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Article Dans Une Revue Journal of the American Mathematical Society Année : 2022

COMPUTING RIEMANN-ROCH POLYNOMIALS AND CLASSIFYING HYPER-K ÄHLER FOURFOLDS

Olivier Debarre
  • Fonction : Auteur
Daniel Huybrechts
  • Fonction : Auteur
Emanuele Macrì
  • Fonction : Auteur

Résumé

We prove that a hyper-Kähler fourfold satisfying a mild topological assumption is of K3 [2] deformation type. This proves in particular a conjecture of O'Grady stating that hyper-Kähler fourfolds of K3 [2] numerical type are of K3 [2] deformation type. Our topological assumption concerns the existence of two integral degree-2 cohomology classes satisfying certain numerical intersection conditions. There are two main ingredients in the proof. We first prove a topological version of the statement, by showing that our topological assumption forces the Betti numbers, the Fujiki constant, and the Huybrechts-Riemann-Roch polynomial of the hyper-Kähler fourfold to be the same as those of K3 [2] hyper-Kähler fourfolds. The key part of the article is then to prove the hyper-Kähler SYZ conjecture for hyper-Kähler fourfolds for divisor classes satisfying the numerical condition mentioned above.
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Dates et versions

hal-03795965 , version 1 (04-10-2022)

Identifiants

  • HAL Id : hal-03795965 , version 1

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Olivier Debarre, Daniel Huybrechts, Emanuele Macrì, Claire Voisin. COMPUTING RIEMANN-ROCH POLYNOMIALS AND CLASSIFYING HYPER-K ÄHLER FOURFOLDS. Journal of the American Mathematical Society, In press. ⟨hal-03795965⟩
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