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.

Dear Jordan,

I don’t remember if you were in Joe Harris’s “Open problems in algebraic geoemtry class” in our first semester of grad school, but if you were, then you might remember that he discused rationality of cubic fourfolds in class. He had the philosophy that every time you can prove a cubic fourfold is rational, a geometric construction involving a K3 enters the picture in some way. He then went on to make the following conjecture:

Just as the moduli space of analytic K3s is 20 dim’l, with a countable family of 19 dim’l analytic subvarieties parametrizing the algebraic K3s, so similarly the 20 dim’l space of cubic fourfolds contains a countable union of 19 dim’l subvarieties that parametrize

the rational cubic fourfolds.

I don’t know if he still believes this, or how it compares with the connections with

K3s that you mentioned above. (But maybe Brendan’s countably many 18 dim’l families are the intersection of Joe’s (conjectural) countably many 19 dim’l families with the

given 19 dim’l divisor that you mention.)

Anyway, an interesting and stimulating post!

Best wishes,

Matt

I am pretty sure Joe (and most experts) do still believe the conjecture Matt described. At one moment there was some feeling that the following conjecture would hold true: for a very general cubic fourfold X, the primitive weight 4 Hodge structure of X is not isomorphic to a sub-Hodge structure of any surface. This would certainly imply X is not rational, by considering how Hodge structures change under blowing up. But Izadi proved that, in fact, the primitive weight 4 Hodge structure does occur in the weight 2 Hodge structure of a surface. Does anybody know if the triangulated category A_X which Kuznetsov considers “occurs” in the derived category of a surface (for a very general cubic fourfold X)?

After reading the MathSciNet review of Izadi’s paper, I think the result I mentioned might have been originally proved by Voisin.

Does anybody know if the triangulated category A_X which Kuznetsov considers “occurs” in the derived category of a surface (for a very general cubic fourfold X)?I think this is precisely what Kuznetsov expects, but cannot yet prove, does

nothappen. Part of the problem is that it’s somewhat tricky to define “occurs” in this context.The theorem of Izadi (or Voisin) that you mention seems to fit well into (what I take to be) Kuznetsov’s point of view — that you might have X where the Hodge structure H^4(X) occurs in (a shift of) the Hodge structure on H^2(surface), but where A_X doesn’t “occur” in the derived category of that surface.