FRG workshop, 2011

E. Markman

Title: Generalized deformations of pairs of a K3 surface and a stable coherent sheaf


A K3 surface admits 22 generalized deformation directions, including non-commutative and gerby deformations. Let E be a stable coherent sheaf on a K3 surface S with a primitive Mukai vector v of positive Mukai self-intersection (v,v). Then E should deform with S in a 21-dimensional family of generalized deformations. The moduli space M, of stable sheaves on S with Mukai vector v, admits a 21-dimensional family of (commutative) Kahler deformations. We propose (joint with S. Mehrotra) an interpretation of the latter 21-dimensional holomorphic family as the universal generalized deformation of the pair (S,E). The construction has two main ingredients.
  1. We reconstruct the bounded derived category D(S), of coherent sheaves on S, from a natural endo-functor F of D(M). This is an application of the Bar-Beck theorem in category theory.
  2. We deform the pair (M, F) along every Kahler deformation of M. This associates to a deformed pair (M', F') a category C(M', F'). When M' is the moduli space of sheaves on a K3 surface, the category C(M', F') is equivalent to D(S').