Stability conditions on triangulated categories

A stability condition (Z, 𝒫) on a triangulated category 𝒟 consists of a group homomorphism Z : K(𝒟) → ℂ called the central charge, and full additive subcategories 𝒫(φ) ⊂ 𝒟 for each φ ∈ ℝ, satisfying the following axioms: (a) if E ∈ 𝒫(φ) then Z(E) = m(E) · exp(iπφ) for some m(E) ∈ ℝ_{>0}; (b) for all φ ∈ ℝ, 𝒫(φ + 1) = 𝒫(φ)[1]; (c) if φ₁ > φ₂ and Aⱼ ∈ 𝒫(φⱼ) then Hom_𝒟(A₁, A₂) = 0; (d) for each nonzero object E ∈ 𝒟 there is a finite sequence of real numbers φ₁ > φ₂ > ⋯ > φₙ and objects E₀, E₁, …, Eₙ ∈ 𝒟 with E₀ = 0, Eₙ = E, together with distinguished triangles Eⱼ₋₁ → Eⱼ → Aⱼ → Eⱼ₋₁[1] for 1 ≤ j ≤ n in which Aⱼ ∈ 𝒫(φⱼ). The same definition is restated more compactly as Definition 5.1 in §5, after slicings (Definition 3.3) are introduced to package axioms (b)–(d).

Let 𝒟 be a triangulated category. For each connected component Σ ⊂ Stab(𝒟) there is a linear subspace V(Σ) ⊂ Hom_ℤ(K(𝒟), ℂ) with a well-defined linear topology and a local homeomorphism Z : Σ → V(Σ) which maps a stability condition (Z, 𝒫) to its central charge Z.

Suppose 𝒟 is numerically finite. For each connected component Σ ⊂ Stab_N(𝒟) there is a subspace V(Σ) ⊂ Hom_ℤ(N(𝒟), ℂ) and a local homeomorphism Z : Σ → V(Σ) which maps a stability condition to its central charge. In particular Σ is a finite-dimensional complex manifold.

A stability function on an abelian category 𝒜 is a group homomorphism Z : K(𝒜) → ℂ such that for all 0 ≠ E ∈ 𝒜 the complex number Z(E) lies in the strict upper half-plane H = { r · exp(iπφ) : r > 0 and 0 < φ ≤ 1 } ⊂ ℂ. The phase of a nonzero object E is then φ(E) = (1/π) · arg Z(E) ∈ (0, 1].

Let Z : K(𝒜) → ℂ be a stability function on an abelian category 𝒜. An object 0 ≠ E ∈ 𝒜 is said to be semistable (with respect to Z) if every subobject 0 ≠ A ⊂ E satisfies φ(A) ≤ φ(E).

Let Z : K(𝒜) → ℂ be a stability function on an abelian category 𝒜. A Harder–Narasimhan filtration of an object 0 ≠ E ∈ 𝒜 is a finite chain of subobjects 0 = E₀ ⊂ E₁ ⊂ ⋯ ⊂ Eₙ₋₁ ⊂ Eₙ = E whose factors Fⱼ = Eⱼ / Eⱼ₋₁ are semistable objects of 𝒜 with φ(F₁) > φ(F₂) > ⋯ > φ(Fₙ). The stability function Z is said to have the Harder–Narasimhan property if every nonzero object of 𝒜 has a Harder–Narasimhan filtration.

A t-structure on a triangulated category 𝒟 is a full subcategory ℱ ⊂ 𝒟, satisfying ℱ[1] ⊂ ℱ, such that if one defines ℱ^⊥ = { G ∈ 𝒟 : Hom_𝒟(F, G) = 0 for all F ∈ ℱ }, then for every object E ∈ 𝒟 there is a triangle F → E → G in 𝒟 with F ∈ ℱ and G ∈ ℱ^⊥. The heart of the t-structure is the full subcategory 𝒜 = ℱ ∩ ℱ^⊥[1] ⊂ 𝒟; the t-structure is bounded if 𝒟 = ⋃_{i,j ∈ ℤ} ℱ[i] ∩ ℱ^⊥[j].

A slicing 𝒫 of a triangulated category 𝒟 consists of full additive subcategories 𝒫(φ) ⊂ 𝒟 for each φ ∈ ℝ satisfying the following axioms: (a) for all φ ∈ ℝ, 𝒫(φ + 1) = 𝒫(φ)[1]; (b) if φ₁ > φ₂ and Aⱼ ∈ 𝒫(φⱼ) then Hom_𝒟(A₁, A₂) = 0; (c) for each nonzero object E ∈ 𝒟 there is a finite sequence of real numbers φ₁ > φ₂ > ⋯ > φₙ and objects E₀, E₁, …, Eₙ ∈ 𝒟 with E₀ = 0, Eₙ = E, together with distinguished triangles Eⱼ₋₁ → Eⱼ → Aⱼ → Eⱼ₋₁[1] for 1 ≤ j ≤ n in which Aⱼ ∈ 𝒫(φⱼ). Writing φ⁺(E) = φ₁ and φ⁻(E) = φₙ for the maximum and minimum phases appearing in this decomposition of E, the full additive subcategory 𝒫(I) ⊂ 𝒟 for an interval I ⊂ ℝ consists of the zero objects together with the 0 ≠ E whose phases satisfy φ⁻(E), φ⁺(E) ∈ I.

Let 𝒫 be a slicing of a triangulated category 𝒟 and let I ⊂ ℝ be an interval of length at most one. Suppose A → E → B → A[1] is a distinguished triangle in 𝒟 all of whose vertices are nonzero objects of 𝒫(I). Then φ⁺(A) ≤ φ⁺(E) and φ⁻(E) ≤ φ⁻(B).

A quasi-abelian category is an additive category 𝒜 with kernels and cokernels such that every pullback of a strict epimorphism is a strict epimorphism, and every pushout of a strict monomorphism is a strict monomorphism. A morphism f in 𝒜 is strict if the canonical map coim f → im f is an isomorphism. A strict short exact sequence in 𝒜 is a sequence 0 → A → B → C → 0 in which A → B is a strict monomorphism with cokernel B → C; equivalently, B → C is a strict epimorphism with kernel A → B.

A stability condition σ = (Z, 𝒫) on a triangulated category 𝒟 consists of a group homomorphism Z : K(𝒟) → ℂ and a slicing 𝒫 of 𝒟 such that if 0 ≠ E ∈ 𝒫(φ) then Z(E) = m(E) · exp(iπφ) for some m(E) ∈ ℝ_{>0}. The linear map Z is called the central charge; the nonzero objects of 𝒫(φ) are the σ-semistable objects of phase φ, and the simple objects of 𝒫(φ) are σ-stable.

If σ = (Z, 𝒫) is a stability condition on a triangulated category 𝒟 then each subcategory 𝒫(φ) ⊂ 𝒟 is abelian.

A slicing 𝒫 of a triangulated category 𝒟 is locally-finite if there exists a real number η > 0 such that for all t ∈ ℝ the quasi-abelian category 𝒫((t − η, t + η)) ⊂ 𝒟 is of finite length. A stability condition (Z, 𝒫) is locally-finite if the corresponding slicing 𝒫 is.

If τ = (W, 𝒬) ∈ B_ε(σ) then there are constants k₁, k₂ > 0 such that k₁ · ‖U‖_σ < ‖U‖_τ < k₂ · ‖U‖_σ for every U ∈ Hom_ℤ(K(𝒟), ℂ).