The Space of augmented stability conditions

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.