Local structure of the stability-condition space

For an old slicing P, write $P(>a)$, P(≤a) for finite extension closures of the indicated phase pieces. Hom vanishing extends to these closures by the long exact Hom sequence. Grouping the finite old HN filtration at a gives a triangle $E_{>a}\to E\to E_{\le a}$. The same Hom vanishing shows uniqueness of this triangle and gives its functoriality: any map between middle objects has a unique compatible map between truncations. Shifts carry the cutoff to a+1. Thus these cutoffs give bounded t-structures. Their hearts $A_a=P((a,a+1])$ are abelian: for a map f:E→F in this interval, form its cone C and truncate C at a+1; the upper truncation shifted by −1 is ker f and the lower truncation is coker f. The octahedral axiom supplies the two image/coimage factorizations and identifies their middle object. The resulting kernel/cokernel universal properties follow from Hom vanishing against the two truncation ranges. This proves the required abelian-heart assertion from the slicing and triangulated axioms. For an interval I of length less than one, P(I) has the induced strict exact structure: its strict short exact sequences are exactly triangles whose three objects lie in P(I). Concretely embed I into a heart $A_a$. For $I=(u,v)$, cut the heart kernel to its $>u$ old-phase part and the heart cokernel to its $<v$ old-phase part. The heart kernel already has highest phase below v, and the heart cokernel already has lowest phase above u. A map from an object of P(I) to the removed ≤u kernel part, or from the removed ≥v cokernel part to an object of P(I), is zero by old Hom vanishing; hence these cuts have the required universal properties. These give the kernel and cokernel in P(I); a sequence is strict exactly when no removed truncation part occurs, equivalently when its triangle stays in P(I). The pullback of a strict epimorphism and the pushout of a strict monomorphism remain strict, by the corresponding heart diagram and the phase truncation bounds. Thus P(I) is quasi-abelian with this exact structure. In particular strict subobject compositions and inverse images of strict quotient subobjects used below exist. For a strict subobject U↪E one has ${\phi }_P^{+}(U)\le {\phi }_P^{+}(E)$; otherwise the highest old HN term of U maps nontrivially into E contrary to old Hom vanishing. Dually a strict quotient has φ⁻ at least that of E. These are the only phase-truncation bounds used below.

If $\mathcal{D}$ is the zero category, its only slicing has every phase subcategory zero. Every $Z\in {\mathrm{Hom}}_{\mathbb{Z}}(\mathrm{\Gamma },\mathbb{C})$ gives a condition, since the nonzero-object and support requirements are vacuous. By the empty-supremum conventions in [F2], both the slicing distance and every charge-perturbation seminorm are zero. Thus each basic neighbourhood is the entire charge space, and both the stability space and its source linear charge space have the indiscrete topology. Their central-charge map is a bijection and a homeomorphism. The charge space is connected (also when it is a singleton), so take $V(\mathrm{\Sigma })={\mathrm{Hom}}_{\mathbb{Z}}(\mathrm{\Gamma },\mathbb{C})$. This proves the zero-category case without choosing any nonzero semistable object. In the following steps take a nonzero category.

Fix σ=(Z,P). Choose $\eta >0$ so the old thin categories of width2η have finite strict length. Choose ${\epsilon }_0>0$ with ${\epsilon }_0<\min (1\mathrm{/}16,\eta \mathrm{/}10)$. Let $0<\epsilon <{\epsilon }_0$ and suppose $\delta =\mathrm{\mid }\mathrm{\mid }W-Z\mathrm{\mid }{\mathrm{\mid }}_{\sigma }<\sin (\pi \epsilon )$. All charges factor through the same finite-rank Γ. To obtain a strict target distance smaller than a prescribed ε, first choose an auxiliary ε' strictly between arcsin(δ)/π and that prescribed ε and run the proof with ε'; below call the auxiliary margin ε. For an old semistable object of phase φ, the estimate $\mathrm{\mid }W(E)-Z(E)\mathrm{\mid }\le \delta \mathrm{\mid }Z(E)\mathrm{\mid }<\sin (\pi \epsilon )\mathrm{\mid }Z(E)\mathrm{\mid }$ gives $\mathrm{\mid }\arg W\mathrm{/}\pi -\phi \mathrm{\mid }<\epsilon$; its argument branch is fixed by the old phase. If $A=P((a,b))$ has $b-a<1-2\epsilon$, all W-values of its nonzero objects lie in the sector (a−ε,b+ε) of width less than one. Therefore W is a strict-exact stability function there. The following finite-lattice argument supplies W-HN filtrations on such an interval. For the interval $I=(a,b)$, put a₀=(a+b)/2, $h=(b-a)\mathrm{/}2$ and $c=\cos (\pi h)>\delta$; use a₀ in place of the centre a in the following estimates. For F∈P(I), let $A_j$ be its σ-HN factors. Then $\mathrm{Re}(e^{-i\pi a_0}Z(F))\ge c{\sum }_j\mathrm{\mid }Z(A_j)\mathrm{\mid }$, $\mathrm{\mid }\mathrm{\mid }v(F)\mathrm{\mid }\mathrm{\mid }\le {\sum }_j\mathrm{\mid }\mathrm{\mid }v(A_j)\mathrm{\mid }\mathrm{\mid }\le C{\sum }_j\mathrm{\mid }Z(A_j)\mathrm{\mid }\le (C\mathrm{/}c)\mathrm{Re}(e^{-i\pi a_0}Z(F))$. The perturbation estimate on the factors gives $\mathrm{Re}(e^{-i\pi a_0}W(F))\ge (c-\mathrm{\mid }\mathrm{\mid }W-Z\mathrm{\mid }{\mathrm{\mid }}_{\sigma }){\sum }_j\mathrm{\mid }Z(A_j)\mathrm{\mid }>0$. Thus W has a continuous phase branch in that half-plane and obeys the strict see-saw property on every strict short exact sequence. If A↪E is a strict subobject in P(I), its strict quotient B lies in P(I); hence $0<\mathrm{Re}(e^{-i\pi a_0}Z(A))\le \mathrm{Re}(e^{-i\pi a_0}Z(E))$. The displayed norm bound puts v(A) in a fixed bounded subset of ${\mathrm{\Gamma }}_R$. A lattice has only finitely many points in a bounded set. Therefore the phases of W(A) over nonzero strict subobjects of a fixed E form a finite nonempty set. Choose a strict subobject A of maximal phase and, among those, maximal $\mathrm{Re}(e^{-i\pi a_0}Z(A))$. Both maxima exist because the attained lattice classes form a finite set. This avoids assuming a Jordan–Hölder length function in a quasi-abelian category. Every strict subobject of A is a strict subobject of E, so A is W-semistable. If a nonzero strict subobject C of E/A had phase at least ${\phi }_W(A)$, pull it back to a strict subobject D of E. If the inequality is strict, see-saw gives ${\phi }_W(D)>{\phi }_W(A)$, a contradiction. If equality holds, D has the same phase and strictly larger $\mathrm{Re}(e^{-i\pi a_0}Z(D))$ than A, since the strict quotient C is nonzero and has positive rotated real Z-part. This again contradicts the second maximum. Thus every subobject of E/A has smaller phase. Repeat on the strict quotient and pull back each chosen term. This process terminates: its cumulative terms are strict subobjects of the original E with strictly increasing rotated real Z-part, but only finitely many lattice classes occur among such subobjects. The resulting finite filtration has semistable factors with strictly decreasing W-phases.

If two slicings P,Q have the same central charge and $d(P,Q)<1$, they coincide. Suppose E∈P(φ) is not in Q(φ). It cannot have all Q-phases in [φ,φ+1) or in (φ−1,φ], since the positive central-charge summands on one side of the ray cannot sum to the ray unless all are on it. Hence its Q-cut at φ gives a nontrivial triangle A→E→B with A∈Q((φ,φ+1)) and B∈Q((φ−1,φ]). A cannot lie in P(≤φ), because distance $<1$ would put its old phases in (φ−1,φ], contradicting the opposite sign of the central-charge imaginary part of its Q-factors relative to φ. Choose the highest old HN term C of A of phase $>\phi$, mapping nontrivially into A. Its composite into E is zero by old Hom vanishing, so it factors through B[−1]. But Q-phases of C are $>\phi -1$ while B[−1] has Q-phases≤φ−1, a contradiction. Thus the slicings coincide. For two lifts near the same old condition the triangle inequality gives distance $<1$, so the lift is unique.

A W-semistable E in a thin category has all old phases in (ψ(E)−ε,ψ(E)+ε). Indeed its highest old HN factor A is a strict subobject and has ψ(A)≤ψ(E), while ψ(A)>φ⁺_P(E)−ε. Apply the dual argument to its lowest old HN quotient. This proves both strict inequalities. Call E enveloped by P((a,b)) if a+ε≤ψ(E)≤b−ε.

If E is enveloped by two thin categories, W-semistability is the same in both. To prove either implication, assume E is W-semistable in the starting category. Step 3.1 then places its old phases in (ψ−ε,ψ+ε); being enveloped alone would not imply this. Thus E belongs to the central category P((ψ−ε,ψ+ε)), which is contained in both envelopes. Compare this smaller category with each original envelope by varying one endpoint, always carrying the semistability hypothesis on the side from which the implication starts. For the upper-endpoint case let $B=P((a,b))\subset C=P((a,c))$, with both enveloping E. Semistability in C implies semistability in B. Conversely suppose 0→A→E→B'→0 is a destabilizing strict sequence in C. The highest-phase bound puts A in B. Truncate B' into $B_1\to B^{\mathrm{\prime }}\to B_2$ with $B_1\in P([b,c))$ and $B_2\in B$. The W-phase of $B_1$ is strictly greater than b−ε≥ψ(E). Let K be its inverse image in E. Then K is in B by the strict-subobject phase bound, and $0\to K\to E\to B_2\to 0$ is strict in B. Since K is an extension of the original destabilizing A and $B_1$, both of W-phase greater than ψ(E), it also destabilizes E in B. This is a contradiction. Changing the lower endpoint is the dual argument with strict quotients. The small central envelope P((ψ−ε,ψ+ε)) lies in both initial envelopes, so these comparisons prove independence in general. No asserted theorem about arbitrary finite-length quasi-abelian stability is used in this step.

Define Q(ψ) to consist of the enveloped W-semistables of W-phase ψ, zero, and their finite direct sums. The envelope-independence just proved makes this intrinsic. Shift by[1] sends Q(ψ) to Q(ψ+1), and W on every nonzero such sum has phase ψ. The old phase bound persists under sums. Let E∈Q(ψ₁), F∈Q(ψ₂), ψ₁>ψ₂. If ψ₁−ψ₂≥2ε, old phase bounds give Hom(E,F)=0 immediately. Otherwise put a=(ψ₁+ψ₂)/2−1/2. Both lie in the old abelian heart P((a,a+1]). If f:E→F were nonzero, let K,I,C be its heart kernel, image and cokernel. The old truncation bounds give K∈P((a,ψ₁+ε)), I∈P((ψ₁−ε,ψ₂+ε)), C∈P((ψ₂−ε,a+1]). The first exact sequence K→E→I is strict in the thin category P((a,ψ₁+ε)) which envelops E. The second I→F→C is strict in P((ψ₂−ε,a+1+ζ)) for sufficiently small ζ>0, which envelops F. Both intervals are thin because ε<1/16 and ψ₁−ψ₂<2ε. Envelope independence and semistability imply ψ₁≤ψ(I)≤ψ₂, a contradiction. Finite sums are handled componentwise. Thus the required global Hom vanishing holds.

For an old semistable E∈P(s), use the W-HN filtration in P((s−4ε,s+4ε)). Its first factor F₁ maps nontrivially into E; the old phase bound for F₁ therefore gives ψ(F₁)<s+ε. Its last factor is a nonzero quotient of E and similarly has phase greater than s−ε. Every intervening factor has phase between these extremes. All factors are enveloped by the chosen interval and hence belong to Q. Thus P(s)⊂Q((s−ε,s+ε)) where the right side is finite extension closure. In particular old phases above t+ε belong to Q(>t), and old phases at most t−ε belong to Q(<t).

Let $T_t=Q(>t)$, $F_t=Q(\le t)$, both finite extension closures. They are orthogonal by step 5.1, with $T_t[1]\subset T_t$ and $F_t[-1]\subset F_t$. For any E group its old HN filtration into the high range $(>t+\epsilon )$, middle range (t−ε,t+ε], and low range (≤t−ε). The high piece is in $T_t$ and the low piece in $F_t$ by step 6.1. The middle piece M lies in P((t−4ε,t+4ε)). Its thin W-HN factors have phases between t−2ε and t+2ε by the same first/last map argument, so are enveloped. Cut that finite filtration at t into $A_0\to M\to B_0$ with $A_0\in T_t,B_0\in F_t$. Two applications of the octahedral axiom glue this to E: first compose $A_0\to M$ into the middle-plus-low truncation, producing a quotient which is an extension of $B_0$ by the old low piece; then compose E to that quotient, whose kernel is an extension of $A_0$ by the old high piece. This gives a triangle A→E→B with $A\in T_t,B\in F_t$. Hence these are actual t-structure cutoffs, with uniqueness and functoriality by their orthogonality.

Every E has a bounded Q-phase range because its finite old HN factors have bounded old phases and step 6.1 puts them into bounded Q ranges. Subdivide that range by finitely many cutoffs of step at most ε, and use the just-constructed truncations. Each resulting piece $E_i$ lies in $T_{t_i}\cap F_{t_i+\epsilon }$. The old bound for Q-pieces and extension closure put it in $P((t_i-\epsilon ,t_i+2\epsilon ))$. Apply thin HN inside $P((t_i-3\epsilon ,t_i+4\epsilon ))$. Its first/last factor estimates make all factors enveloped, so in Q. Orthogonality forces their phases into $(t_i,t_i+\epsilon ]$: a higher first factor would have no map to $E_i\in F_{t_i+\epsilon }$, and a lower last factor no quotient from $E_i\in T_{t_i}$. Concatenate these finite filtrations in decreasing cutoff order. This gives a finite Q-HN filtration of E. Thus Q satisfies all slicing axioms.

The inclusion Q(ψ)⊂P((ψ−ε,ψ+ε)) implies d(P,Q)≤ε. To see the converse phase bound involved in this metric assertion, if E lies in Q(≤t), it lies in P(≤t+ε); if E does not lie in Q(≤t), a higher Q-HN term maps nontrivially to E and prevents E from lying in P(≤t−ε). This bounds largest phases in both directions; the smallest-phase argument is dual. The initially chosen smaller auxiliary margin makes the final prescribed distance strict. Local finiteness holds: Q((t−ρ,t+ρ)) lies in P((t−ρ−ε,t+ρ+ε)), and any strict sequence of the former is a triangle in the latter, hence strict there. Choose ρ>0 small so that the latter interval has finite old strict length. Therefore the former has finite strict length.

For a Q-semistable E of phase ψ let $A_j$ be its old HN factors. Their phases lie within ε of ψ. Hence $\mathrm{\mid }\mathrm{\mid }v(E)\mathrm{\mid }\mathrm{\mid }\le C\sum \mathrm{\mid }Z(A_j)\mathrm{\mid }$, $\mathrm{\mid }W(E)\mathrm{\mid }\ge \mathrm{Re}(e^{-i\pi \psi }W(E))\ge (\cos (\pi \epsilon )-\delta )\sum \mathrm{\mid }Z(A_j)\mathrm{\mid }$. The denominator is positive. Thus the new condition has support constant C/(cos(πε)−δ), with the same finite-rank class map. This proves the required support assertion, not just existence of an unqualified slicing.

All linear charge perturbations on Γ have finite source seminorm, because |U(vA)|≤||U||op C|Z(vA)| on old semistables. For the constructed nearby condition, the estimates above give ||U||(W,Q)≤(cos(πε)−δ)^{-1}||U||_σ. Applying the same sector estimate in reverse (with the already finite new perturbation seminorm and small margin) gives the opposite inequality up to a constant. Therefore the source linear topologies agree locally and the central-charge map is continuous. These comparisons use the common kernel on charges vanishing on v(K(D)); if the class map has redundant directions, retain the source seminorm topology exactly as in the Statement rather than silently replacing it by an inequivalent Hausdorff norm.

Take a sufficiently small source-seminorm ball V around Z and the open set U of stability conditions with charge in V and slicing distance less than ε₀/2 from P. Existence above makes U→V surjective, and uniqueness makes it injective. It is a homeomorphism: at a lift τ=(W,Q), repeat the same construction based at τ; seminorm comparisons make a sufficiently small charge neighbourhood inside V, and uniqueness identifies its nearby lifts with the original inverse. These neighbourhoods prove continuity of that inverse. V is convex in the stated source linear topology, so its lift stays in the connected component of σ. In the finite-Γ support setting one may take the source finite-seminorm linear space V(Σ) (in fact every Γ-charge has finite seminorm); the same local comparison identifies its topology over the component. Thus the central-charge map on the component is a local homeomorphism as claimed.