19. [19] M. V. Nori, Algebraic cycles and Hodge-theoretic connectivity, Invent. Math.111 (1993), 349–373. Zbl0822.14008MR1198814
  20. [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. Zbl0961.14004MR1710511
  21. [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. MR1703267
  22. [22] K. H. Paranjape, Cohomological and cycle-theoretic connectivity, Ann. of Math.139 (1994), 641–660. Zbl0828.14003MR1283872
  23. [23] C. Peters, Bloch-type conjectures and an example of a three-fold of general type, Commun. Contemp. Math.12 (2010), 587–605. Zbl1200.14015MR2678942
  24. [24] A. A. Rojtman, The torsion of the group of 0 -cycles modulo rational equivalence, Ann. of Math.111 (1980), 553–569. Zbl0504.14006MR577137
  25. [25] S. Saito, Motives and filtrations on Chow groups, Invent. Math.125 (1996), 149–196. Zbl0897.14001MR1389964
  26. [26] C. Schoen, On Hodge structures and nonrepresentability of Chow groups, Compositio Math.88 (1993), 285–316. Zbl0802.14004MR1241952
  27. [27] A. J. Sommese, Submanifolds of Abelian varieties, Math. Ann.233 (1978), 229–256. MR466647
  28. [28] T. Terasoma, Infinitesimal variation of Hodge structures and the weak global Torelli theorem for complete intersections, Ann. of Math.132 (1990), 213–235. Zbl0732.14005MR1070597
  29. [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. MR1205880
  30. [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. MR1397993
  31. [31] C. Voisin, Sur les groupes de Chow de certaines hypersurfaces, C. R. Acad. Sci. Paris Sér. I Math.322 (1996), 73–76. MR1390824
  32. [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. MR1967689
  33. [33] C. Voisin, Coniveau 2 complete intersections and effective cones, Geom. Funct. Anal.19 (2010), 1494–1513. Zbl1205.14009MR2585582
  34. [34] C. Voisin, Lectures on the Hodge and Grothendieck-Hodge conjectures, Rend. Semin. Mat. Univ. Politec. Torino69 (2011), 149–198. MR2931228

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.