Computing Riemann-Roch polynomials and classifying hyper-Kähler 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.

arXiv comment34 pages. v3: Minor corrections, references updated

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