Math Conjectures

Kuznetsov’s Rationality Conjecture for Cubic Fourfolds

OPENMajorConjectureProposed 2010 · Full conjecture

Canonical statement

Let X⊂PC5X\subset\mathbb P^5_\mathbb C be a smooth cubic fourfold. In the bounded derived category Db(Coh⁡X)D^b(\operatorname{Coh}X) of coherent sheaves on XX, put
AX=⟨OX,OX(1),OX(2)⟩⊥, \mathcal A_X= \langle\mathcal O_X,\mathcal O_X(1),\mathcal O_X(2)\rangle^\perp,
where ⊥{}^\perp denotes the full right-orthogonal subcategory. Then XX is rational if and only if there is a complex K3 surface SS with an exact C\mathbb C-linear triangulated equivalence AX≃Db(Coh⁡S)\mathcal A_X\simeq D^b(\operatorname{Coh}S).
View source LaTeX
Let \(X\subset\mathbb P^5_\mathbb C\) be a smooth cubic fourfold. In the bounded derived category \(D^b(\operatorname{Coh}X)\) of coherent sheaves on \(X\), put
\[
  \mathcal A_X=
    \langle\mathcal O_X,\mathcal O_X(1),\mathcal O_X(2)\rangle^\perp,
\]
where \({}^\perp\) denotes the full right-orthogonal subcategory. Then \(X\) is rational if and only if there is a complex K3 surface \(S\) with an exact \(\mathbb C\)-linear triangulated equivalence \(\mathcal A_X\simeq D^b(\operatorname{Coh}S)\).
AG-RAT-018:[Kuznetsov2010Cubic][AddingtonThomas2014Hodge][Huybrechts2025K3Cubic]
Algebraic geometryAlgebra and representation theory

Notes

Kuznetsov conjectured in 2010 a categorical criterion for rationality of cubic fourfolds. For a smooth cubic X⊂PC5X\subset\mathbb P^5_{\mathbb C}, the right orthogonal AX\mathcal A_X to the line bundles OX,OX(1),OX(2)\mathcal O_X,\mathcal O_X(1),\mathcal O_X(2) in Db(Coh⁡X)D^b(\operatorname{Coh}X) has many formal properties of the derived category of a K3 surface. The conjecture asserts that XX is rational if and only if AX\mathcal A_X is equivalent, as a C\mathbb C-linear triangulated category, to Db(Coh⁡S)D^b(\operatorname{Coh}S) for an actual complex K3 surface SS [Kuznetsov2010Cubic].

The criterion agrees with the known rational examples. Addington and Thomas showed that the categorical condition is tightly linked with Hassett's Hodge-theoretic divisors of special cubic fourfolds, closely aligning the two approaches [AddingtonThomas2014Hodge]. An up-to-date account of the K3 category AX\mathcal A_X is given by Huybrechts [Huybrechts2025K3Cubic].

A resolution would tie the classical rationality problem for cubic fourfolds definitively to the theory of K3 categories. At present, however, neither implication has been established for every smooth cubic fourfold, and the conjecture remains open in both directions.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.