Stability conditions and $\mu$-stable sheaves on K3 surfaces with Picard number one

Abstract

In this article, we show that some semi-rigid $\mu$-stable sheaves on a projective K3 surface $X$ with Picard number $1$ are stable under Bridgeland's stability condition. As a consequence of our work, we show that the special set $U(X) \subset \mathrm{Stab}(X)$ introduced by Bridgeland reconstructs $X$ itself. This gives a sharp contrast to the case of an abelian surface.