Mathematische Zeitschrift

, Volume 284, Issue 3–4, pp 1245–1278 | Cite as

Küchle fivefolds of type c5

  • Alexander KuznetsovEmail author


We show that Küchle fivefolds of type (c5)—subvarieties of the Grassmannian \({{\mathrm{\mathsf {Gr}}}}(3,7)\) parameterizing 3-subspaces that are isotropic for a given 2-form and are annihilated by a given 4-form—are birational to hyperplane sections of the Lagrangian Grassmannian \({{\mathrm{\mathsf {SGr}}}}(3,6)\) and describe in detail these birational transformations. As an application, we show that the integral Chow motive of a Küchle fivefold of type (c5) is of Lefschetz type. We also discuss Küchle fourfolds of type (c5)—hyperplane sections of the corresponding Küchle fivefolds—an interesting class of Fano fourfolds, which is expected to be similar to the class of cubic fourfolds in many aspects.



I am very grateful to Atanas Iliev, Grzegorz and Michał Kapustka, Laurent Manivel, Dmitri Orlov, and Kristian Ranestad for useful discussions. I would also like to thank the referee for his comments.


  1. 1.
    Debarre, O., Kuznetsov, A.: Gushel–Mukai varieties: classification and birationalities, arXiv preprint arXiv:1510.05448 (2015)
  2. 2.
    Harvey, F.R.: Spinors and Calibrations, Perspectives in Mathematics 9. Academic Press, New York (1990)Google Scholar
  3. 3.
    Hassett, B.: Special cubic fourfolds. Compositio Math. 120(1), 1–23 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Iliev, A., Ranestad, K.: Geometry of the Lagrangian Grassmannian LG(3,6) with applications to Brill-Noether loci. Michigan Math. J 53(2), 383–417 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Iliev, A., Manivel, L.: Fano manifolds of Calabi-Yau Hodge type. J. Pure Appl. Algebra 219(6), 2225–2244 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kapustka, M., Ranestad, K.: Vector bundles on Fano varieties of genus ten. Math. Ann. 356(2), 439–467 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Küchle, O.: On Fano 4-fold of index 1 and homogeneous vector bundles over Grassmannians. Math. Z. 218(4), 563–575 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kuznetsov, A.G.: Hyperplane sections and derived categories. Izv. Ross. Akad. Nauk Ser. Mat 70(3), 23–128 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kuznetsov, A.: Derived categories of cubic fourfolds. In: Cohomological and Geometric Approaches to Rationality Problems, Progr. Math., vol. 282, pp. 219–243. Birkhäuser Boston, Inc., Boston (2010)Google Scholar
  10. 10.
    Kuznetsov, A.: Semiorthogonal decompositions in algebraic geometry. In: Proceedings of the International Congress of Mathematicians: Seoul 2014 (S.Y. Jang, Y.R. Kim, D.W. Lee, and I. Yie, eds.), vol. 2, pp. 635–660. Kyung Moon (2014)Google Scholar
  11. 11.
    Kuznetsov, A.: Calabi–Yau and fractional Calabi–Yau categories, arXiv preprint arXiv:1509.07657 (2015)
  12. 12.
    Kuznetsov, A.: Derived categories view on rationality problems, arXiv preprint arXiv:1509.09115 (2015)
  13. 13.
    Kuznetsov, A.: On Küchle varieties with Picard number greater than 1. Izv. Ross. Akad. Nauk Ser. Mat. 79(4), 57–70 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kuznetsov, A., Perry, A.: Derived categories of Gushel–Mukai varieties, arXiv preprint arXiv:1605.06568 (2016)
  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.
    Manivel, L.: On Fano manifolds of Picard number one. Math. Z. 281(3–4), 1129–1135 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Marcolli, M., Tabuada, G.: From exceptional collections to motivic decompositions via noncommutative motives. J. Reine Angew. Math. 701, 153–167 (2015)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Morin, U.: Sulla razionalità dell’ipersuperficie cubica generale dello spazio lineare \(S_5\). Rend. Sem. Mat. Univ. Padova 11, 108–112 (1940)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Mukai, S.: Biregular classification of Fano 3-folds and Fano manifolds of coindex 3. Proc. Nat. Acad. Sci. USA 86(9), 3000–3002 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Samokhin, A.: On the derived category of coherent sheaves on a 5-dimensional Fano variety. C. R. Math. Acad. Sci. Paris 340(12), 889–893 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Algebraic Geometry SectionSteklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

Personalised recommendations