Stable sheaves on K3 surfaces via wall-crossing

We give a new proof of the following theorem: moduli spaces of stable complexes on a complex projective K3 surface, with primitive Mukai vector and with respect to a generic Bridgeland stability condition, are hyper-Kähler varieties of K3n-type of expected dimension. We use derived equivalences, deformations, and wall-crossing for Bridgeland stability to reduce to the case of the Hilbert scheme of points.