Stability of Gieseker stable sheaves on K3 surfaces in the sense of Bridgeland and some applications

We show that some Gieseker stable sheaves on a projective K3 surface $X$ are stable with respect to a stability condition of Bridgeland on the derived category of $X$ if the stability condition is in explicit subsets of the space of stability conditions depending on the sheaves. Furthermore we shall give two applications of the result. As a part of these applications, we show that the fine moduli space of Gieseker stable torsion free sheaves on a K3 surface with Picard number one is the moduli space of $\mu$-stable locally free sheaves if the rank of the sheaves is not a square number.Comment: 34 pages, v2:we added one figure in page 2

Stability conditions on morphisms in a category

Let $\mathrm{h}\mathscr{C}$ be the homotopy category of a stable infinity category $\mathscr{C}$. Then the homotopy category $\mathrm{h}\mathscr{C}^{\Delta^{1}}$ of morphisms in the stable infinity category $\mathscr{C}$ is also triangulated. Hence the space $\mathsf{Stab}\,{ \mathrm{h}\mathscr{C}^{\Delta^{1}}}$ of stability conditions on $\mathrm{h}\mathscr{C}^{\Delta^{1}}$ is well-defined though the non-emptiness of $\mathsf{Stab}\,{ \mathrm{h}\mathscr{C}^{\Delta^{1}}}$ is not obvious. Our basic motivation is a comparison of the homotopy type of $\mathsf{Stab}{\mathrm{h}\mathscr{C}}$ and that of $\mathsf{Stab}{\mathrm{h}\mathscr{C}^{\Delta^{1}}}$. Under the motivation we show that functors $d_{0}$ and $d_{1} \colon \mathscr{C}^{\Delta^{1}} \rightrightarrows \mathscr{C}$ induce continuous maps from $\mathsf{Stab} {\mathrm{h}\mathscr{C}}$ to $\mathsf{Stab}{\mathrm{h}\mathscr{C}^{\Delta^{1}}}$ contravariantly where $d_{0}$ (resp. $d_{1}$) takes a morphism to the target (resp. source) of the morphism. As a consequence, if $\mathsf{Stab}{\mathrm{h}\mathscr{C}}$ is nonempty then so is $\mathsf{Stab}{\mathrm{h}\mathscr{C}^{\Delta^{1}}}$. Assuming $\mathscr{C}$ is the derived infinity category of the projective line over a field, we further study basic properties of $d_{0}^{*}$ and $d_{1}^{*}$. In addition, we give an example of a derived category which does not have any stability condition.Comment: 26 pages, comments are welcome. For v2, added subsection 2.2 which gives a description of a Serre functor of the category of morphisms in $\mathbf D$. For v3, the proof of Proposition 3.3 has been updated. For v5, Section 6 was added. For v6, modified the proof of Proposition 6.1. For v7, minor revision for Proposition 6.1, final versio

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

In this article, we show that some semi-rigid $\mu$-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) \subset \Stab (X) reconstructs $X$ itself. This gives a sharp contrast to the case of an abelian surface.Comment: 31 pages, 2 figures. Claim 4.3 was deleted and added some references;v3. some typos were correcte