The aim of the talk is to study smooth projective hyperkahler fourfolds which are deformations of Hilbert squares of K3 surfaces and are equipped with a polarization of fixed degree and divisibility. These are parametrized by a quasi-projective irreducible 20-dimensional moduli space and Verbitksys Torelli theorem implies that their period map is an open embedding. Our main result is that the complement of the image of the period map is a finite union of explicit Heegner divisors that we describe. We will also comment on the higher dimensional case. The key technical ingredient is the description of the nef and movable cone for projective hyperkahler manifolds (deformation equivalent of Hilbert schemes of K3 surfaces) by Bayer, Hassett, and Tschinkel. As an application we will present a new short proof (by Bayer and Mongardi) for the celebrated result by Laza and Looijenga on the image of the period map for cubic fourfolds. If time permits, as second application, we will show that infinitely many Heegner divisors in a given period space have the property that their general points correspond to fourfolds which are isomorphic to Hilbert squares of a K3 surfaces, or to double EPW sextics. This is joint work with Olivier Debarre. |

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