Skip to main content
SpringerLink
Log in

On a multiplicative version of Bloch’s conjecture

  • Original Paper
  • Published:
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry Aims and scope Submit manuscript

Abstract

A theorem of Esnault, Srinivas and Viehweg asserts that if the Chow group of 0-cycles of a smooth complete complex variety decomposes, then the top-degree coherent cohomology group decomposes similarly. In this note, we prove that (a weak version of) the converse holds for varieties of dimension at most 5 that have finite-dimensional motive and satisfy the Lefschetz standard conjecture. The proof is based on Vial’s construction of a refined Chow–Künneth decomposition for these varieties.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. It is somewhat frustrating that it is not known unconditionally whether (P1) implies (P2), i.e. without assuming the generalized Hodge conjecture. Apparently Esnault, Srinivas and Viehweg had claimed to prove this in an earlier version of their paper, but the argument was found to be incomplete (Esnault et al. 1993, remark 2).

  2. To be precise, Nori’s result is more general, as the notion of k-decomposability in Esnault et al. (1993) is broader than the notion discussed here: in loc. cit., an element \(a\in A^nV\) is defined to be k-decomposable if it can be written as \(a=\sum _{j=1}^k a_j\cdot b_j\) with \(a_j\cdot b_j\) homogeneous of degree n.

References

  • André, Y.: Motifs de dimension finie (d’après S.-I. Kimura, P. O’Sullivan...), Séminaire Bourbaki. vol. 2003/2004. Astérisque 299 Exp. No. 929, viii, pp. 115–145 (2004)

  • Ayoub, J.: Motives and algebraic cycles: a selection of conjectures and open questions (2016, preprint)

  • Bloch, S.: Some elementary theorems about algebraic cycles on abelian varieties. Invent. Math. 37, 215–228 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  • Bloch, S.: On an argument of Mumford in the theory of algebraic cycles. In: Beauville, A. (ed.) Géométrie algébrique. Sijthoff and Noordhoff, Angers (1979)

    Google Scholar 

  • Bloch, S.: Lectures on algebraic cycles. Duke Univ. Press, Durham (1980)

    MATH  Google Scholar 

  • Bloch, S., Ogus, A.: Gersten’s conjecture and the homology of schemes. Ann. Sci. Ecole Norm. Sup. 4, 181–202 (1974)

    MathSciNet  MATH  Google Scholar 

  • Bloch, S., Srinivas, V.: Remarks on correspondences and algebraic cycles. Am. J. Math. 105(5), 1235–1253 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  • Esnault, H., Srinivas, V., Viehweg, E.: Decomposability of Chow groups implies decomposability of cohomology. In:“Journées de Géométrie Algébrique d’Orsay, Juillet 1992”. Astérisque 218, 227–242 (1993)

  • Guletskiĭ, V., Pedrini, C.: The Chow motive of the Godeaux surface. In: Beltrametti, M.C., Catanese, F., Ciliberto, C., Lanteri, A., Pedrini, C. (eds.) Algebraic Geometry, a volume in memory of Paolo Francia. Walter de Gruyter, Berlin, New York (2002)

    Google Scholar 

  • Guletskiĭ, V., Pedrini, C.: Finite-dimensional motives and the conjectures of Beilinson and Murre. K-Theory 30(3), 243–263 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • 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, Volume III, Panoramas et synthèses, Société mathématique de France (2011)

  • Iyer, J.: Murre’s conjectures and explicit Chow-Künneth projectors for varieties with a nef tangent bundle. Transactions of the Amer. Math. Soc. 361, 1667–1681 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Iyer, J.: Absolute Chow-Künneth decomposition for rational homogeneous bundles and for log homogeneous varieties. Mich. Math. J. 60(1), 79–91 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Jannsen, U.: Motives, numerical equivalence, and semi-simplicity. Invent. Math. 107(3), 447–452 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Jannsen, U.: Equivalence relations on algebraic cycles. In: Gordon, B., et al. (eds.) The arithmetic and geometry of algebraic cycles. Kluwer, Banff Conf. (1998)

    Google Scholar 

  • Jannsen, U.: On finite–dimensional motives and Murre’s conjecture. In: Algebraic cycles and motives, London Math. Soc. Lecture Note Ser., vol. 344. Cambridge Univ. Press, Cambridge (2007)

  • Kahn, B.: Equivalences rationnelle et numérique sur certaines variétés de type abélien sur un corps fini. Ann. Sci. Ec. Norm. Sup. 36, 977–1002 (2003)

    MathSciNet  Google Scholar 

  • Kahn, B., Murre, J.P., Pedrini, C.: On the transcendental part of the motive of a surface. In: Algebraic cycles and motives. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 344, pp. 143–202. Cambridge Univ. Press, Cambridge (2007)

  • Kahn, B., Sujatha, R.: Birational motives, I: pure birational motives (2009). arXiv:0902.4902v4

  • Kimura, S.: Chow groups are finite dimensional, in some sense. Math. Ann. 331, 173–201 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Kleiman, S.: Algebraic cycles and the Weil conjectures. In: Dix exposés sur la cohomologie des schémas, pp. 359—386. North–Holland Amsterdam, (1968)

  • Kleiman, S.: The standard conjectures. In: Jannsen, U., et al, (eds.) Motives. Proceedings of Symposia in Pure Mathematics, vol. 55, Part 1 (1994)

  • Laterveer, R.: A brief note concerning hard Lefschetz for Chow groups. Can. Math. Bull. (2015). doi:10.4153/CMB-2015-046-x

    MathSciNet  MATH  Google Scholar 

  • Mumford, D.: Rational equivalence of \(0\)-cycles on surfaces. J. Math. Kyoto Univ. 9(2), 195–204 (1969)

    MathSciNet  MATH  Google Scholar 

  • Murre, J., Nagel, J., Peters, C.: Lectures on the theory of pure motives. Amer. Math. Soc. University Lecture Series, vol. 61 (2013)

  • Pedrini, C.: On the finite dimensionality of a \(K3\) surface. Manuscr. Math. 138, 59–72 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Shen, M., Vial, C.: The Fourier transform for certain hyperKähler fourfolds. Mem. AMS 240(1139) (2016)

  • Tankeev, S.: On the standard conjecture of Lefschetz type for complex projective threefolds. II. Izvestiya Math. 75(5), 1047–1062 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Vial, C.: Algebraic cycles and fibrations. Documenta Math. 18, 1521–1553 (2013)

    MathSciNet  Google Scholar 

  • Vial, C.: Projectors on the intermediate algebraic Jacobians. N. Y. J. Math. 19, 793–822 (2013)

    MathSciNet  MATH  Google Scholar 

  • Vial, C.: Remarks on motives of abelian type (2011). arXiv:1112.1080v2

  • Vial, C.: Niveau and coniveau filtrations on cohomology groups and Chow groups. Proc. LMS 106(2), 410–444 (2013)

    MathSciNet  MATH  Google Scholar 

  • Vial, C.: Chow-Künneth decomposition for 3- and 4-folds fibred by varieties with trivial Chow group of zero-cycles. J. Algebraic Geom. 24, 51–80 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Voevodsky, V.: A nilpotence theorem for cycles algebraically equivalent to zero. Internat Math. Res. Not. 4, 187–198 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  • Voisin, C.: Remarks on zero-cycles on self-products of varieties, in: Moduli of vector bundles (Proceedings of the Taniguchi Congress), Maruyama Ed., Decker (1994)

  • Voisin, C.: Chow rings, decomposition of the diagonal, and the topology of families. Princeton University Press, Princeton and Oxford (2014)

    Book  MATH  Google Scholar 

  • Yin, Q.: Finite-dimensionality and cycles on powers of K3 surfaces. Comment. Math. Helv. 90, 503–511 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This note was stimulated by the Strasbourg 2014–2015 “groupe de travail” based on the monograph Voisin (2014). I want to thank all the participants of this groupe de travail for the very pleasant and stimulating atmosphere, and their interesting lectures. Thanks to Charles Vial and the referee for helpful comments. Many thanks to Yasuyo, Kai and Len for providing excellent working conditions at home in Schiltigheim.

Author information

Authors and Affiliations

  1. CNRS-IRMA, Université de Strasbourg, 7 rue René Descartes, 67084, Strasbourg Cedex, France

    Robert Laterveer

Authors
  1. Robert Laterveer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert Laterveer.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Laterveer, R. On a multiplicative version of Bloch’s conjecture. Beitr Algebra Geom 57, 723–734 (2016). https://doi.org/10.1007/s13366-016-0296-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13366-016-0296-4

Keywords

Mathematics Subject Classification

Search

Navigation