SOME NEW EXAMPLES OF SMASH-NILPOTENT ALGEBRAIC CYCLES
Glasgow mathematical journal, Tome 59 (2017) no. 3, pp. 623-634

Voir la notice de l'article provenant de la source Cambridge University Press

Voevodsky has conjectured that numerical equivalence and smash-equivalence coincide for algebraic cycles on any smooth projective variety. Building on work of Vial and Kahn–Sebastian, we give some new examples of varieties where Voevodsky's conjecture is verified.
LATERVEER, ROBERT. SOME NEW EXAMPLES OF SMASH-NILPOTENT ALGEBRAIC CYCLES. Glasgow mathematical journal, Tome 59 (2017) no. 3, pp. 623-634. doi: 10.1017/S0017089516000434
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[1] 1. André, Y., Motifs de dimension finie (d'après S.-I. Kimura, P. O'Sullivan, . . .), Séminaire Bourbaki 2003/2004, Astérisque 299 Exp. No. 929, viii, 115–145. Google Scholar

[2] 2. Beauville, A., Sur l'anneau de Chow d'une variété abélienne, Math. Ann. 273 (1986), 647–651. Google Scholar | DOI

[3] 3. Bloch, S., Some elementary theorems about algebraic cycles on abelian varieties, Invent. Math. 37 (1976), 215–228. Google Scholar

[4] 4. Bloch, S., Lectures on algebraic cycles (Duke Univ. Press, Durham, 1980). Google Scholar

[5] 5. Bloch, S., Kas, A. and Lieberman, D., Zero cycles on surfaces with p = 0, Comp. Math. 33 (2) (1976), 135–145. Google Scholar

[6] 6. Bloch, S. and Ogus, A., Gersten's conjecture and the homology of schemes, Ann. Sci. Ecole Norm. Sup. 4 (1974), 181–202. Google Scholar | DOI

[7] 7. Bloch, S. and Srinivas, V., Remarks on correspondences and algebraic cycles, Am. J. Math. 105 (5) (1983), 1235–1253. Google Scholar

[8] 8. Bonfanti, M., On the cohomology of regular surfaces isogenous to a product of curves with χ( ) = 2, arXiv:1512.03168v1. Google Scholar

[9] 9. Brion, M., Log homogeneous varieties, in Actas del XVI Coloquio Latinoamericano de Algebra, Revista Matemática Iberoamericana (Madrid, 2007). Google Scholar

[10] 10. De Cataldo, M. and Migliorini, L., The Chow groups and the motive of the Hilbert scheme of points on a surface, J. Algebra 251 (2) (2002), 824–848. Google Scholar

[11] 11. Charles, F. and Markman, E., The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces, Comp. Math. 149 (2013), 481–494. Google Scholar

[12] 12. Cynk, S. and Hulek, K., Higher–dimensional modular Calabi–Yau manifolds, Canad. Math. Bull. 50 (4) (2007), 486–503. Google Scholar | DOI

[13] 13. Deligne, P., La conjecture de Weil pour les surfaces K3, Invent. Math. 15 (1972), 206–226. Google Scholar

[14] 14. Deninger, C. and Murre, J., Motivic decomposition of abelian schemes and the Fourier transform, J. Reine u. Angew. Math. 422 (1991), 201–219. Google Scholar

[15] 15. Garbagnati, A. and Penegini, M., K3 surfaces with a non–symplectic automorphism and product–quotient surfaces with cyclic groups, Rev. Mat. Iberoam. 31 (4) (2015), 1277–1310. Google Scholar

[16] 16. Guletskiĭ, V. and Pedrini, C., The Chow motive of the Godeaux surface, in Algebraic geometry, a volume in memory of Paolo Francia (Beltrametti, M. C., Catanese, F., Ciliberto, C., Lanteri, A. and Pedrini, C. Editors) (Walter de Gruyter, Berlin, New York, 2002), 179–196. Google Scholar

[17] 17. Ivorra, F., Finite dimensional motives and applications, following S.-I. Kimura, P. O'Sullivan and others, in Autour des motifs, Asian-French summer school on algebraic geometry and number theory, (Saito, T. et al., Editors) Volume III (Panoramas et synthèses, Société mathématique de France, 2011), 65–100. Google Scholar

[18] 18. Iyer, J., Murre's conjectures and explicit Chow–Künneth projectors for varieties with a nef tangent bundle, Trans. Amer. Math. Soc. 361 (2008), 1667–1681. Google Scholar

[19] 19. Iyer, J., Absolute Chow–Künneth decomposition for rational homogeneous bundles and for log homogeneous varieties, Michigan Math. J. 60 (1) (2011), 79–91. Google Scholar

[20] 20. Jannsen, U., Motivic sheaves and filtrations on Chow groups, in Motives (Jannsen, U. et al. Editors) Proceedings of Symposia in Pure Mathematics, vol. 55 (1994), Part 1, Amer. Math. Soc., 245–302. Google Scholar

[21] 21. Jannsen, U., Equivalence relations on algebraic cycles, in The arithmetic and geometry of algebraic cycles (Gordon, B. et al. Editors) (Banff Conference, Kluwer, 1998), 225–260. Google Scholar

[22] 22. Jannsen, U., On finite–dimensional motives and Murre's conjecture, in Algebraic cycles and motives (Nagel, J. and Peters, C. Editors) (Cambridge University Press, Cambridge, 2007), 112–142. Google Scholar

[23] 23. Kahn, B. and Sebastian, R., Smash–nilpotent cycles on abelian 3–folds, Math. Res. Lett. 16 (2009), 1007–1010. Google Scholar

[24] 24. Kimura, S., Chow groups are finite dimensional, in some sense, Math. Ann. 331 (2005), 173–201. Google Scholar

[25] 25. Kimura, S., Surjectivity of the cycle map for Chow motives, in Motives and algebraic cycles (de Jeu, R. and Lewis, J. Editors) (Amer. Math. Soc., Providence, 2009), 157–165. Google Scholar

[26] 26. Kimura, S. and Vistoli, A., Chow rings of infinite symmetric products, Duke Math. J. 85 (1996), 411–430. Google Scholar | DOI

[27] 27. Kleiman, S., Algebraic cycles and the Weil conjectures, in Dix exposés sur la cohomologie des schémas (North Holland Publishing, Amsterdam, 1968), 359–386. Google Scholar

[28] 28. Kleiman, S., The standard conjectures, in Motives (Jannsen, U. et al. Editors) Proceedings of Symposia in Pure Mathematics, vol. 55 (1994), Part 1, Amer. Math. Soc., 3–20. Google Scholar

[29] 29. Laterveer, R., Some desultory remarks concerning algebraic cycles and Calabi–Yau threefolds, Rend. Circ. Mat. Palermo 65 (2) (2016), 333–344. Google Scholar

[30] 30. Laterveer, R., A family of Calabi–Yau threefolds with finite–dimensional motive, submitted to Tokyo Math. J. Google Scholar

[31] 31. Murre, J., On a conjectural filtration on the Chow groups of an algebraic variety, parts I and II, Indag. Math. 4 (1993), 177–201. Google Scholar

[32] 32. Murre, J., Nagel, J. and Peters, C., Lectures on the theory of pure motives, Amer. Math. Soc. University Lecture Series, vol. 61 (Amer. Math. Soc., Providence, 2013). Google Scholar | DOI

[33] 33. Paranjape, K., Abelian varieties associated to certain K3 surfaces, Comp. Math. 68 (1988), 11–22. Google Scholar

[34] 34. Pedrini, C., On the finite dimensionality of a K3 surface, Manuscr. Math. 138 (2012), 59–72. Google Scholar

[35] 35. Pedrini, C. and Weibel, C., Some surfaces of general type for which Bloch's conjecture holds, in Recent Advances in Hodge Theory, Period Domains, Algebraic Cycles and Arithmetic (Kerr, M. and Pearlstein, G. Editors) (Cambridge University Press, Cambridge, 2016). Google Scholar

[36] 36. Samuel, P., Relations d'équivalence en géométrie algébrique, in Proc. Int. Congress Math. 1958 (Cambridge Univ. Press, New York, 1960), 470–487. Google Scholar

[37] 37. Scholl, T., Classical motives, in Motives (Jannsen, U. et al. Editors), Proceedings of Symposia in Pure Mathematics, vol. 55 (1994), Part 1, Amer. Math. Soc., 163–187. Google Scholar

[38] 38. Sebastian, R., Smash nilpotent cycles on varieties dominated by products of curves, Comp. Math. 149 (2013), 1511–1518. Google Scholar

[39] 39. Sebastian, R., Examples of smash nilpotent cycles on rationally connected varieties, J. Algebra 438 (2015), 119–129. Google Scholar

[40] 40. Shioda, T., The Hodge conjecture for Fermat varieties, Math. Ann. 245 (1979), 175–184. Google Scholar

[41] 41. Tankeev, S., On the standard conjecture of Lefschetz type for complex projective threefolds. II, Izv. Math. 75 (5) (2011), 1047–1062. Google Scholar | DOI

[42] 42. Vial, C., Algebraic cycles and fibrations, Doc. Math. 18 (2013), 1521–1553. Google Scholar

[43] 43. Vial, C., Projectors on the intermediate algebraic Jacobians, New York J. Math. 19 (2013), 793–822. Google Scholar

[44] 44. Vial, C., Remarks on motives of abelian type, to appear in Tohoku Math. J. Google Scholar

[45] 45. Vial, C., Niveau and coniveau filtrations on cohomology groups and Chow groups, Proc. LMS 106 (2) (2013), 410–444. Google Scholar

[46] 46. Vial, C., Chow–Künneth decomposition for 3– and 4–folds fibred by varieties with trivial Chow group of zero–cycles, J. Alg. Geom. 24 (2015), 51–80. Google Scholar

[47] 47. Voevodsky, V., A nilpotence theorem for cycles algebraically equivalent to zero, Internat. Math. Res. Not. 4 (1995), 187–198. Google Scholar

[48] 48. Voisin, C., Remarks on zero–cycles of self–products of varieties, in Moduli of vector bundles, Proceedings of the Taniguchi Congress (Maruyama, M. Editor) (Marcel Dekker, New York, Basel Hong Kong, 1994), 265–285. Google Scholar

[49] 49. Voisin, C., Bloch's conjecture for Catanese and Barlow surfaces, J. Differ. Geom. 97 (2014), 149–175. Google Scholar

[50] 50. Voisin, C., Chow rings, decomposition of the diagonal, and the topology of families (Princeton University Press, Princeton and Oxford, 2014). Google Scholar

[51] 51. Voisin, C., The generalized Hodge and Bloch conjectures are equivalent for general complete intersections, II, J. Math. Sci. Univ. Tokyo 22 (2015), 491–517. Google Scholar

[52] 52. Xu, Z., Algebraic cycles on a generalized Kummer variety, arXiv:1506.04297v1. Google Scholar

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