Hodge theory and derived categories of cubic fourfolds
N. Addington, R. P. Thomas
TL;DR
The paper develops a precise geometric correspondence between Hassett’s Hodge-theoretic notion of an associated K3 and Kuznetsov’s derived-category notion for cubic fourfolds. It constructs a Mukai lattice for the noncommutative K3 category $\mathcal A_X$ via topological K-theory, relates it to $H^4(X,\mathbb Z)$, and translates the Hassett discriminant condition into a lattice-theoretic statement about $K_{\rm top}(\mathcal A_X)$. Through a deformation-theoretic framework based on Hochschild (co)homology and the HKR isomorphism, it shows that for discriminants $d$ meeting a numerical condition, the set of cubic fourfolds with $\mathcal A_X$ geometric is Zariski-open and dense inside the Hassett divisor $\mathcal C_d$, while generic $X$ in $\mathcal C_d$ carries a K3 surface $S$ with a derived-equivalence $D(S)\simeq \mathcal A_X$. The work further provides algebraic realizations of the Hodge isometries via cycles in $S\times X$ and establishes Hodge-conjecture-type statements in this setting, advancing a noncommutative Torelli-type picture for cubic fourfolds and their associated K3 data.
Abstract
Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics - conjecturally, the ones that are rational - have specific K3s associated to them geometrically. Hassett has studied cubics with K3s associated to them at the level of Hodge theory, and Kuznetsov has studied cubics with K3s associated to them at the level of derived categories. These two notions of having an associated K3 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 non-empty, Zariski open subset in each divisor.