Formality conjecture for K3 surfaces

We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the DG algebra RHom(F,F) is formal for any sheaf F polystable with respect to an ample line bundle. Our main tool is the uniqueness of DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition.