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

Olivier Debarre (1) , Daniel Huybrechts (2) , Claire Voisin (3) , Emanuele Macrì (4)

1 UPCité - Université Paris Cité
2 Universität Bonn = University of Bonn
3 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
4 Université Paris-Saclay

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.

Mots clés: 2020 Mathematics Subject Classification. 14C20 14J35 14J42 14J60 Hyper-Kähler manifolds Lagrangian fibrations moduli spaces fourfolds; 2020 Mathematics Subject Classification. 14C20; 14J35; 14J42; 14J60 Hyper-Kähler manifolds; Lagrangian fibrations; moduli spaces; fourfolds

Domaines: Mathématiques [math]

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⟩