A global Torelli theorem for rigid hyperholomorphic sheaves

(Joint work with S. Mehrotra). We construct moduli spaces M of marked triples (X,f,A), where X is an irreducible holomorphic symplectic manifold, f is a marking of X, and A is a stable and infinitesimally rigid reflexive sheaf of Azumaya algebras over the cartesian product Xd, such that the second Chern class of A is invariant under the diagonal action a finite index subgroup of the monodromy group of X. We prove a global Torelli theorem: The period map from the moduli space M to the period domain is a local homeomorphism, surjective, and generically injective.
The main example is the rank 2n-2 sheaf of Azumaya algebras constructed in arXiv:1105.3223 over the Cartesian square XхX of an irreducible holomorphic symplectic manifold of K3[n] deformation type. The characteristic classes of A were used to prove the standard conjectures when X is algebraic (joint with F. Charles). The sheaf A is also used to associate to X a generalized (non-commutative) deformation of the derived category of a K3 surface (joint work in progress with S. Mehrotra).