On the Motive of Codimension 2 Linear Sections of $\mathrm{Gr}(3,6)$

Abstract

We consider Fano sevenfolds $Y$ obtained by intersecting the Grassmannian $\mathrm{Gr}(3,6)$ with a codimension 2 linear subspace (with respect to the Pl\"ucker embedding). We prove that the motive of $Y$ is Kimura finite-dimensional. We also prove the generalized Hodge conjecture for all powers of $Y$.