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

International audienceWe 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