A Fourier-Mukai Transform for Stable Bundles on K3 Surfaces

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We define a Fourier-Mukai transform for sheaves on K3 surfaces over, 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 K3 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.