# The period map for polarized hyperkahler fourfold

##
Emanuele MacrÃ¬

created by risa on 15 Jun 2018

21 jun 2018
-- 14:30

Aula 211, Dip.Matematica, UniversitÃ "Roma Tre", Roma

**Abstract.**

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 Verbitksy's 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.