The space of augmented stability conditions A Stab(C) is a partial compactification of the quotient of a triangulated category's Bridgeland stability manifold by the category's action, built to expose the structure of its semiorthogonal decompositions. A point is an augmented stability condition: a multiscale decomposition (a new homological structure that, generically, encodes a semiorthogonal decomposition together with a configuration of points) equipped with stability conditions on the associated subquotient categories. Near certain boundary points the space is conjectured to look like a manifold with corners; when this holds it yields proper good moduli spaces of Bridgeland semistable objects for smooth proper C.
Halpern-Leistner and Robotis construct a partial compactification, written A Stab(C), of the quotient of the Bridgeland stability manifold of a triangulated category C by the action of C. Its purpose…