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.
Categories: math.AG