On the irrationality of cubic fourfolds

TL;DR

This paper investigates the irrationality of cubic fourfolds, establishing a link between rational cubic fourfolds and the Hodge structures of K3 surfaces, thereby advancing understanding of their geometric properties.

Contribution

It proves that rational smooth complex cubic fourfolds have primitive cohomology isomorphic to that of a K3 surface, revealing a new structural connection.

Findings

Rational cubic fourfolds have Hodge structures matching those of K3 surfaces.

The work extends previous results on the irrationality of very general cubic fourfolds.

Provides a criterion linking rationality to K3 surface cohomology.

Abstract

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.