|Location:||MSRI: Simons Auditorium|
The hyperkahler manifolds are higher dimensional generalizations of the K3 surfaces. It is expected, as in the case of K3 surfaces, that their moduli space can be understood via the period map. However, little is known in higher dimensions. I will discuss a case of hyperkahler manifolds, the Fano varieties of cubic fourfolds, where we do have a good understanding
of the moduli space and of the period map. Using the special geometry of cubics, one can prove the global Torelli theorem (Voisin) and the surjectivity of the period map (Laza, Looijenga) for such manifolds.