Deformations of twisted sheaves and formality results

We show that infinitesimal deformations of twisted sheaves are controlled by the DG Lie algebra of their derived automorphisms. We prove that such DG Lie algebra is formal for polystable twisted sheaves on minimal surfaces of Kodaira dimension 0 and for projectively hyper-holomorphic locally free twisted sheaves on hyper-Kähler manifolds.