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

Components of the moduli space of sheaves on a K3 surface are parametrized by a lattice, the (algebraic) Mukai lattice. Isometries of the Mukai lattic e often lift to symplectic birational isomorphisms of the collection of com ponents. 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 integ rable system: the union of Jacobians of hyperplane sections (curves) of gen us 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 (Theor em 2.4). It applies to a holomorphic-symplectic variety M with a stratifica tion, where the first stratum is a P-n bundle, but lower strata are Grassma nnian bundles. The resulting (transformed) symplectic variety W admits a st ratification by the dual Grassmannian bundles. (2) The isometry group of the Mukai lattice contains two natural commuting reflections, which act trivially on the second cohomology of the K3 surface . These reflections act on moduli spaces of sheaves (with "minimal" first C hern class) as birational stratified elementary transformations (Theorem 1. 1). (3) We derive a Picard-Lefschetz type formula identifying the isomorphism o f cohomology rings of a holomorphic-symplectic variety M and its stratified transform W as the cup product with an algebraic correspondence (Theorem 1 .2).