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Journal Articles Journal of the American Mathematical Society Year : 2024

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

Abstract

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 and versions

hal-04252833 , version 1 (21-10-2023)

Identifiers

  • HAL Id : hal-04252833 , version 1

Cite

Olivier Debarre, Daniel Huybrechts, Claire Voisin, Emanuele Macrì. COMPUTING RIEMANN-ROCH POLYNOMIALS AND CLASSIFYING HYPER-K ÄHLER FOURFOLDS. Journal of the American Mathematical Society, 2024, 37 (1), pp.151-185. ⟨hal-04252833⟩
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