On the derived category of the adjoint Grassmannian of type F

TL;DR

This paper constructs the first full exceptional collection in the derived category of the adjoint Grassmannian of type F4, linking it to quantum cohomology and advancing understanding of derived categories for non-simply laced groups.

Contribution

It provides the first full exceptional collection for the adjoint Grassmannian of type F4, confirming a conjecture relating derived categories to quantum cohomology.

Findings

Established a full rectangular Lefschetz collection in the derived category.

First example of a full exceptional collection on this variety.

Confirmed a conjecture connecting derived categories and quantum cohomology.

Abstract

We construct a full rectangular Lefschetz collection in the derived category of the adjoint Grassmannian in type $\mathrm{F}_4$. This gives the first example of a full exceptional collection on this variety and also completes the proof of a conjecture due to Alexander Kuznetsov and the author that relates the structure of the derived category of coherent sheaves to the small quantum cohomology in the case of adjoint varieties for non-simply laced groups.