Birational Invariants and Decomposition of the Diagonal

  • Claire VoisinEmail author
Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 26)


We give a rather detailed account of cohomological and Chow-theoretic methods in the study of the stable version of the Lroth problem, which ask how to distinguish (stably) rational varieties from general unirational varieties. In particular, we study the notion of Chow or cohomological decomposition of the diagonal, which is a necessary criterion for stable rationality. Having better stability properties than the previously known obstructions under specialization with mildly singular central fibers, it has been very useful in the recent study of rationality questions.


  1. 1.
    N. Addington, On two rationality conjectures for cubic fourfolds. Math. Res. Lett. 23(1), 1–13 (2016)MathSciNetzbMATHGoogle Scholar
  2. 2.
    N. Addington, R. Thomas, Hodge theory and derived categories of cubic fourfolds. Duke Math. J. 163(10), 1885–1927 (2014)MathSciNetzbMATHGoogle Scholar
  3. 3.
    M. Artin, D. Mumford, Some elementary examples of unirational varieties which are not rational. Proc. Lond. Math. Soc. 25(3), 75–95 (1972)MathSciNetzbMATHGoogle Scholar
  4. 4.
    M.F. Atiyah, F. Hirzebruch, Analytic cycles on complex manifolds. Topology 1, 25–45 (1962)MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. Auel, J.-L. Colliot-Thélène, R. Parimala, Universal unramified cohomology of cubic fourfolds containing a plane, in Brauer Groups and Obstruction Problems. Progress in Mathematics, vol. 320 (Birkhäuser/Springer, Cham, 2017), pp. 29–55Google Scholar
  6. 6.
    L. Barbieri-Viale, Cicli di codimensione 2 su varietà unirazionali complesse, in K-Theory (Strasbourg 1992). Astérisque, vol. 226 (Société mathématique de France, Paris, 1994), pp. 13–41Google Scholar
  7. 7.
    L. Barbieri-Viale, On the Deligne-Beilinson cohomology sheaves. Ann. K-Theory 1(1), 3–17 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    A. Beauville, Variétés de Prym et jacobiennes intermédiaires. Ann. Sci. Éc. Norm. Sup. 10, 309–391 (1977)zbMATHGoogle Scholar
  9. 9.
    A. Beauville, The Lüroth problem, in Rationality Problems in Algebraic Geometry. Springer Lecture Notes, vol. 2172 (Springer, Berlin, 2016), pp. 1–27Google Scholar
  10. 10.
    A. Beauville, R. Donagi, La variété des droites d’une hypersurface cubique de dimension 4. C. R. Acad. Sci. Paris Sér. I Math. 301, 703–706 (1985)MathSciNetzbMATHGoogle Scholar
  11. 11.
    S. Bloch, A. Ogus, Gersten’s conjecture and the homology of schemes. Ann. Sci. Éc. Norm. Supér. IV. Sér. 7, 181–201 (1974)MathSciNetzbMATHGoogle Scholar
  12. 12.
    S. Bloch, V. Srinivas, Remarks on correspondences and algebraic cycles. Am. J. Math. 105, 1235–1253 (1983)MathSciNetzbMATHGoogle Scholar
  13. 13.
    L. Borisov, The class of the affine line is a zero divisor in the Grothendieck ring. J. Algebraic Geom. 27, 203–209 (2018)MathSciNetzbMATHGoogle Scholar
  14. 14.
    H. Clemens, P. Griffiths, The intermediate Jacobian of the cubic threefold. Ann. Math. Second Ser. 95(2), 281–356 (1972)MathSciNetzbMATHGoogle Scholar
  15. 15.
    J.-L. Colliot-Thélène, Quelques cas d’annulation du troisième groupe de cohomologie non ramifiée, in Regulators. Contemporary Mathematics, vol. 571 (American Mathematical Society, Providence, 2012), pp. 45–50Google Scholar
  16. 16.
    J.-L. Colliot-Thélène, M. Ojanguren, Variétés unirationnelles non rationnelles: au-delà de l’exemple d’Artin et Mumford. Invent. Math. 97(1), 141–158 (1989)MathSciNetzbMATHGoogle Scholar
  17. 17.
    J.-L. Colliot-Thélène, A. Pirutka, Hypersurfaces quartiques de dimension 3: non-rationalité stable. Ann. Sci. Éc. Norm. Supér. 49(2), 371–397 (2016)MathSciNetzbMATHGoogle Scholar
  18. 18.
    J.-L. Colliot-Thélène, C. Voisin, Cohomologie non ramifiée et conjecture de Hodge entière. Duke Math. J. 161(5), 735–801 (2012)MathSciNetzbMATHGoogle Scholar
  19. 19.
    A. Conte, J. Murre, The Hodge conjecture for fourfolds admitting a covering by rational curves. Math. Ann. 238(1), 79–88 (1978)MathSciNetzbMATHGoogle Scholar
  20. 20.
    S. Endrass, On the divisor class group of double solids. Manuscripta Math. 99, 341–358 (1999)MathSciNetzbMATHGoogle Scholar
  21. 21.
    W. Fulton, Intersection Theory: Ergebnisse der Math. und ihrer Grenzgebiete 3 Folge, Band 2 (Springer, Berlin, 1984)Google Scholar
  22. 22.
    S. Galkin, E. Shinder, The Fano variety of lines and rationality problem for a cubic hypersurface (2014). arXiv:1405.5154Google Scholar
  23. 23.
    T. Graber, J. Harris, J. Starr, Families of rationally connected varieties. J. Am. Math. Soc. 16(1), 57–67 (2003)MathSciNetzbMATHGoogle Scholar
  24. 24.
    P. Griffiths, On the periods of certain rational integrals I, II. Ann. Math. 90, 460–495 (1969); ibid. (2) 90 (1969) 496–541Google Scholar
  25. 25.
    B. Hassett, Special cubic fourfolds. Compos. Math. 120(1), 1–23 (2000)MathSciNetzbMATHGoogle Scholar
  26. 26.
    B. Hassett, A. Pirutka, Y. Tschinkel, Stable rationality of quadric surface bundles over surfaces. Acta Math. 220(2), 341–365 (2018)MathSciNetzbMATHGoogle Scholar
  27. 27.
    A. Höring, C. Voisin, Anticanonical divisors and curve classes on Fano manifolds. Pure Appl. Math. Quart. 7(4), 1371–1393 (2011)MathSciNetzbMATHGoogle Scholar
  28. 28.
    V.A. Iskovskikh, Y. Manin, Three-dimensional quartics and counterexamples to the Lüroth problem. Mat. Sb. (N.S.) 86(128), 140–166 (1971)Google Scholar
  29. 29.
    B. Kahn, Torsion order of smooth projective surfaces. Comment. Math. Helv. 92, 839–857 (2017)MathSciNetzbMATHGoogle Scholar
  30. 30.
    M. Kerz, The Gersten conjecture for Milnor K-theory. Invent. Math. 175(1), 1–33 (2009)MathSciNetzbMATHGoogle Scholar
  31. 31.
    J. Kollár, Nonrational hypersurfaces. J. Am. Math. Soc. 8(1995), 241–249 (1990)MathSciNetzbMATHGoogle Scholar
  32. 32.
    J. Kollár, Lemma, in Classification of Irregular Varieties, ed. by E. Ballico, F. Catanese, C. Ciliberto. Lecture Notes in Mathematics, vol. 1515 (Springer, Berlin, 1992), p. 134Google Scholar
  33. 33.
    A. Kuznetsov, Derived categories of cubic fourfolds, in Cohomological and Geometric Approaches to Rationality Problems. Progress in Mathematics, vol. 282 (Birkhäuser, Boston, 2010)Google Scholar
  34. 34.
    A. Kuznetsov, Scheme of lines on a family of 2-dimensional quadrics: geometry and derived category. Math. Z. 276(3–4), 655–672 (2014)MathSciNetzbMATHGoogle Scholar
  35. 35.
    M. Larsen, V. Lunts, Motivic measures and stable birational geometry. Mosc. Math. J. 3(1), 85–95 (2003)MathSciNetzbMATHGoogle Scholar
  36. 36.
    D. Markushevich, A. Tikhomirov, The Abel–Jacobi map of a moduli component of vector bundles on the cubic threefold. J. Algebraic Geometry 10, 37–62 (2001)MathSciNetzbMATHGoogle Scholar
  37. 37.
    T. Matsusaka, On a characterization of a Jacobian variety. Memo. Coll. Sci. Univ. Kyoto. Ser. A. Math. 32, 1–19 (1959)MathSciNetzbMATHGoogle Scholar
  38. 38.
    T. Matsusaka, Algebraic deformations of polarized varieties. Nagoya Math. J. 31, 185–245 (1968)MathSciNetzbMATHGoogle Scholar
  39. 39.
    A. Merkur’ev, A. Suslin, K-cohomology of Severi-Brauer varieties and the norm residue homomorphism. Izv. Akad. Nauk SSSR Ser. Mat. 46(5), 1011–1046, 1135–1136 (1982)Google Scholar
  40. 40.
    G. Mongardi, J. Ottem, Curve classes on irreducible holomorphic symplectic varieties (2018). arXiv:1806.09598Google Scholar
  41. 41.
    D. Mumford, Rational equivalence of 0-cycles on surfaces. J. Math. Kyoto Univ. 9, 195–204 (1968)MathSciNetzbMATHGoogle Scholar
  42. 42.
    J.P. Murre, Reduction of the proof of the non-rationality of a non-singular cubic threefold to a result of Mumford. Compos. Math. 27, 63–82 (1973)MathSciNetzbMATHGoogle Scholar
  43. 43.
    J.P. Murre, Applications of algebraic K-theory to the theory of algebraic cycles, in Proceedings Conference on Algebraic Geometry, Sitjes 1983. Lecture Notes in Mathematics, vol. 1124 (Springer, Berlin, 1985), pp. 216–261Google Scholar
  44. 44.
    A. Roitman, The torsion of the group of zero-cycles modulo rational equivalence. Ann. Math. 111, 553–569 (1980)MathSciNetGoogle Scholar
  45. 45.
    S. Schreieder, Stably irrational hypersurfaces of small slopes (2018). arXiv:1801.05397Google Scholar
  46. 46.
    S. Schreieder, On the rationality problem for quadric bundles. Duke Math. J. 168, 187–223 (2019)MathSciNetzbMATHGoogle Scholar
  47. 47.
    J.-P. Serre, On the fundamental group of a unirational variety. J. Lond. Math. Soc. 34, 481–484 (1959)MathSciNetzbMATHGoogle Scholar
  48. 48.
    C. Soulé, C. Voisin, Torsion cohomology classes and algebraic cycles on complex projective manifolds. Adv. Math. 198(1), 107–127 (2005)MathSciNetzbMATHGoogle Scholar
  49. 49.
    T.A. Springer, Sur les formes quadratiques d’indice zéro. C. R. Acad. Sci. Paris 234, 1517–1519 (1952)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Z. Tian, R. Zong, One-cycles on rationally connected varieties. Compos. Math. 150(3), 396–408 (2014)MathSciNetzbMATHGoogle Scholar
  51. 51.
    B. Totaro, Hypersurfaces that are not stably rational. J. Am. Math. Soc. 29(3), 883–891 (2016)MathSciNetzbMATHGoogle Scholar
  52. 52.
    V. Voevodsky, A nilpotence theorem for cycles algebraically equivalent to zero. Int. Math. Res. Not. 4, 187–198 (1995)MathSciNetzbMATHGoogle Scholar
  53. 53.
    V. Voevodsky, Motivic cohomology with \(\mathbb {Z}/l\)-coefficients. Ann. Math. 174, 401–438 (2011)Google Scholar
  54. 54.
    C. Voisin, Remarks on zero-cycles of self-products of varieties, in Moduli of Vector Bundles (Proceedings of the Taniguchi Congress on Vector Bundles), ed. by M.E. Decker (1994), pp. 265–285Google Scholar
  55. 55.
    C. Voisin, Hodge Theory and Complex Algebraic Geometry II. Cambridge Studies in Advanced Mathematics, vol. 77 (Cambridge University Press, Cambridge, 2003)Google Scholar
  56. 56.
    C. Voisin, On integral Hodge classes on uniruled and Calabi-Yau threefolds, in Moduli Spaces and Arithmetic Geometry. Advanced Studies in Pure Mathematics, vol. 45 (2006), pp. 43–73Google Scholar
  57. 57.
    C. Voisin, Some aspects of the Hodge conjecture. Jpn. J. Math. 2(2), 261–296 (2007)MathSciNetzbMATHGoogle Scholar
  58. 58.
    C. Voisin, Abel–Jacobi map, integral Hodge classes and decomposition of the diagonal. J. Algebraic Geom. 22, 141–174 (2013)MathSciNetzbMATHGoogle Scholar
  59. 59.
    C. Voisin, Remarks on curve classes on rationally connected varieties. Clay Math. Proc. 18, 591–599 (2013)MathSciNetzbMATHGoogle Scholar
  60. 60.
    C. Voisin, Unirational threefolds with no universal codimension 2 cycle. Invent. Math. 201(1), 207–237 (2015)MathSciNetzbMATHGoogle Scholar
  61. 61.
    C. Voisin, On the universal CH 0 group of cubic hypersurfaces. J. Eur. Math. Soc. 19(6), 1619–1653 (2017)MathSciNetzbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Collège de FranceParisFrance

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