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Mathematische Zeitschrift

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

Küchle fivefolds of type c5

  • Alexander Kuznetsov
Article

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.

Notes

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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

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

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