On the motive of codimension 2 linear sections of $\hbox{Gr}(3,6)$

TL;DR

This paper studies Fano sevenfolds formed by intersecting the Grassmannian Gr(3,6) with a codimension 2 linear space, proving their motive is finite-dimensional and confirming the generalized Hodge conjecture for their powers.

Contribution

It establishes the finite-dimensionality of the motive and verifies the generalized Hodge conjecture for these specific Fano sevenfolds.

Findings

Motive of the Fano sevenfolds is Kimura finite-dimensional.

Generalized Hodge conjecture holds for all powers of these sevenfolds.

Provides new insights into the geometry of codimension 2 linear sections of Grassmannians.

Abstract

We consider Fano sevenfolds $Y$ obtained by intersecting the Grassmannian $\hbox{Gr}(3,6)$ 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 $Y$.