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 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.