Computing Riemann–Roch polynomials and classifying hyper-Kähler fourfolds

We prove that a hyper-Kähler fourfold satisfying a mild topological assumption is of K3 [ 2 ] ^{[2]} deformation type. This proves in particular a conjecture of O’Grady stating that hyper-Kähler fourfolds of K3 [ 2 ] ^{[2]} numerical type are of K3 [ 2 ] ^{[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 ] ^{[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.