Topology on Stab(D) #
Lemma 6.4 (local injectivity) and supporting topology lemmas.
The basis neighborhoods, the topology instance, connected-component types,
and the Theorem 1.2 statement are in StabilityCondition.Defs.
Two positive polar representatives whose phases differ by less than two have the same phase.
Equal central charges determine the phase of a common nonzero semistable object inside a branch of width two.
Central charge imaginary part helper #
For a σ-semistable nonzero object of phase ψ, the imaginary part of
Z(F) · exp(-iπφ) equals b · sin(π(ψ - φ)) for some b > 0. This factors
out the repeated "divide by exp, rewrite to sin" computation in the Lemma 6.4 proofs.
Lemma 6.4: Local injectivity #
One-sided phase impossibility for Lemma 6.4 (below). If σ and τ have the same
central charge, E is nonzero and τ-semistable of phase φ, and all nonzero factors
in a σ-HN filtration of E have phase strictly below φ (and above φ - 1), then
we reach a contradiction.
The proof decomposes Z(E) = Σ Z(Fᵢ) via K₀ additivity, divides by exp(iπφ),
and shows the imaginary part is both zero (since Z(E) = m · exp(iπφ) is on a ray)
and strictly negative (since each nonzero factor contributes a negative sin term).
One-sided phase impossibility for Lemma 6.4 (above). Symmetric version of
false_of_all_hn_phases_below: if all nonzero factors have phase strictly above φ
(and below φ + 1), we also reach a contradiction.
Lemma 6.4 for P-semistable objects. If σ and τ have the same central charge,
d(P, Q) < 1, E is nonzero and τ-semistable of phase φ, and E is also
σ-semistable of some phase c, then c = φ and E is σ-semistable of phase φ.
This handles the case of Lemma 6.4 where E has a trivial σ-HN filtration (single
factor), combining the metric phase bound with phase_eq_of_same_Z.
Phase straddling for Lemma 6.4 (intrinsic). If σ.Z = τ.Z, d(P, Q) < 1,
E is nonzero and τ-semistable of phase φ, then the intrinsic σ-phases satisfy
phiMinus_σ(E) ≤ φ ≤ phiPlus_σ(E).
Combined with phiPlus_σ(E), phiMinus_σ(E) ∈ (φ-1, φ+1) from
intervalProp_of_semistable_slicingDist, this pins φ between the extreme σ-phases.
The proof applies false_of_all_hn_phases_below / false_of_all_hn_phases_above to
the canonical HN filtration from exists_both_nonzero.
Cross-slicing imaginary part contradiction. If σ.Z = τ.Z, X is nonzero,
all σ-HN phases of X are in (φ, φ + 1), and all τ-HN phases are in (φ - 1, φ],
then False.
The proof decomposes Z(X) via both σ- and τ-HN filtrations. From the σ-decomposition,
Im(Z(X)/exp(iπφ)) > 0. From the τ-decomposition, Im(Z(X)/exp(iπφ)) ≤ 0.
One-sided phase impossibility (below with equality). If σ and τ have the same
central charge, E is τ-semistable at φ, and all σ-HN phases are ≤ φ with at
least one strictly below, then False. This extends false_of_all_hn_phases_below
to allow some phases equal to φ.
Auxiliary: for a nonzero object with s.gtProp φ, the intrinsic minimum phase
is > φ.
Auxiliary: for a nonzero object with s.leProp φ, the intrinsic maximum phase
is ≤ φ.
Bridgeland's Lemma 6.4 (statement and proof). If two stability conditions have the
same central charge and d(P, Q) < 1, then their slicings agree on all phases.
The proof proceeds as follows. Given E ∈ Q(φ), we show E ∈ P(φ):
- Split
Eviaσ's t-structure atφ:X → E → Y → X⟦1⟧withX ∈ P(>φ)andY ∈ P(≤φ), all σ-phases in(φ-1, φ+1). - Show
τ.phiPlus(X) ≤ φby contradiction: ifτ's top factorUofXhad phase> φ, thenHom(U, E) = 0(τ-hom-vanishing) forcesU → Xto factor throughY⟦-1⟧, butHom(U, Y⟦-1⟧) = 0too, givingU → X = 0, contradictingU ≠ 0. - Now
Xhas σ-phases> φand τ-phases≤ φ: the cross-slicing imaginary part argument (false_of_gt_and_le_phases) givesX = 0. - So
E ≅ Yhas all σ-phases≤ φ. The imaginary part argument then forces all phases= φ, makingEσ-semistable atφ.