Motives and the Pfaffian-Grassmannian equivalence

TL;DR

This paper explores the motivic aspects of the Pfaffian-Grassmannian equivalence, establishing relationships at the level of Chow motives and verifying several conjectures for specific Calabi-Yau threefolds and linear sections.

Contribution

It provides the first motivic perspective on the Pfaffian-Grassmannian equivalence and verifies multiple conjectures for new classes of varieties.

Findings

Verification of Orlov's conjecture for Borisov's Calabi-Yau threefolds

Verification of Kimura's finite-dimensionality conjecture

Examples of Fano varieties with infinite-dimensional Griffiths group

Abstract

We consider the Pfaffian-Grassmannian equivalence from the motivic point of view. The main result is that under certain numerical conditions, both sides of the equivalence are related on the level of Chow motives. The consequences include a verification of Orlov's conjecture for Borisov's Calabi-Yau threefolds, and verifications of Kimura's finite-dimensionality conjecture, Voevodsky's smash conjecture and the Hodge conjecture for certain linear sections of Grassmannians. We also obtain new examples of Fano varieties with infinite-dimensional Griffiths group.