The space of augmented stability conditions

TL;DR

The paper develops a comprehensive framework of augmented stability conditions $\mathcal{A}\mathrm{Stab}(\mathcal{C})$ for a stable $dg$-category $\mathcal{C}$, extending Bridgeland’s stability by incorporating multiscale decompositions and markings that reflect semiorthogonal structures. It introduces multiscale lines and decompositions, coarsenings, and the notion of augmented stability conditions, endowing $\mathcal{A}\mathrm{Stab}(\mathcal{C})$ with a Hausdorff topology that reconciles interior Bridgeland stability with boundary degenerations via a weak and a convergent-net topology; a key construct is the moduli spaces of marked multiscale lines $\mathcal{A}_n$ and their real blowups $\mathcal{A}_n^{\mathbb{R}}$, which parameterize degenerating central charges and configurations. A central conjecture—the manifold-with-corners conjecture—predicts local real-analytic coordinates near generic admissible boundary points, realized at generic points via gluing techniques, and the authors prove a partial generic-point version. The framework connects to noncommutative MMP by offering a mechanism to produce multiscale decompositions and stability data on graded pieces, and it yields boundedness results with implications for the existence of proper moduli spaces of semistable objects in smooth, proper settings; the paper also provides foundational examples on points, $\mathbb{P}^1$, and higher-genus curves to illustrate the boundary geometry and potential obstacles. Overall, this work lays groundwork for systematic control over stability structures and moduli in noncommutative and geometric settings, with potential applications to birational geometry and noncommutative MMP programs.

Abstract

Given a triangulated category $\mathcal{C}$, we construct a partial compactification, denoted $\mathcal{A}\mathrm{Stab}(\mathcal{C})$, of the quotient of its stability manifold by $\mathbb{C}$. The purpose of $\mathcal{A}\mathrm{Stab}(\mathcal{C})$ is to shed light on the structure of semiorthogonal decompositions of $\mathcal{C}$. A point of $\mathcal{A}\mathrm{Stab}(\mathcal{C})$, called an augmented stability condition on $\mathcal{C}$, consists of a newly introduced homological structure called a multiscale decomposition, along with stability conditions on subquotient categories of $\mathcal{C}$ associated to this multiscale decomposition. A generic multiscale decomposition corresponds to a semiorthogonal decomposition along with a configuration of points in $\mathbb{C}$. We give a conjectural description of open neighborhoods of certain boundary points, called the "manifold-with-corners conjecture," and we prove it in a special case. We show that this conjecture implies the existence of proper good moduli spaces of Bridgeland semistable objects in $\mathcal{C}$ when $\mathcal{C}$ is smooth and proper, and discuss some first examples where the manifold-with-corners conjecture holds.

Figures (9)

• Figure 1: A sluice, rendered with ChatGPT.
• Figure 2: This illustrates the partial order on terminal vertices of a multiscale line. We say $v \leq_{\zeta,t} w$ if the point $\frac{\mathfrak{p}(v,w)}{\zeta}$ on the unit circle lies in the interior of the shaded cone shown.
• Figure 3: Given distinct terminal components $u$ and $w$ of a multiscale line, this diagram shows all of the possible ways that the path from the root $r$ to a third terminal component $v$ can meet the sub-tree spanned by $u$, $w$, and $r$.
• Figure 4: A picture of a two level multiscale line, its dual tree, the configuration of node points on $\Sigma_{\rm{root}}\setminus \{p_\infty\}$, and the resulting filtered semiorthogonal decomposition. For concreteness, we have taken $k=6$.
• Figure 5: The cone with boundary on the left represents $\mathcal{A}\mathop{\mathrm{Stab}}\nolimits(\mathbb{P}^1)/\mathop{\mathrm{Aut}}\nolimits(\mathbb{P}^1)$. The boundary circle on the right side of the surface labeled "$\partial$" depicts the admissible boundary. The horizontal arrow is the quotient map which contracts the admissible boundary to a point. The gray interior of both pictures corresponds to $\mathop{\mathrm{Aut}}\nolimits(\mathbb{P}^1)\backslash \mathop{\mathrm{Stab}}\nolimits(\mathbb{P}^1)/\mathbb{C}$, which is biholomorphic to an open cylinder.

Theorems & Definitions (240)

• Theorem A
• Theorem B: =\ref{['T:moduli_spaces']}, simplified
• Remark 1.1
• Example 1.2
• Definition 1.3
• Definition 1.4: Convergence in $\mathcal{A}\mathop{\mathrm{Stab}}\nolimits(\mathcal{C})$
• Remark 1.5
• Conjecture A: Manifold-with-corners, simple version
• Theorem C: =\ref{['T:genericmanifoldwithcorners']}
• Definition 2.1