FRG workshop, 2011
E. Markman
Title: Generalized deformations of pairs of a K3 surface and a stable coherent sheaf
Abstract:
A K3 surface admits 22 generalized deformation directions, including
noncommutative and gerby deformations. Let E be a stable coherent sheaf on a
K3 surface S with a primitive Mukai vector v of positive Mukai selfintersection (v,v).
Then E should deform with S in a 21dimensional family of generalized deformations.
The moduli space M, of stable sheaves on S with Mukai vector v,
admits a 21dimensional family of (commutative) Kahler deformations.
We propose (joint with S. Mehrotra) an interpretation of the latter 21dimensional
holomorphic family as the universal generalized deformation of the pair (S,E).
The construction has two main ingredients.

We reconstruct the bounded derived category D(S), of coherent sheaves on S,
from a natural endofunctor F of D(M). This is an application of the
BarBeck theorem in category theory.
 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').