Hodge Theory of Cubic Fourfolds, Their Fano Varieties, and Associated K3 Categories

Abstract

These are notes of lectures given at the school 'Birational Geometry of Hypersurfaces' in Gargnano in March 2018. The main goal was to discuss the Hodge structures that come naturally associated with a cubic fourfold. The emphasis is on the Hodge and lattice theoretic aspects with many technical details worked out explicitly. More geometric or derived results are only hinted at.The primitive Hodge structure of a smooth cubic fourfold X ⊂ P 5 is concentrated in degree four and it is of a very particular type. Once a Tate twist is applied and the sign of the intersection form is changed, it reveals its true nature. It very much looks like the Hodge structure of a K3 surface. In his thesis Hassett [Ha00] studied this curious relation and the intricate lattice theory behind it in greater detail. He established a transcendental correspondence between polarized K3 surfaces of certain degrees and special cubic fourfolds, some aspects of which are reminiscent of the Kuga-Satake construction. The geometric nature of the Hassett correspondence is still not completely understood but it seems that derived categories are central for its understanding. Work of Addington and Thomas [AT14] represents an important step in this direction, combining Hassett's Hodge theory with Kuznetsov's categorical approach to hypersurfaces.

Stability conditions in families

“…The category Ku(X) shares many properties with the derived category of K3 surfaces. Its foundations were developed in [Kuz10,AT14,Huy17]; see [Huy19,MS19a] for surveys of those results. In particular, we recall:…”

Categorical Torelli theorems: results and open problems

“…Before moving to the sketch of the proof, let us briefly clarify the notation in the statement. If X is a cubic fourfold, the middle cohomology H 4 (X, Z) has a natural lattice and Hodge structure (see [52] for an excellent introduction). If H is the class of a hyperplane section then the selfintersection H 2 is an algebraic class in H 4 (X, Z).…”

On lattice polarizable cubic fourfolds

“…This section reviews some important aspects on lattice theory and Hodge theory of cubic fourfolds and mainly focuses on special cubic fourfolds; we refer to Hassett [Has00,Has16] and Huybrechts [Huy19,Huy20] for more detailed discussions.…”

“…The rationality of cubic fourfolds is a long standing open problem; we refer to the excellent surveys [Has16,Huy19]. In 1972, Clemens-Griffiths [CG72] proved that all smooth cubic threefolds are irrational via Hodge theory.…”