K\"uchle fivefolds of type c5

TL;DR

This paper demonstrates that K"uchle fivefolds of type (c5), defined as specific subvarieties of Grassmannians, are birational to hyperplane sections of Lagrangian Grassmannians, revealing their Chow motives and exploring related Fano fourfolds.

Contribution

It establishes a birational equivalence between K"uchle fivefolds of type (c5) and hyperplane sections of Lagrangian Grassmannians, and analyzes their Chow motives and related Fano fourfolds.

Findings

K"uchle fivefolds of type (c5) are birational to hyperplane sections of LGr(3,6).

The integral Chow motive of these fivefolds is of Lefschetz type.

K"uchle fourfolds of type (c5) are hyperplane sections of the fivefolds, similar to cubic fourfolds.

Abstract

We show that K\"uchle 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 K\"uchle fivefold of type (c5) is of Lefschetz type. We also discuss K\"uchle fourfolds of type (c5) --- hyperplane sections of the corresponding K\"uchle fivefolds --- an interesting class of Fano fourfolds, which is expected to be similar to the class of cubic fourfolds in many aspects.