Walls and asymptotics for Bridgeland stability conditions on 3-folds

Abstract

We consider Bridgeland stability conditions for three-folds conjectured by Bayer-Macrì-Toda in the case of Picard rank one. We study the differential geometry of numerical walls, characterizing when they are bounded, discussing possible intersections, and showing that they are essentially regular. Next, we prove that walls within a certain region of the upper half plane that parametrizes geometric stability conditions must always intersect the curve given by the vanishing of the slope function and, for a fixed value of s, have a maximum turning point there. We then use these facts to prove that Gieseker semistability is equivalent to a strong form of asymptotic semistability along a class of paths in the upper half plane, and show how to find large families of walls. We illustrate how to compute all of the walls and describe the Bridgeland moduli spaces for the Chern character (2, 0, −1, 0) on complex projective 3-space in a suitable region of the upper half plane.

Vertical asymptotics for Bridgeland stability conditions on 3-folds

“…This notion of stability is the first step to identify a possible Gieseker chamber in the stability manifold. Indeed, as proven in [JM19] and [Pre21], there are paths (with α fixed and β → −∞), for which asymptotic stability coincides with Gieseker stability. However, given a Chern character v, a bound for the Bridgeland walls for v intersecting the path γ and depending only v is still unknown.…”

“…In this section, we will focus on the characterization of asymptotically λ α,β,ssemistable objects along vertical lines. In contrast with the case of horizontal lines studied in [JM19], the set of asymptotically λ α,β,s -semistable objects along a vertical line {β = β} with fixed Chern character v does depend both on β and on the parameter s.…”

“…There is no general argument proving that for some triple (α, β, s) (or any stability condition), the only λ α,β,s -semistable objects of a given Chern character are precisely the Gieseker semistable sheaves. At the moment, the closest we can get to a result of this type is through the concept of asymptotic stability introduced in [JM19].…”

“…In this article, we will concentrate in the study of asymptotic stability along the paths of fixed β = β. Contrary to the case of the paths considered in [JM19] and [Pre21], these paths provide a set up closer to the set up for surfaces in the following sense:…”

Zero Rank Asymptotic Bridgeland Stability

“…It is important to note that in [9] there is a different version of asymptotic stability, a stronger variation. This stronger version is equivalent to saying that whenever we choose an unbounded curve γ, an object A is strongly asymptotic λ γ -(semi)stable if exists a last actual wall intersecting γ and A is asymptotic λ γ -(semi)stable, that is, if there exists t 0 such that A is λ γ(t),s -(semi)stable for all t > t 0 .…”