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Quantum cohomology and irrationality of Gushel-Mukai fourfolds

Vladimiro Benedetti, Laurent Manivel, Nicolas Perrin

Abstract

We compute the small quantum cohomology of Gushel-Mukai fourfolds. Following [13], our computations imply that the very general ones are not rational. Following [8], and thanks to a suitable deformation of the small quantum cohomology ring, we also deduce that a rational Gushel-Mukai fourfold has the same rational cohomology as some K3 surface.

Quantum cohomology and irrationality of Gushel-Mukai fourfolds

Abstract

We compute the small quantum cohomology of Gushel-Mukai fourfolds. Following [13], our computations imply that the very general ones are not rational. Following [8], and thanks to a suitable deformation of the small quantum cohomology ring, we also deduce that a rational Gushel-Mukai fourfold has the same rational cohomology as some K3 surface.
Paper Structure (8 sections, 11 theorems, 74 equations)

This paper contains 8 sections, 11 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Gushel-Mukai fourfolds
  3. Lines and conics on Gushel-Mukai fourfolds
  4. Ambient cohomology
  5. Quantum multiplication by the hyperplane class
  6. Irrationality of the very general Gushel-Mukai fourfold
  7. The ambient quantum cohomology
  8. On rational Gushel-Mukai fourfolds

Key Result

Theorem 1

The quantum multiplication by the hyperplane class is given by the following matrix: For $q$ generic, it has four nonzero simple eigenvalues and a two dimensional kernel.

Theorems & Definitions (17)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • proof
  • Theorem 4
  • proof : Proof, following kkpy
  • Lemma 5
  • Theorem 6
  • proof
  • Lemma 7
  • ...and 7 more