Title: The hyperKummer construction

Abstract: Any hyperKähler sixfold K of generalized Kummer type (Kum^3-type) has a naturally associated manifold Y_K of K3^[3]-type, defined as crepant resolution of the quotient K/G by the action of a finite group G=(Z/2)^5. I will report on joint work with Lie Fu in which we explore this 'hyperKummer' construction, guided by the analogy with the classical Kummer construction in dimension 2. We obtain a birational characterization of hyperKummer varieties of K3^[3]-type as well as relations between K and Y_K at the level of motives and derived categories. As I will explain, our study suggests how to construct very general projective varieties of Kum^3-type as rational covers of suitable moduli spaces of sheaves on certain K3 surfaces. If time permits, I will also discuss a somewhat surprising connection with hyperKähler varieties of OG6-type.