Hodge theory and derived categories of cubic fourfolds

Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics—conjecturally, the ones that are rational—have specific K3 surfaces associated to them geometrically. Hassett has studied cubics with K3 surfaces associated to them at the level of Hodge theory, and Kuznetsov has studied cubics with K3 surfaces associated to them at the level of derived categories. These two notions of having an associated K3 surface should coincide. We prove that they coincide generically: Hassett’s cubics form a countable union of irreducible Noether–Lefschetz divisors in moduli space, and we show that Kuznetsov’s cubics are a dense subset of these, forming a nonempty, Zariski-open subset in each divisor.