Stability conditions on higher dimensional varieties and moduli spaces

Abstract

Moduli theory has been a central theme in Algebraic Geometry. To construct moduli spaces, one usually needs the concept of stability conditions. Motivated by the work in string theory, Bridgeland introduced the stability conditions on a triangulated category and proposed the existence of a stability manifold. Moreover, there exists a locally finite wall and chamber structure on this manifold such that moduli spaces for fixed numerical invariants are constant in each chamber. This breakthrough has not only revolutionized the study of moduli spaces but also provided further connections to many other fields, including the representation theory, counting invariants, and the classical Algebraic Geometry. There are two main goals in this project. The first is to investigate the existence of stability conditions on varieties with higher dimensions, which is a highly non-trivial problem even for some threefolds. Geometers have made considerable contributions to this problem, but it is still widely open. We defined a family of "Euler" stability conditions on projective spaces of any dimension using the twisted Euler characteristic and its derivatives. The candidate worked on this family of stability with some applications during his Ph.D. We attempt to generalize this idea to other varieties whose bounded derived category can be generated by any kind of exceptional collection. The starting examples would be quadrics and Grassmannians. The second goal is to study the moduli space of Bridgeland stable complexes from the viewpoint of quiver representations. In the case that a variety owns an exceptional collection, the moduli space of Bridgeland stable complexes on that variety is equivalent to King's construction of quiver moduli. The wall-crossing phenomenon on the stability manifold is then related to the "classical" variation of GIT. Some conjectures and applications, for instance, the Bridgeland stability converges to Gieseker stability in some certain directions, could be proved by identifying moduli spaces. (AU)