stabilityconditions

Stability conditions, often simply stability conditions in the mathematical literature, provide a framework for studying objects in triangulated categories by decomposing them into semistable pieces. Introduced by Tom Bridgeland in 2007, they generalize classical stability notions for coherent sheaves to derived categories and give rise to a rich parameter space that encodes how objects may stabilize as one changes data.

A stability condition on a triangulated category D consists of a central charge Z: K(D) → C, a group homomorphism, together with a slicing P: R → D that assigns to each real number φ a full additive subcategory P(φ) of D whose nonzero objects are called semistable of phase φ. These data satisfy: P(φ+1) = P(φ)[1]; if φ1 > φ2 then Hom(P(φ1), P(φ2)) = 0; every nonzero E ∈ D admits a finite filtration with factors A_i ∈ P(φ_i) with φ1 > φ2 > … > φn; and for every nonzero A ∈ P(φ) one has Z([A]) ∈ R_{>0} e^{iπφ}.

A stability condition imposes a Harder–Narasimhan property and, with a support finiteness condition, yields a well-behaved moduli-like structure. The space Stab(D) of stability conditions is endowed with a natural topology and, in favorable cases, a complex manifold structure; it carries an action of the group of exact autoequivalences of D via pullback of stability data.

Examples and significance: on the derived category of coherent sheaves on a smooth projective variety, stability recovers slope stability and its tilts; stability conditions have been constructed in many settings, including curves and Calabi–Yau categories. They are central to wall-crossing phenomena, moduli of semistable objects, and connections to Donaldson–Thomas theory and string theory.