Skip to main content
SpringerLink
Log in

Küchle fivefolds of type c5

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

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.

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. Debarre, O., Kuznetsov, A.: Gushel–Mukai varieties: classification and birationalities, arXiv preprint arXiv:1510.05448 (2015)

  2. Harvey, F.R.: Spinors and Calibrations, Perspectives in Mathematics 9. Academic Press, New York (1990)

  3. Hassett, B.: Special cubic fourfolds. Compositio Math. 120(1), 1–23 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Iliev, A., Manivel, L.: Fano manifolds of Calabi-Yau Hodge type. J. Pure Appl. Algebra 219(6), 2225–2244 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kapustka, M., Ranestad, K.: Vector bundles on Fano varieties of genus ten. Math. Ann. 356(2), 439–467 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Küchle, O.: On Fano 4-fold of index 1 and homogeneous vector bundles over Grassmannians. Math. Z. 218(4), 563–575 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kuznetsov, A.G.: Hyperplane sections and derived categories. Izv. Ross. Akad. Nauk Ser. Mat 70(3), 23–128 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  11. Kuznetsov, A.: Calabi–Yau and fractional Calabi–Yau categories, arXiv preprint arXiv:1509.07657 (2015)

  12. Kuznetsov, A.: Derived categories view on rationality problems, arXiv preprint arXiv:1509.09115 (2015)

  13. Kuznetsov, A.: On Küchle varieties with Picard number greater than 1. Izv. Ross. Akad. Nauk Ser. Mat. 79(4), 57–70 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kuznetsov, A., Perry, A.: Derived categories of Gushel–Mukai varieties, arXiv preprint arXiv:1605.06568 (2016)

  15. Landsberg, J.M., Manivel, L.: The projective geometry of Freudenthal’s magic square. J. Algebra 239(2), 477–512 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Manivel, L.: On Fano manifolds of Picard number one. Math. Z. 281(3–4), 1129–1135 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Marcolli, M., Tabuada, G.: From exceptional collections to motivic decompositions via noncommutative motives. J. Reine Angew. Math. 701, 153–167 (2015)

    MathSciNet  MATH  Google Scholar 

  18. Morin, U.: Sulla razionalità dell’ipersuperficie cubica generale dello spazio lineare \(S_5\). Rend. Sem. Mat. Univ. Padova 11, 108–112 (1940)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

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.

Author information

Authors and Affiliations

  1. Algebraic Geometry Section, Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkin str, Moscow, 119991, Russia

    Alexander Kuznetsov

Authors
  1. Alexander Kuznetsov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Kuznetsov.

Additional information

This work is supported by the Russian Science Foundation under grant 14-50-00005.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kuznetsov, A. Küchle fivefolds of type c5. Math. Z. 284, 1245–1278 (2016). https://doi.org/10.1007/s00209-016-1707-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-016-1707-9

Keywords

Search

Navigation