The aim of the talk is to study smooth projective hyperkähler 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 Verbitksy’s Torelli theorem implies that their period map is an open
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 hyperkähler 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.