An episode of GAGLeS organized by Raymond Cheng during the fall of 2018
Cubic fourfolds, i.e. nonsingular cubic hypersurface in projective five space, pose one of the most tantalizing challenges to the modern algebraic geometer: show that the very general cubic fourfold is not rational. Maybe somewhat embarrassingly, there is not a single known example of a cubic fourfold which is provably not rational!
This episode of GAGLeS will be focussed around cubic fourfolds and, to a large extent, the techniques that have been developed to address their rationality. Along the way, we will learn something about Hodge theory, period maps, K3 surfaces, hyperkähler manifolds, and derived categories.
Below is a skewed and non-exhaustive collection of papers touching on the geometry, moduli, and rationality of cubic fourfolds. This will hopefully serve as a guide to the topics discussed in the seminar.
Cubic fourfolds are classical and surely would have been studied by the Italians. Perhaps one of the most noteworthy classical studies on cubic fourfolds was done by Fano, who, at some point, studied the variety of lines on such hypersurfaces and who exhibited special cubic fourfolds containing quartic scrolls and which are rational. The MathSciNet review of the following paper is delightfully alien and at least makes me wonder what algebraic geometry looked back then.
A modern classic in the theory of cubic fourfolds is one by Arnaud Beauville and Ron Donagi on the Fano variety of lines on a cubic fourfold. They show that the Fano variety is a hyperkähler manifold, deformation equivalent to the Hilbert square of a K3 surface. Beauville–Donagi achieve this by explicitly constructing a K3 surface whose Hilbert square is the Fano variety of lines of a Pfaffian cubic fourfold, and the rest is deformation theory. This paper perhaps marks the beginning of a long story in so much modern algebraic geometry.
Cubic fourfolds, being hypersurfaces in a projective space, admit a simple moduli space given as the set of cubic forms in 6 variables with non-vanishing discriminant. Perhaps a more interesting way to parameterize cubic fourfolds, and one which is more amenable to the construction of a compact moduli space, is via period mappings and Hodge theory. Quite remarkably, as Claire Voisin established in her thesis, a global Torelli theorem holds for cubic fourfolds, perhaps giving the first indication of the importance and power of Hodge theory in the study of cubic fourfolds. The geometry of the period domain and the properties of the period map have since been clarified by Radu Laza, Eduard Looijenga, and others.
There are now several different proofs of the Torelli theorem for cubic fourfolds. Fraçois Charles has given a short proof based on Verbitsky’s Torelli theorem for hyperkähler manifolds; Daniel Huybrechts and Jørgen Rennemo have given a proof by using the derived Torelli theorem for K3 surfaces; and Bayer et al. have given a variation of the Huybrechts–Rennemo proof using their work on stability conditions on the Kuznetsov component of a cubic fourfold.
Toward something more familiar, I think Brendan Hassett is one of the first modern algebraic geometers who systematically studied the rationality problem for cubic fourfolds. In his thesis, Hassett used Hodge theoretic methods to construct certain divisors in the moduli of cubic fourfolds. These divisors of special cubic fourfolds contain an unexpected algebraic cycle and are expected to contain all rational examples. An important construction made here is that of the K3 associated with a cubic fourfold; this K3, or rather its primitive Hodge structure, enters as the orthogonal complement of the extra algebraic class in the primitive cohomology of the cubic fourfold.
For a more recent and up-to-date account of these ideas, see also Hassett’s survey:
A more recent perspective comes from a derived category point of view. Specifically, Alexander Kuznetsov, partly inspired by the classical result of Clemens–Griffiths on the non-rationality of a cubic threefold, has constructed a derived category invariant which may be a new obstruction to rationality. In the case of a cubic fourfold, this invariant looks like the derived category of a K3 surface and Kuznetsov conjectures that a cubic fourfold is rational precisely when this category can be realized as the derived category of an honest K3 surface.
A very readable guide to these ideas is Kuznetsov’s survey article below. For a further reference for derived category stuff, K3 categories, and relations to rationality problems, see also the notes by Emanuele Macrì and Paolo Stellari. Also, notes by Asher Auel and Marcello Bernardara give a good account of how Kuznetsov’s approach to rationality fits into the larger picture of things, and how it is, for instance, related to the Clemens–Griffiths result.
The Hodge theoretic viewpoint and the derived category viewpoint are closely related. Nicolas Addington and Richard Thomas seem to be the first to really work out the precise relationship. Daniel Huybrechts has since then expanded on this relation and have proved some basic structure results for the Kuznetsov component of a cubic fourfold.
One more approach to the rationality problem for cubic problems is a baby motivic one due to Sergey Galkin and Evgeny Shinder. Their approach is to find a relation between the class of a cubic fourfold and its associated Fano variety of lines in the Grothendieck ring of varieties. This relation together with the hypothesis that a certain element in the Grothendieck ring is not a zero divisor implies that a very general cubic fourfold is not rational. Unfortunately, the non-zero divisor hypothesis seems likely to be false in view of Lev Borisov’s example which shows that the Lefschetz motive is actually a zero divisor in the Grothendieck ring. Nonetheless, the work of Galkin–Shinder suggest some further conditions for when a cubic fourfold might be rational through geometric properties of the associated Fano variety of lines, a direction further clarified by Nicolas Addington.
Finally, a helpful guide to the relationships between cubic fourfolds and K3 surfaces is contained in a set of slides by Daniel Huybrechts:
For a more up-to-date and more polished reference, see also Daniel Huybrechts’s notes in progress:
We meet Mondays in Mathematics Room 622 between 1:15PM and 2:30PM.