Eyal Markman (UMass Amherst), Monodromy of generalized Kummers and Triality for Spin(8)


We will describe the computation of the monodromy group of compact hyperkahler varieties deformation equivalent to the generalized Kummer variety associated to an abelian surface. As an application we will explain O'Grady's recent observation that complete families of abelian fourfolds of Weil type (precisely those abelian fourfolds for which the Hodge conjecture is open) arise as intermediate Jacobians of generalized Kummers.

Mukai, Orlov, and Polishchuk have shown that the group of autoequivalences of the derived category of coherent sheaves on a g dimensional abelian variety acts on its cohomology via the natural action of Spin(4g). This action plays the role of a symmetry group for nice moduli spaces of stable sheaves on the abelian variety. The generalized Kummers are fibers of the Albanese map from moduli spaces of stable sheaves on an abelian surface (g=2). In this case an integral version of triality for Spin(8) is instrumental in computing the monodromy group for generalized Kummers.