• Yu-Hsiang Liu: Donaldson-Thomas theory for quantum Fermat quintic threefolds is an interesting preprint I would like to read more carefully. Donaldson–Thomas invariants can be defined for CY3 categories, and a quintic hypersurface in a noncommutative $\mathbb{P}^4$ of a specific type gives rise to such a category which is not of purely geometric origin.

    An interesting feature of DT theory is that it should be deformation-invariant. But these CY3 categories are not the deformation of the derived category of a quintic 3-fold. Hence the invariants one obtains are distinct from the ones in the commutative setting, as

    R.<t> = PowerSeriesRing(ZZ)
    chi = -200
    print prod([1 / (1 - (-t)^n)^n for n in range(1, 20)])^chi

    suggests that in the commutative case series from corollary 6.13 is (if I didn't misinterpret things, please tell me if I did!) \begin{equation*} \sum_{n=0}^4\operatorname{DT}^n(Y)t^n = 1 + 200t + 19500t^2 + 1234000t^3 + 56923950t^4 \end{equation*}

    I was hoping to see 2875 here, but maybe this is a different setting?

  • Anton Fonarev: Full exceptional collections on Lagrangian Grassmannians shows that the exceptional collection of expected length in $\mathbf{D}^{\mathrm{b}}(\operatorname{LGr}(n,2n))$ constructed by Kuznetsov–Polishchuk is a full exceptional collection. Here $\operatorname{LGr}(n,2n)$ is the Lagrangian Grassmannian, the isotropic Grassmannian associated to the Dynkin diagram of type $\mathrm{C}_n$ and the maximal parabolic at the "special" vertex. Hence another family bites the dust!

    I think a good way to waste my time would be to make an overview table of all the known results, saying which partial flag varieties have

    • a (known) full exceptional collection
    • a (known) Lefschetz decomposition (which is stronger, and more useful)
    • a conjectural exceptional collection

    Let me know if you have any suggestions!

  • Alexander Kuznetsov, Yuri Prokhorov: Rationality of Fano threefolds over non-closed fields discusses a complete picture for (uni)rationality for forms of Fano 3-folds of Picard rank 1. Aside from the obvious geometric interest, an important aspect is the structure of the derived category of such varieties. For instance, they prove that a rationally connected threefold with a semiorthogonal decomposition containing only exceptional objects and derived categories of a curve has an intermediate Jacobian which is isomorphic (and not merely isogenous) to the Jacobian of said curve. Cool!