A Fourier-Mukai transform for stable bundles on K 3 surfaces

We define a Fourier-Mukai transform for sheaves on K3 surfaces over C, and show that it maps polystable bundles to polystable ones. The role of ''dual'' variety to the given K3 surface X is here played by a suitable component X of the moduli space of stable sheaves on X. For a wide class of K 3 surfaces X can be chosen to be isomorphic to X; then the Fourier-Mukai transform is invertible, and the image of a zero-degree stable bundle F is stable and has the same Euler characteristic as F.