Vertical asymptotics for Bridgeland stability conditions on 3-folds

TL;DR

This paper studies the behavior of Bridgeland stability conditions on threefolds with Picard number one, characterizing limit semistable objects at large volume and analyzing wall structures, especially in projective space.

Contribution

It provides a general characterization of limit Bridgeland semistable objects on threefolds and analyzes the structure of walls in the stability space for specific cases.

Findings

Finite number of nested walls in the $(eta,s)$-plane for projective space.

Only Gieseker semistable sheaves are semistable in the outermost chamber when $R=0$.

Limit semistable objects are precisely shifts of instanton sheaves.

Abstract

Let $X$ be a smooth projective threefold of Picard number one for which the generalized Bogomlov-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 $ch^\beta(E)=(-R,0,D,0)$, 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.