Zero-cycles on Cancian–Frapporti surfaces

  • Robert LaterveerEmail author


An old conjecture of Voisin describes how 0-cycles on a surface S should behave when pulled-back to the self-product \(S^m\) for \(m>p_g(S)\). We show that Voisin’s conjecture is true for a 3-dimensional family of surfaces of general type with \(p_g=q=2\) and \(K^2=7\) constructed by Cancian and Frapporti, and revisited by Pignatelli–Polizzi.


Algebraic cycles Chow groups Motives Voisin conjecture surfaces of general type Abelian varieties Prym varieties 

Mathematics Subject Classification

Primary 14C15 14C25 14C30 



I am grateful to a referee who kindly suggested substantial simplifications of the main argument. Thanks to Kai and Len, my dedicated coworkers at the Alsace Center for Advanced Lego-Building and Mathematics.


  1. 1.
    Abdulali S.: Tate twists of Hodge structures arising from abelian varieties. In: Kerr M., Pearlstein G. (eds.) Recent Advances in Hodge Theory: Period Domains, Algebraic Cycles, and Arithmetic , London Math. Society Lecture Note Series 427, pp. 292–307. Cambridge University Press, Cambridge (2016)Google Scholar
  2. 2.
    Beauville, A.: Sur l’anneau de Chow d’une variété abélienne. Math. Ann. 273, 647–651 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bini, G., Laterveer R., Pacienza, G.: Voisin’s conjecture for zero-cycles on Calabi–Yau varieties and their mirrors. Adv. Geom (to appear) Google Scholar
  4. 4.
    Bloch, S.: Lectures on algebraic cycles. Duke University, Press Durham (1980)zbMATHGoogle Scholar
  5. 5.
    Bloch, S., Srinivas, V.: Remarks on correspondences and algebraic cycles. Am. J. Math. 105(5), 1235–1253 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Burek, D.: Higher-dimensional Calabi–Yau manifolds of Kummer type. arXiv:1810.11084
  7. 7.
    Cancian, N., Frapporti, D.: On semi-isogenous mixed surfaces. Math. Nachr. 291(2–3), 264–283 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Deninger, Ch., Murre, J.: Motivic decomposition of abelian schemes and the Fourier transform. J. Reine Angew. Math. 422, 201–219 (1991)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fulton, W.: Intersection Theory. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
  10. 10.
    Fu, L., Vial, Ch.: Distinguished cycles on varieties with motive of abelian type and the section property. J. Algebraic GeomGoogle Scholar
  11. 11.
    Jannsen, U.: Motivic sheaves and filtrations on Chow groups. In: Jannsen, U., et al. (eds.) Motives, Proceedings of Symposia in Pure Mathematics, Part 1, vol. 55 (1994)Google Scholar
  12. 12.
    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
  13. 13.
    Kahn, B., Murre, J., Pedrini, C.: On the transcendental part of the motive of a surface. In: Nagel, J., Peters, C. (eds.) Algebraic Cycles and Motives. Cambridge University Press, Cambridge (2007)Google Scholar
  14. 14.
    Kimura, S.: Chow groups are finite dimensional, in some sense. Math. Ann. 331, 173–201 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Laterveer, R.: Some results on a conjecture of Voisin for surfaces of geometric genus one. Boll. Unione Mat. Ital. 9(4), 435–452 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Laterveer, R.: Some desultory remarks concerning algebraic cycles and Calabi–Yau threefolds. Rend. Circ. Mat. Palermo 65(2), 333–344 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Laterveer, R.: Algebraic cycles on surfaces with \(p_g=1\) and \(q=2\). Comment. Math. Univ. St. Pauli 65(2), 121–130 (2016)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Laterveer, R.: Algebraic cycles and Todorov surfaces. Kyoto J. Math. 58(3), 493–527 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Laterveer, R.: On Voisin’s conjecture for zero-cycles on hyperkähler varieties. J. Korean Math. Soc. 54(6), 1841–1851 (2017)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Laterveer, R.: Some Calabi–Yau fourfolds verifying Voisin’s conjecture. Ricerche Mat. 67, 401–411 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Laterveer, R.: Zero-cycles on self-products of surfaces: some new examples verifying Voisin’s conjecture. Rend. Circ. Mat. Palermo 2, 1–13 (2018). Google Scholar
  22. 22.
    Laterveer, R., Vial, Ch.: On the Chow ring of Cynk–Hulek Calabi–Yau varieties and Schreieder varieties. arXiv:1712.03070,
  23. 23.
    Murre, J.: On a conjectural filtration on the Chow groups of an algebraic variety, parts I and II. Indag. Math. 4, 177–201 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Murre, J., Nagel, J., Peters, C.: Lectures on the theory of pure motives, Am. Math. Soc. University Lecture Series 61, Providence 2013Google Scholar
  25. 25.
    Pignatelli, R.,: Quotients of the square of a curve by a mixed action, further quotients and Albanese morphisms. arXiv:1708.01750,
  26. 26.
    Pignatelli, R., Polizzi, F.: A family of surfaces with \(p_g = q = 2\), \(K^2 = 7\) and Albanese map of degree 3. Math. Nachr. 290(16), 2684–2695 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rito, C.:New surfaces with \(K^2 = 7\) and \(p_g = q\le 2\). arXiv:1506.09117. (To appear in Asian Journal of Math)
  28. 28.
    Rojtman, A.A.: The torsion of the group of 0-cycles modulo rational equivalence. Ann. Math. 111, 553–569 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Scholl, T.: Classical motives. In: Jannsen, U., et al. (eds.) Motives, Proceedings of Symposia in Pure Mathematics, Part 1, vol. 55 (1994)Google Scholar
  30. 30.
    Shen, M., Vial, Ch.: The Fourier transform for certain hyperKähler fourfolds. Mem. AMS 240, 1139 (2016)zbMATHGoogle Scholar
  31. 31.
    Vial, Ch.: Remarks on motives of Abelian type. Tohoku Math. J. 69(2), 195–220 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Vial, Ch.: Generic cycles, Lefschetz representations and the generalized Hodge and Bloch conjectures for abelian varieties. arXiv:1803.00857v1
  33. 33.
    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 (1994)Google Scholar
  34. 34.
    Voisin, C.: Chow Rings, Decomposition of the Diagonal, and the Topology of Families. Princeton University Press, Princeton (2014)CrossRefzbMATHGoogle Scholar

Copyright information

© Università degli Studi di Ferrara 2019

Authors and Affiliations

  1. 1.Institut de Recherche Mathématique AvancéeCNRS – Université de StrasbourgStrasbourg CedexFrance

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