Claire Voisin: A topological characterization of hyper-Kähler fourfolds of Hilb²(K3) type

There are two known deformations types of hyper-Kähler (HK) fourfolds, namely Hilb²(K3) (Beauville, Fujiki) and the generalized Kummer variety K₂(A) (Beauville). It is however still unknown whether there are other topological types or deformation types of HK fourfolds. Some strong restrictions on the Betti numbers of HK fourfolds are known by work of Beauville, S. Salamon, Verbitsky and Guan. In this talk, I will sketch the proof of the following:

Theorem. A hyper-Kähler fourfold X is a deformation of Hilb²(K3) if and only if it has two integral degree 2 cohomology classes satisfying the conditions l⁴=0, m⁴=0, l^2m2=2. In particular, a HK fourfold which is homeomorphic to Hilb²(K3) is a deformation of Hilb²(K3).

This is joint work with Debarre, Huybrechts and Macrì.

This video was produced by the Japan-US Mathematics Institute and forms part of JAMI Conference 2022.