ncts

Varieties of generalized Kummer type (Kumⁿ-type) are one of the two infinite series of known hyper-Kähler varieties, the other one being given by deformations of Hilbert schemes of points of K3 surfaces. Via Hodge theory and the Torelli theorem, it is possible to associate a K3 surface S_Kwith any variety K of Kumⁿ type, naturally defined as the K3 surface whose transcendental lattice is Hodge isometric to that of K with the form rescaled by a factor 2. In my talk I will explain how K and the associated K3 surface are related geometrically through a moduli space of sheaves on S_K. Building upon the work of O'Grady, Markman, Voisin and Varesco, we are able to prove the Hodge conjecture for all powers of these K3 surfaces. As a corollary we obtain the Hodge conjecture for all powers of any abelian fourfold of Weil type with discriminant 1, strenghtening a result of Markman.