Gluing Stability Conditions

Abstract

We define and study a gluing procedure for Bridgeland stability conditions in the situation when a triangulated category has a semiorthogonal decomposition. As an application, we construct stability conditions on the derived categories of $Z_2$-equivariant sheaves associated with ramified double coverings of P3. Also, we study the stability space for the derived category of $Z_2$-equivariant coherent sheaves on a smooth curve $X$, associated with a degree 2 map $X → Y$ , where $Y$ is another smooth curve. In the case when the genus of $Y is ≥ 1$ we give a complete description of the stability space.