On the Derived Category of the Cayley Grassmannian

TL;DR

This paper constructs a full exceptional collection of vector bundles in the derived category of the Cayley Grassmannian, a special subvariety of the Grassmannian, using innovative methods involving quadric bundles.

Contribution

It introduces a full exceptional collection for the derived category of the Cayley Grassmannian, advancing understanding of its geometric and categorical structure.

Findings

Constructed a full exceptional collection of vector bundles.

Developed a novel method involving self-dual vector bundles and quadric bundle operations.

Proved the collection's fullness in the derived category.

Abstract

We construct a full exceptional collection consisting of vector bundles in the derived category of coherent sheaves on the so-called Cayley Grassmannian, the subvariety of the Grassmannian $\mathrm{Gr}(3, 7)$ parameterizing 3-subspaces that are annihilated by a general 4-form. The main step in the proof of fullness is a construction of two self-dual vector bundles which is obtained from two operations with quadric bundles that might be interesting in themselves.