Brill-Noether duality for moduli spaces of sheaves on K3 surfaces

Abstract

Components of the Moduli space of sheaves on a K3 surface are parametrized by a lattice; the (algebraic) Mukai lattice. Isometries of the Mukai lattice often lift to symplectic birational isomorphisms of the collection of components. An example of such a birational isomorphism is the Abel-Jacobi map relating the Hilbert scheme of g points on a K3 of degree 2g-2 to an integrable system: the union of Jacobians of hyperplane sections (curves) of genus g. The main results are: 1) We construct a stratified version of a Mukai elementary transformation modeled after dual pairs of Springer resolutions of nilpotent orbits. It applies to a holomorphic-symplectic variety M with a stratification where the first stratum is a P^n bundle, but lower strata are Grassmannian bundles. The resulting (transformed) symplectic variety W admits a stratification by the dual Grassmannian bundles. 2) The group of reflections of the Mukai lattice, which act trivially on the second cohomology of the K3 surface, acts on moduli spaces of sheaves (with ``minimal'' first Chern class) as birational stratified elementary transformations. 3) We derive a Picard-Lefschetz type formula identifying the isomorphism of cohomology rings of a holomorphic-symplectic variety M and its stratified transform W as the cup product with an algebraic correspondence.