Homological projective duality for Grassmannians of lines

TL;DR

This paper explores the homological projective duality for Grassmannians, revealing noncommutative resolutions of Pfaffian varieties and analyzing derived categories of their linear sections, including implications for cubic 4-folds.

Contribution

It establishes the homological projective duality for Grassmannians of lines and describes the derived categories of their linear sections, including Pfaffian varieties and cubic 4-folds.

Findings

Derived category of Pfaffian cubic 4-fold has a semiorthogonal decomposition with 3 line bundles and a K3-surface.

Mutually orthogonal Calabi-Yau linear sections of Gr(2,7) and Pfaffian varieties are derived equivalent.

Conjecture on rationality criterion for cubic 4-folds based on 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.