Gluing stability conditions

TL;DR

This paper develops a gluing method for Bridgeland stability conditions in triangulated categories with semiorthogonal decompositions and applies it to construct stability conditions on derived categories of equivariant sheaves, providing a detailed description of their stability spaces.

Contribution

It introduces a new gluing procedure for stability conditions and applies it to equivariant sheaves on ramified covers, expanding the understanding of stability spaces.

Findings

Constructed stability conditions on derived categories of ${f Z}_2$-equivariant sheaves.

Provided a complete description of the stability space for certain equivariant sheaves on curves.

Extended the theory of stability conditions to categories with semiorthogonal decompositions.

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 ${\mathbb{Z}}_2$-equivariant sheaves associated with ramified double coverings of ${\mathbb{P}}^3$. Also, we study the stability space for the derived category of ${\mathbb{Z}}_2$-equivariant coherent sheaves on a smooth curve $X$, associated with a degree 2 map $X\to Y$, where $Y$ is another smooth curve. In the case when the genus of $Y$ is $\geq 1$ we give a complete description of the stability space.