Skip to main content
SpringerLink
Log in

A 9-dimensional family of K3 surfaces with finite-dimensional motive

  • Published:
Rendiconti del Circolo Matematico di Palermo Series 2 Aims and scope Submit manuscript

Abstract

Let S be a K3 surface obtained as triple cover of a quadric branched along a genus 4 curve. Using the relation with cubic fourfolds, we show that S has finite-dimensional motive, in the sense of Kimura. We also establish the Kuga–Satake Hodge conjecture for S, as well as Voisin’s conjecture concerning zero-cycles. As a consequence, we obtain Kimura finite-dimensionality, the Kuga–Satake Hodge conjecture, and Voisin’s conjecture for 2 (9-dimensional) irreducible components of the moduli space of K3 surfaces with an order 3 non-symplectic automorphism.

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

References

  1. Artebani, M., Sarti, A.: Non-symplectic automorphisms of order 3 on K3 surfaces. Math. Ann. 342(4), 903–921 (2008)

    Article  MathSciNet  Google Scholar 

  2. Beauville, A.: Sur l’anneau de Chow d’une variété abélienne. Math. Ann. 273, 647–651 (1986)

    Article  MathSciNet  Google Scholar 

  3. Bloch, S., Gillet, H., Soulé, C.: Non-Archimedean Arakelov theory. J. Algebr. Geom. 4(3), 427–485 (1995)

    MathSciNet  Google Scholar 

  4. Bolognesi, M., Pedrini, C.: The transcendental motive of a cubic fourfold. J. Pure Appl. Algebra 224(8), 106333 (2020)

    Article  MathSciNet  Google Scholar 

  5. Boissière, S., Camere, C., Sarti, A.: Classification of automorphisms on a deformation family of hyperkähler fourfolds by \(p\)-elementary lattices. Kyoto J. Math. 56(3), 465–499 (2016)

    Article  MathSciNet  Google Scholar 

  6. Boissière, S., Heckel, T., Sarti, A.: The Fano variety of lines of a cuspidal cyclic cubic fourfold, arXiv:2209.11022

  7. de Cataldo, M., Migliorini, L.: The Chow groups and the motive of the Hilbert scheme of points on a surface. J. Algebra 251, 824–848 (2002)

    Article  MathSciNet  Google Scholar 

  8. Deligne, P.: La conjecture de Weil pour les surfaces K3. Invent. Math. 15, 206–226 (1972)

    Article  MathSciNet  Google Scholar 

  9. Deninger, C., Murre, J.: Motivic decomposition of abelian schemes and the Fourier transform. J. Reine Angew. Math. 422, 201–219 (1991)

    MathSciNet  Google Scholar 

  10. Floccari, S.: Sixfolds of generalized Kummer type and K3 surfaces, arXiv:2210.02948

  11. Fu, L., Laterveer, R., Vial, Ch.: The generalized Franchetta conjecture for some hyper-Kähler varieties II. J. l’Ecole Polytechnique Math. 8, 1065–1097 (2021)

    Article  MathSciNet  Google Scholar 

  12. Fu, L., Vial, Ch.: A motivic global Torelli theorem for isogenous K3 surfaces. Adv. Math. 383, 107674 (2021)

    Article  MathSciNet  Google Scholar 

  13. Fulton, W.: Intersection Theory. Springer-Verlag Ergebnisse der Mathematik, Berlin Heidelberg New York Tokyo (1984)

    Book  Google Scholar 

  14. Galkin, S., Shinder, E.: The Fano variety of lines and rationality problem for a cubic hypersurface, arXiv:1405.5154

  15. van Geemen, B.: Kuga–Satake varieties and the Hodge conjecture. In: The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), Kluwer Academic, Dordrecht (2000)

  16. van Geemen, B., Izadi, E.: Half twists and the cohomology of hypersurfaces. Math. Zeit. 242, 279–301 (2002)

    Article  MathSciNet  Google Scholar 

  17. Huybrechts, D.: Motives of isogenous K3 surfaces. Comment. Math. Helv. 94(3), 445–458 (2019)

    Article  MathSciNet  Google Scholar 

  18. Huybrechts, D.: Lectures on K3 Surfaces. Cambridge University Press, Cambridge (2016)

    Book  Google Scholar 

  19. Guletski, 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 

  20. Hassett, B.: Special cubic fourfolds, (English summary). Compos. Math. 120(1), 1–23 (2000)

    Article  Google Scholar 

  21. Hirschowitz, A., Iyer, J.: Hilbert schemes of fat \(r\)-planes and the triviality of Chow groups of complete intersections. In: Vector Bundles and Complex Geometry, pp. 53—70 (2010)

  22. Ingalls, C., Logan, A., Patashnick, O.: Explicit coverings of elliptic surfaces by squares of curves, arXiv:2009.07807

  23. Jannsen, U.: On finite-dimensional motives and Murre’s conjecture. In: Nagel, J., Peters, C. (eds.) Algebraic Cycles and Motives. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  24. Kimura, S.-I.: Fractional intersection and bivariant theory. Commun. Algebra 20, 285–302 (1992)

    Article  MathSciNet  Google Scholar 

  25. Kimura, S.-I.: Chow groups are finite dimensional, in some sense. Math. Ann. 331(1), 173–201 (2005)

    Article  MathSciNet  Google Scholar 

  26. Kondo, S.: The moduli space of curves of genus 4 and Deligne–Mostow’s complex reflection groups. In: Advanced Studies in Pure Math. Algebraic Geometry, Azumino, vol. 36, pp. 383–400 (2000)

  27. Kuga, M., Satake, I.: Abelian varieties attached to polarized K3 surfaces. Math. Ann. 169(239–242), 72 (1967)

    Google Scholar 

  28. Kuznetsov, A.: Derived categories of cubic fourfolds, "Cohomological and Geometric Approaches to Rationality Problems. New Perspectives" Series: Progress in Mathematics, Bogomolov, Fedor; Tschinkel, Yuri (Eds.), X, Vol. 282, p. 314, Birkhauser (2010)

  29. Laterveer, R.: Algebraic varieties with small Chow groups. J. Math. Kyoto Univ. 38(4), 673–694 (1998)

    MathSciNet  Google Scholar 

  30. Laterveer, R.: A family of K3 surfaces having finite-dimensional motive. Arch. Math. 106(6), 515–524 (2016)

    Article  MathSciNet  Google Scholar 

  31. Laterveer, R.: Some results on a conjecture of Voisin for surfaces of geometric genus one. Boll. Unione Mat. Italiana 9(4), 435–452 (2016)

    Article  MathSciNet  Google Scholar 

  32. Laterveer, R.: Some desultory remarks concerning algebraic cycles and Calabi–Yau threefolds. Rend. Circ. Mat. Palermo 65(2), 333–344 (2016)

    Article  MathSciNet  Google Scholar 

  33. Laterveer, R.: A remark on the motive of the Fano variety of lines of a cubic. Ann. Math. Québec 41(1), 141–154 (2017)

    Article  MathSciNet  Google Scholar 

  34. Laterveer, R.: Algebraic cycles and Todorov surfaces. Kyoto J. Math. 58(3), 493–527 (2018)

    Article  MathSciNet  Google Scholar 

  35. Laterveer, R.: A family of cubic fourfolds with finite-dimensional motive. J. Math. Soc. Jpn. 70(4), 1453–1473 (2018)

    Article  MathSciNet  Google Scholar 

  36. Laterveer, R., Vial, Ch.: On the Chow ring of Cynk–Hulek Calabi–Yau varieties and Schreieder varieties. Can. J. Math. 72(2), 505–536 (2020)

    Article  MathSciNet  Google Scholar 

  37. Laterveer, R.: Zero-cycles on self-products of varieties: some elementary examples verifying Voisin’s conjecture. Boll. Unione Mat. Ital. 14(2), 323–329 (2021)

    Article  MathSciNet  Google Scholar 

  38. Lewis, J.: Cylinder homomorphisms and Chow groups. Math. Nachr. 160, 205–221 (1993)

    Article  MathSciNet  Google Scholar 

  39. Livné, R., Schütt, M., Yui, N.: The modularity of K3 surfaces with non-symplectic group actions. Math. Ann. 348(2), 333–355 (2010)

    Article  MathSciNet  Google Scholar 

  40. Miura, K., Taki, S.: Quartic surfaces with a Galois point and Eisenstein K3 surfaces. arxiv:2308.11102 (2023)

  41. Murre, J., Nagel, J., Peters, C.: Lectures on the Theory of Pure Motives. University Lecture Series, American Mathematical Society, Providence (2013)

    Book  Google Scholar 

  42. Otwinowska, A.: Remarques sur les groupes de Chow des hypersurfaces de petit degré. C. R. Acad. Sci. Paris Sér. I Math. 329(1), 51–56 (1999)

    Article  MathSciNet  Google Scholar 

  43. Paranjape, K.: Abelian varieties associated to certain K3 surfaces. Compos. Math. 68, 11–22 (1988)

    MathSciNet  Google Scholar 

  44. Paranjape, K.: Cohomological and cycle-theoretic connectivity. Ann. Math. 139(2), 641–660 (1994)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  46. Pedrini, C., Weibel, C.: Some surfaces of general type for which Bloch’s conjecture holds. In: Period Domains, Algebraic Cycles, and Arithmetic, Cambridge University Press, Cambridge (2015)

  47. Scholl, T.: Classical motives. In: Jannsen, U. et al. (ed.) Motives. Proceedings of Symposia in Pure Mathematics, Part 1, vol. 55 (1994)

  48. Shen, M., Vial, C.: The Fourier Transform for Certain HyperKähler Fourfolds, vol. 240. American Mathematical Society, Providence (2016)

    Google Scholar 

  49. B. Totaro, Chow groups, Chow cohomology and linear varieties. In: Forum of Mathematics, Sigma, vol. 1, p. e1 (2014)

  50. Vial, Ch.: Projectors on the intermediate algebraic Jacobians. New York J. Math. 19, 793–822 (2013)

    MathSciNet  Google Scholar 

  51. Vial, Ch.: Remarks on motives of abelian type. Tohoku Math. J. 69, 195–220 (2017)

    Article  MathSciNet  Google Scholar 

  52. Vial, Ch.: Chow-Küneeth decomposition for 3- and 4-folds fibred by varieties with trivial Chow group of zero-cycles. J. Algebr. Geom. 24, 51–80 (2015)

    Article  Google Scholar 

  53. Voisin, C.: Remarks on zero-cycles of self-products of varieties. In: Maruyama, M. (ed.) Moduli of Vector Bundles, Proceedings of the Taniguchi Congress. Marcel Dekker, New York Basel Hong Kong (1994)

    Google Scholar 

  54. Voisin, C.: Unirational threefolds with no universal codimension 2 cycle. Invent. Math. 201(1), 207–237 (2015)

    Article  MathSciNet  Google Scholar 

  55. Voisin, C.: Chow Rings, Decomposition of the Diagonal, and the Topology of Families. Princeton University Press, Princeton and Oxford (2014)

    Book  Google Scholar 

  56. Xu, Z.: Algebraic cycles on a generalized Kummer variety. Int. Math. Res. Not. IMRN 2018(3), 932–948 (2018)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Thanks to Samuel Boissière for an inspiring talk on [6] in Strasbourg (May 2023). RL thanks MB for an invitation to sunny Montpellier (June 2023), where this work was initiated.

Author information

Authors and Affiliations

  1. Institut Montpellierain Alexander Grothendieck, Université de Montpellier, CNRS, Case Courrier 051 - Place Eugène Bataillon, 34095, Montpellier Cedex, France

    Michele Bolognesi

  2. Institut de Recherche Mathématique Avancée, CNRS – Université de Strasbourg, 7 Rue René Des-car-tes, 67084, Strasbourg Cedex, France

    Robert Laterveer

Authors
  1. Michele Bolognesi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert Laterveer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert Laterveer.

Ethics declarations

Conflict of interest

The authors have no Conflict of interest to declare that are relevant to the content of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Michele Bolognesi and Robert Laterveer are supported by ANR Grant ANR-20-CE40-0023.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bolognesi, M., Laterveer, R. A 9-dimensional family of K3 surfaces with finite-dimensional motive. Rend. Circ. Mat. Palermo, II. Ser (2024). https://doi.org/10.1007/s12215-024-01036-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12215-024-01036-0

Keywords

Mathematics Subject Classification

Search

Navigation