By chance, Wisconsin had two seminars about cubic fourfolds on consecutive days last week, by means of which I learned much more about cubic fourfolds than I had in my life up to now. Summary follows — based on my understanding of what was said, so all mistakes belong to me and not the speakers.
Radu Laza talked about his work on the moduli space of cubic fourfolds. One way you can study families of varieties is via the period map, which sends a variety X to the Hodge structure on H^i(X), for some degree i. It follows from a 1985 theorem of Claire Voisin that the period map from the space of cubic fourfolds X to Hodge structures on H^4(X) is injective; so the moduli space of cubic fourfolds is a subvariety of the moduli space of Hodge structures of the right shape, which is a nice 20-dimensional ball quotient. Laza goes further, computing the precise image of the period map, and thus giving a very clean description of the moduli space.
Alexander Kuznetsov gave a talk the following day on the rationality of cubic fourfolds. Some cubic fourfolds are rational; for instance, if X is a cubic fourfold containing two skew planes P and Q, then you get a map from P x Q to X sending (p,q) to the third point of intersection of the line pq with X, and this map is birational. The generic cubic fourfold, on the other hand, is conjectured to be non-rational; but I believe that no example of a provably non-rational cubic fourfold is known.
Kuznetsov’s idea is to approach this problem from the viewpoint of the derived category. The derived category of a smooth cubic fourfold has a “semi-orthogonal decomposition” into a bunch of simple pieces, which don’t depend on X, and one interesting piece, a subcategory we call A_X. The category A_X isn’t a birational invariant, but it does behave nicely under basic birational operations — when you blow up a smooth subvariety Z of X (necessarily of dimension 0,1, or 2) you find that A_X simply “picks up a copy of A_Z.” In particular, if X is birational to P^4, A_X must be “made out of” pieces coming from derived categories of varieties of dimension at most 2. Kuznetsov believes this criterion can be used in practice to obstruct the rationality of X.
If this sounds familiar, it’s because it’s explicitly modeled on the Clemens-Griffiths obstruction to rationality of a cubic threefold Y. There, the role of A_X is played by the intermediate Jacobian J(Y); and Clemens and Griffiths prove that if J is not isomorphic to the Jacobian of a curve, Y can’t be rational. A critical role is played here by the semistability of the category of principally polarized abelian varieties; this doesn’t hold for triangulated categories, but Kuznetsov believes that a suitable version of semistability should apply to some class of categories including A_X.
If X is a cubic fourfold, the Fano variety F(X) parametrizing lines in X is again a fourfold, and is deformation equivalent to the Hilbert scheme parametrizing pairs of points on a K3. This fact is omnipresent in Laza’s work, and it shows up for Kuznetsov too: it turns out that in the cases where X is known to be rational (for instance, the infinite families produced by Brendan Hassett in his thesis) the Fano variety is not just a deformation of, but actually is, the Hilbert scheme Hilb^2 S for some K3 surface S. And in this case, Kuznetsov’s A_X is nothing but the derived category D^b(S).
This might lead one to ask whether one could make a conjecture that dispensed with derived categories entirely (though I feel guilty and antique for suggesting that this might be a good thing!) and guess that X is rational exactly when F(X) is Hilb^2 of a K3 surface S. I think would be pretty close to a kind of Hodge-theoretic criterion for rationality, as in the cubic threefold case.
But if I understand correctly, Kuznetsov doesn’t think it can be that simple. There is a divisor in the moduli space of cubic fourfolds which is naturally identified with the moduli space of K3 surfaces of some given degree. Hassett shows that on this 19-dimensional variety there is a countable union of 18-dimensional families of rational cubic fourfolds. Kuznetsov says that if X is a point on this divisor, and S the corresponding K3, that A_X is not the derived category D^b(S), but a twisted derived category D^b(S,alpha) for some Brauer class alpha on S which is generically nonvanishing (but vanishes on Hassett’s locus.) And this twist, he believes, obstructs rationality — though without a good enough “semistability” property for the categories involved, nothing can yet be proved.