Existence of μ-stable bundles on K3 surfaces and the Fourier-Mukai transform

We consider a family of polarized K3 complex surfaces X which includes all generic Kummer surfaces. We prove a non-emptiness result for the moduli space of stable vector bundles on X for small values of the second Chern class. For K3 surfaces in this family, the Fourier-Mukai transform preserves the mu-semistability of coherent sheaves. New results on quasi-homogeneous sheaves, and a novel algebraic proof of preservation of mu-stability are given as well.