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

Journal of the American Mathematical Society, 2024-01, Vol.37 (1), p.151-185 [Peer Reviewed Journal]

Distributed under a Creative Commons Attribution 4.0 International License ;ISSN: 0894-0347 ;EISSN: 1088-6834

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  • Title:
    COMPUTING RIEMANN-ROCH POLYNOMIALS AND CLASSIFYING HYPER-K ÄHLER FOURFOLDS
  • Author: Debarre, Olivier ; Huybrechts, Daniel ; Voisin, Claire ; Macrì, Emanuele
  • Subjects: Mathematics
  • Is Part Of: Journal of the American Mathematical Society, 2024-01, Vol.37 (1), p.151-185
  • Description: 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.
  • Publisher: American Mathematical Society
  • Language: English
  • Identifier: ISSN: 0894-0347
    EISSN: 1088-6834
  • Source: Hyper Article en Ligne (HAL) (Open Access)
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