Homological projective duality for Grassmannians of lines
Alexander Kuznetsov
TL;DR
The paper proves that the Grassmannians ${\mathsf{Gr}}(2,6)$ and ${\mathsf{Gr}}(2,7)$ have Homological Projective Duals given by certain noncommutative resolutions of Pfaffian varieties ${\mathsf{Pf}}(4,W^*)$. It constructs the noncommutative resolution $(Y,{\mathcal{R}})$ and develops the required Lefschetz and dual Lefschetz decompositions to realize HPD, including a detailed kernel construction that yields the HPD functor. As applications, it derives semiorthogonal decompositions for linear sections of the Grassmannians and their Pfaffian duals; in particular, it isolates a Pfaffian cubic 4-fold with a K3 residual category and proves derived equivalences between mutually orthogonal Calabi–Yau linear sections. The work further conjectures a derived-categorical criterion for the rationality of cubic 4-folds and outlines extensions to larger $\dim W$ and generalized Pfaffian varieties. Overall, the paper provides a concrete HPD framework linking classical dual varieties to noncommutative resolutions and illuminates rich interplays between Calabi–Yau, K3, and Pfaffian geometries via derived categories.
Abstract
We show that homologically projectively dual varieties for Grassmannians Gr(2,6) and Gr(2,7) are given by certain noncommutative resolutions of singularities of the corresponding Pfaffian varieties. As an application we describe the derived categories of linear sections of these Grassmannians and Pfaffians. In particular, we show that (1) the derived category of a Pfaffian cubic 4-fold admits a semiorthogonal decompositions consisting of 3 exceptional line bundles, and of the derived category of a K3-surface; (2) mutually orthogonal Calabi-Yau linear sections of Gr(2,7) and of the corresponding Pfaffian variety are derived equivalent. We also conjecture a rationality criterion for cubic 4-folds in terms of their derived categories.