Nielsen realization problem for Bridgeland stability conditions on generic K3 surfaces

Yu-Wei Fan, Kuan-Wen Lai

Published 2023-02-24Version 1

The original Nielsen realization problem asks whether a finite subgroup of the mapping class group of an oriented surface has a fixed point on the Teichm\"uller space or not. As a categorical analogue, one can ask whether a finite group of autoequivalences of a triangulated category has a fixed point on the space of Bridgeland stability conditions or not. We can also modify this question by only requiring the group to be finite modulo shift functors and considering its action on a suitable quotient. The main objects of study in this paper are K3 surfaces of Picard number one, where the realization problem leads to classifications of finite subgroups in various settings. We also discuss the realization problem for curves, standard autoequivalences on surfaces, twisted abelian surfaces, and generic twisted K3 surfaces.