Küchle fivefolds of type c5

Abstract

We show that Küchle fivefolds of type (c5) --- subvarieties of the Grassmannian 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 LGr(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.

34 pages

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