The period map for polarized hyperkaehler manifolds
Emanuele Macrì
(Northeastern University)

The aim of the talk is to study smooth projective hyperkaehler manifolds which are deformations of Hilbert schemes of points on K3 surfaces and are equipped with a polarization of fixed type. These are parametrized by a quasi-projective 20-dimensional moduli space and Verbitsky Torelli theorem implies that their period map is an open embedding when restricted to each irreducible component. 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. The key technical ingredient is the description of the nef and movable cone for projective hyperkaehler manifolds (deformation equivalent to Hilbert schemes of points on 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 projective hyperkaehler manifolds which are isomorphic to Hilbert schemes of points on K3 surfaces. This is joint work with Olivier Debarre.