Computing Riemann–Roch polynomials and classifying hyper-Kähler fourfolds

Debarre, Olivier; Huybrechts, Daniel; Macrì, Emanuele; Voisin, Claire

doi:10.1090/jams/1016

Computing Riemann–Roch polynomials and classifying hyper-Kähler fourfolds
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by Olivier Debarre, Daniel Huybrechts, Emanuele Macrì and Claire Voisin;

J. Amer. Math. Soc. 37 (2024), 151-185

DOI: https://doi.org/10.1090/jams/1016

Published electronically: February 17, 2023

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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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