Bridgeland stability conditions and group actions

Bridgeland stability conditions have been constructed on curves, surfaces, and in some higher dimensional examples. In several cases, there are only so-called "geometric" stability conditions, which are constructed using slope stability for sheaves, whereas in other cases, there are more (for example if there is an equivalence with quiver representations). Lie Fu, Chunyi Li, and Xiaolei Zhao were the first to provide a general result explaining this phenomenon. In particular, they showed that if a variety has a finite map to an abelian variety, then all stability conditions are geometric. In this talk, we test the converse in two ways on surfaces that arise as free quotients by finite groups. One method is via Le Potier functions which characterize the existence of slope-semistable sheaves. The second method uses equivariant categories. This is joint work with Edmund Heng and Anthony Licata, based on arxiv:2307.00815 and arxiv:2311.06857.