Hodge theory and derived categories of cubic fourfolds

Abstract

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.