Irrational cubic fourfolds with Hodge-theoretic and categorical K3 associations

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For every sufficiently large admissible Hassett discriminant, a very general smooth complex cubic fourfold is irrational despite having both an untwisted geometric K3 category and an integral Hodge-theoretic K3 association. This disproves Kuznetsov’s rationality conjecture and the sufficiency of the associated-K3 criterion for rationality; the discriminant threshold is ineffective.

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What the papers state

Theorem 1.1 (Irrationality on large admissible Hassett divisors)

In Irrational cubic fourfolds with geometric K3 categories, 24 September 2026

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There is an integer d_0 such that, for every admissible d>d_0, a very general cubic fourfold X in \mathcal C_d is irrational and admits an exact \C-linear equivalence

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

for a smooth projective K3 surface S.

Corollary 1.2 (Failure of associated-K3 sufficiency)

In Irrational cubic fourfolds with geometric K3 categories, 24 September 2026

Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.

Let d_0 be the threshold in Theorem 1.1. For every admissible d>d_0, a very general X\in\mathcal C_d is irrational and admits a labelling K of discriminant d and a primitively polarized smooth projective K3 surface (S_{\mathrm{Hdg}},L), with L^2=d, for which there is an integral Hodge isometry

K^\perp \simeq L^\perp(-1).

Here L^\perp\subset H^2(S_{\mathrm{Hdg}},\Z), and the surface cup pairing is reversed. Thus an associated untwisted polarized K3 surface in this Hodge-theoretic sense is not sufficient for rationality.

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Two more possible matches

Corollary 1.4 (Integral Chow decompositions for the irrational cubics) in Irrational cubic fourfolds with geometric K3 categories, 24 September 2026.
Corollary 1.3 (Failure of Hilbert-square sufficiency) in Irrational cubic fourfolds with geometric K3 categories, 24 September 2026.

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Papers

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Irrational cubic fourfolds with geometric K3 categories

24 September 2026, current edition.

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For every sufficiently large admissible Hassett discriminant, we prove that a very general cubic fourfold of that discriminant is irrational, although its Kuznetsov component is equivalent to the ordinary derived category of a projective K3 surface. The discriminant threshold is ineffective. This disproves Kuznetsov’s rationality conjecture. The same cubics have associated untwisted polarized K3 surfaces in the Hodge-theoretic sense, so they also disprove the sufficiency direction of the associated-K3 rationality prediction.

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66 references in Irrational cubic fourfolds with geometric K3 categories

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