Yesterday Sasha Kuznetsov, Maxim Smirnov and I uploaded a preprint to the arXiv, titled Derived categories of the Cayley plane and the coadjoint Grassmannian of type F. In it we discuss the derived categories of two homogeneous varieties (in exceptional type): the Cayley plane $\mathrm{E}_6/\mathrm{P}_1$ and the coadjoint Grassmannian in type F, i.e. $\mathrm{F}_4/\mathrm{P}_4$. Here I am referring to Dynkin types, and maximal parabolic subgroups in the Bourbaki labelling.

The variety $\mathrm{E}_6/\mathrm{P}_1$ is a smooth projective Fano variety of dimension 16 and index 12. The variety $\mathrm{F}_4/\mathrm{P}_4$ is a smooth projective Fano variety of dimension 15 and index 11. Conjecturally, every $G/P$ (of which these are 2 exceptional cases) has a full exceptional collection in its derived category.

In the paper we show that

  1. the residual category of the Faenzi–Manivel Lefschetz collection is generated by 3 completely orthogonal exceptional objects;
  2. $\mathbf{D}^{\mathrm{b}}(\mathrm{F}_4/\mathrm{P}_4)$ has a full Lefschetz collection (so one more case of the conjecture bites the dust!);
  3. the residual category of this Lefschetz collection is equivalent to $\mathbf{D}^{\mathrm{b}}(\mathrm{A}_2)$.

The reason that this list somewhat ridiculously starts at 2 and not 1, is because the obvious first element is

  1. $\mathbf{D}^{\mathrm{b}}(\mathrm{E}_6/\mathrm{P}_1)$ has a full Lefschetz collection, due to Faenzi–Manivel.

The relationship between points 1 and 3 comes from a very convenient coincidence: it is possible to describe the variety $\mathrm{F}_4/\mathrm{P}_4$ as a hyperplane section of $\mathrm{E}_6/\mathrm{P}_1$.

The relationship between 2 and 4 is part of a more general picture: whenever you have a residual category which is generated by a completely orthogonal exceptional collection, if taking a hyperplane does not introduce any new objects in the derived category when restricting the Lefschetz collection, the residual category is a bunch of Dynkin quivers in type A.

All of these statements involve some familiarity with derived categories, and homological projective duality in particular. The notion of Lefschetz collection and residual category especially are not (yet) in the standard toolkit for the working algebraic geometer or representation theorist, but the margin of this book this announcement blogpost is too small to contain a detailed discussion. I will likely revisit the latter topic, which is very exciting, in some other blogposts soon.

Maybe I should retroactively introduce some other recent preprints, as a form of blatant self-promotion.