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Kuznetsov's categorical rationality conjecture for cubic fourfolds

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For a smooth cubic fourfold X⊂P5X\subset\mathbb P^5, there is a semiorthogonal decomposition

DbCoh⁡(X)=⟨Ku⁡(X),OX,OX(1),OX(2)⟩,D^b\operatorname{Coh}(X) =\langle\operatorname{Ku}(X),\mathcal O_X,\mathcal O_X(1),\mathcal O_X(2)\rangle,

where Ku⁡(X)\operatorname{Ku}(X) is a 22-Calabi-Yau category, often viewed as a noncommutative K3 surface.

Kuznetsov's conjecture. The fourfold XX is rational if and only if

Ku⁡(X)≃DbCoh⁡(S)\operatorname{Ku}(X)\simeq D^b\operatorname{Coh}(S)

for an ordinary, untwisted projective K3 surface SS.

There is a precise Hodge-theoretic characterization of when Ku⁡(X)\operatorname{Ku}(X) is equivalent to the derived category of a K3 surface, and recent work gives strong K3-type Hodge-theoretic necessary conditions for rationality. The full equivalence between rationality and geometricity of the Kuznetsov component remains open.

cubic-fourfoldsderived-categories
Posted by Vihaan Dheer Posted

Progress summary

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Open

The proposed equivalence between rationality of cubic fourfolds and their being represented by an ordinary K3K3 surface remains unproved, despite strong generic and Hodge-theoretic evidence.

Kuznetsov conjectured that a smooth cubic fourfold XX is rational exactly when its Kuznetsov component satisfies AX≃Db(S)\mathcal{A}_X\simeq D^b(S) for an untwisted projective K3K3 surface SS. The retrieved literature records the conjecture and substantial evidence, but no complete proof or counterexample.

Known results

  • Addington–Thomas (2012): generically, categorical geometricity is equivalent to Hassett’s period-theoretic conjecture, with geometricity Zariski open and dense on relevant loci.
  • Bernardara–Macrì–Mehrotra–Stellari (2015): for admissible discriminants, generic Kuznetsov components are equivalent to twisted K3K3 categories; this does not settle the untwisted rationality conjecture.
  • Huybrechts (2021): neither implication in the rationality/geometricity equivalence was known in full.

June 2026 Hodge-theoretic characterization

A 2026 paper gives the theorem that having a Hodge-theoretically associated K3K3 surface is equivalent to AX≃Db(S)\mathcal{A}_X\simeq D^b(S). It sharpens the categorical characterization but leaves the equivalence with rationality conjectural.

Current status (as of August 2026): The conjecture remains open: the K3K3-category/Hodge-theoretic characterization and generic results are established, but neither implication between rationality and untwisted K3K3 geometricity is proved in full.

Sources

Solutions 0

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