Hyperkähler manifolds from the Tits–Freudenthal magic square

  • Atanas IlievEmail author
  • Laurent Manivel
Research Article


We suggest a way to associate to each Lie algebra of type \(G_2,D_4,F_4,E_6\), \(E_7,E_8\) a family of polarized hyperkähler fourfolds, constructed as parametrizing certain families of cycles of hyperplane sections of certain homogeneous or quasi-homogeneous varieties. These cycles are modeled on the Legendrian varieties studied by Freudenthal in his geometric approach to the celebrated Tits–Freudenthal magic square of Lie algebras.


Fano varieties Hyperkähler varieties Legendrian varieties Tits–Freudenthal square 

Mathematics Subject Classification

14J28 14J45 14M15 



We thank Alexander Kuznetsov, Dmitri Orlov, Kieran O’Grady, Grzegorz Kapustka and Michał Kapustka for their useful hints and comments.


  1. 1.
    Beauville, A., Donagi, R.: La variété des droites d’une hypersurface cubique de dimension 4. C. R. Acad. Sci. Paris Sér. I Math. 301(14), 703–706 (1985)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bourbaki, N.: Groupes et algèbres de Lie. Hermann, Paris (1954)zbMATHGoogle Scholar
  3. 3.
    Buczyński, J.: Legendrian subvarieties of projective space. Geom. Dedicata 118, 87–103 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cayley, A.: A memoir on quartic surfaces. Proc. London Math. Soc. 3, 19–69 (1869–1871)Google Scholar
  5. 5.
    Chaput, P.E., Perrin, N.: On the quantum cohomology of adjoint varieties. Proc. London Math. Soc. 103(2), 294–330 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Debarre, O., Voisin, C.: Hyper-Kähler fourfolds and Grassmann geometry. J. Reine Angew. Math. 649, 63–87 (2010)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Deligne, P.: La série exceptionnelle de groupes de Lie. C. R. Acad. Sci. Paris Sér. I Math. 322(4), 321–326 (1996)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Deligne, P., Gross, B.H.: On the exceptional series, and its descendants. C. R. Acad. Sci. Paris Sér. I Math. 335, 877–881 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ferretti, A.: Special subvarieties of EPW sextics. Math. Z. 272(3–4), 1137–1164 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Festi, D., Garbagnati, A., van Geemen, B., van Luijk, R.: The Cayley–Oguiso automorphism of positive entropy on a K3 surface. J. Mod. Dyn. 7(1), 75–97 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants, and Multidimensional Determinants. Modern Birkhäuser Classics. Birkhäuser, Birkhäuser, Boston (2008)zbMATHGoogle Scholar
  12. 12.
    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)CrossRefGoogle Scholar
  13. 13.
    Iliev, A., Manivel, L.: Fano manifolds of Calabi–Yau Hodge type. J. Pure Appl. Algebra 219(6), 2225–2244 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Iskovskikh, V.A., Prokhorov, Yu.A.: Fano varieties. In: Parshin, A.N., Shafarevich, I.R. (eds.) Algebraic Geometry V, 1–247. Encyclopaedia of Mathematical Sciences, vol. 47. Springer, Berlin (1999)Google Scholar
  15. 15.
    Landsberg, J.M., Manivel, L.: The projective geometry of Freudenthal’s magic square. J. Algebra 239(2), 477–512 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Landsberg, J.M., Manivel, L.: Triality, exceptional Lie algebras and Deligne dimension formulas. Adv. Math. 171(1), 59–85 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Landsberg, J.M., Manivel, L.: The sextonions and \(E_{7\frac{1}{2}}\). Adv. Math. 201(1), 143–179 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Landsberg, J.M., Manivel, L.: Legendrian varieties. Asian J. Math. 11(3), 341–359 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Snow, D.M.: Homogeneous vector bundles. In: Group Actions and Invariant Theory: Proceedings of the 1988 Montreal Conference, pp. 193–205. American Mathematical Society, Providence, RI (1989)Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsSeoul National UniversitySeoulKorea
  2. 2.Institut de Mathématiques de Toulouse, UMR 5219Université de Toulouse, CNRS, UPS IMTToulouse Cedex 9France

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