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arXiv · 2112.00923

Vertical asymptotics for Bridgeland stability conditions on 3-folds

Marcos Jardim·Antony Maciocia·Cristian Martinez
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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.

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Marcos Jardim, Antony Maciocia, Cristian Martinez. 2021-12-02. Vertical asymptotics for Bridgeland stability conditions on 3-folds. https://arxiv.org/abs/2112.00923

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