Stability conditions and μ-stable sheaves on K3 surfaces with Picard number one

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In this article, we show that some semi-rigid μ-stable sheaves on a projective K3 surface X with Picard number 1 are stable in the sense of Bridgeland's stability condition. As a consequence of our work, we show that the special set U(X) ⊂ (X) reconstructs X itself. This gives a sharp contrast to the case of an abelian surface.