On the irrationality of cubic fourfolds

Following the work of Katzarkov–Kontsevich–Pantev–Yu concerning the irrationality of the very general complex cubic fourfold, we prove the following: for every rational smooth complex cubic fourfold, the primitive cohomology is isomorphic as a Hodge structure to the (twisted) middle cohomology of a projective K3 surface.

💡 Research Summary

The paper addresses the long‑standing problem of the irrationality of complex cubic fourfolds by establishing a precise Hodge‑theoretic relationship between any rational smooth cubic fourfold and a K3 surface. Building on the recent work of Katzarkov, Kontsevich, Pantev, and Yu, which used Gromov–Witten theory to distinguish very general cubic fourfolds from rational ones, the author introduces a new birational invariant, denoted by the symbol ♥, that remains unchanged under blow‑ups at smooth centers up to dimension four. This invariant is defined via quantum cohomology and an endomorphism κτ constructed from the Euler vector field associated to a formal Hodge class τ∈H²(X)Hdg.

The paper begins with a careful review of Hodge structures, Deligne twists, and Hochschild degree, then develops the formal machinery of the Novikov ring, formal power series, and the quantum product φi⋆τφj expressed through genus‑zero Gromov–Witten correlators. The Euler vector field Euτ is defined as c₁(X) plus a weighted sum of the τ‑coefficients, and κτ=Euτ⋆τ acts on the quantum‑cohomology module H⁎(X)R(X). Proposition 14 proves that κτ is a degree‑2 morphism of Hodge structures, preserving both the Hodge decomposition and the Hochschild grading.

Two key examples illustrate the behavior of κτ. For a smooth surface with nef canonical class, κτ is computed explicitly; its matrix is block‑triangular with nilpotent off‑diagonal blocks, showing that κτ−f₀ is nilpotent on the cohomology. For a smooth cubic fourfold X⊂ℙ⁵, the author takes τ=0. The primitive part of H⁴(X) is annihilated by κτ, while on the ambient part κτ coincides with a concrete 5×5 matrix M (originally computed by Givental). The eigenvalue analysis of M reveals a one‑dimensional zero eigenspace and a two‑dimensional generalized eigenspace, confirming that κτ acts non‑trivially only on the non‑primitive cohomology.

Section 2 introduces a non‑Archimedean field F=ℚ((a^ℚ)) with trivial absolute value and shows how the blow‑up formula for quantum cohomology can be interpreted in this setting. The author proves that the property ♥ is preserved under birational transformations, including blow‑ups, by exploiting the deformation invariance of Gromov–Witten invariants and the compatibility of the Euler vector field with the blow‑up map. This contrasts with the earlier property ♣ used in