Vertical Asymptotics for Bridgeland Stability Conditions on 3-Folds

Abstract

Abstract Let $X$ be a smooth projective threefold of Picard number one for which the generalized Bogomolov–Gieseker inequality holds. We characterize the limit Bridgeland semistable objects at large volume in the vertical region of the geometric stability conditions associated to $X$ in complete generality and provide examples of asymptotically semistable objects. In the case of the projective space and $\operatorname {ch}^\beta (E)=(-R,0,D,0)$, we prove that there are only a finite number of nested walls in the $(\alpha ,s)$-plane. Moreover, when $R=0$ the only semistable objects in the outermost chamber are the 1-dimensional Gieseker semistable sheaves, and when $\beta =0$ there are no semistable objects in the innermost chamber. In both cases, the only limit semistable objects of the form $E$ or $E[1]$ (where $E$ is a sheaf) that do not get destabilized until the innermost wall are precisely the (shifts of) instanton sheaves.