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On the irrationality of cubic fourfolds

Jérémy Guéré

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.

On the irrationality of cubic fourfolds

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.
Paper Structure (20 sections, 27 theorems, 232 equations)

This paper contains 20 sections, 27 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. Quantum cohomology
  3. Hodge structures
  4. Formal power series
  5. Quantum product
  6. Examples
  7. Birational transformation
  8. Invariant properties
  9. Non-archimedean geometry
  10. Blow-up formula
  11. Examples regarding Property $\clubsuit$
  12. Basic cases
  13. Surface with non-negative canonical class
  14. Low dimension
  15. Cubic fourfold
  16. ...and 5 more sections

Key Result

Theorem 1

If $X$ is a rational smooth complex cubic fourfold, then there exists a projective K3 surface $S$ and an isomorphism of Hodge structures

Theorems & Definitions (82)

  • Theorem : see Theorem \ref{['rat cubic K3']}
  • Remark
  • Definition 1: Hodge structure
  • Example 2: Deligne twist
  • Definition 3: Hodge classes and Hochschild degree
  • Example 4
  • Definition 5: Novikov ring
  • Definition 6
  • Definition 7: $K$-span of morphisms of Hodge structures
  • Definition 8: Quantum product
  • ...and 72 more