Partial compactification of stability manifolds by massless semistable objects

Abstract

We introduce two partial compactifications of the space of Bridgeland stability conditions of a triangulated category. First we consider lax stability conditions where semistable objects are allowed to have mass zero but still have a phase. The subcategory of massless objects is thick and there is an induced classical stability on the quotient category. We study deformations of lax stability conditions. Second we consider the space arising by identifying lax stability conditions which are deformation-equivalent with fixed charge. This second space is stratified by stability spaces of Verdier quotients of the triangulated category by thick subcategories of massless objects. We illustrate our results through examples in which the Grothendieck group has rank 2. For these, our partial compactification can be explicitly described and related to the wall-and-chamber structure of the stability space. Contents 1. Introduction 2. Notation and preliminaries 3. Restriction, descent and glueing of slicings 4. Lax stability conditions and quotient categories 5. The space of lax stability conditions 6. Deforming lax stability conditions 7. The topology of the space of lax stability conditions 8. The space of quotient stability conditions 9. Codimension one strata 10. Finite type components 11. Closures of G-orbits 12. Two-dimensional stability spaces 13. Comparisons with other constructions 14. Open questions References Ginzburg algebra C = D b (Γ 2 A 2 ) Bound path algebra C = D b (Λ 1,2,0 ) Examples of quotient stability spaces Stab Q (C) * up to C action; see Figure 2.