Vertcal Asymptotics for Bridgeland Stability Conditions on 3-folds

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 chβ(E)=(−R,0,D,0), we prove that there are only a finite number of nested walls in the (α,s)-plane. Moreover, when R=0 the only semistable objects in the outermost chamber are the 1-dimensional Gieseker semistable sheaves, and when β=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.