Stability Seminorm #
Metric properties of slicingDist, seminorm properties of stabilitySeminorm,
the finite seminorm subgroup, and sector bound / comparison lemmas.
The definitions of slicingDist and stabilitySeminorm are in StabilityCondition.Defs.
A nearby semistable object has both intrinsic phases in the corresponding open phase window.
Projecting a finite collection of rays onto the bisector of a sector bounds their total length by the norm of their sum.
The subspace V(σ) of group homomorphisms with finite seminorm.
This is an AddSubgroup of Λ →+ ℂ consisting of those U for which
‖U‖_σ < ∞. On a connected component of Stab(D), this subspace is independent
of the chosen σ (by Lemma 6.2).
Equations
- One or more equations did not get rendered due to their size.
Instances For
Sector bound (Lemma 6.2 core) #
Sector bound (Lemma 6.2 core). For a stability condition σ = (Z, P) and a group
homomorphism U : Λ →+ ℂ, if every semistable factor satisfies
‖U([A])‖ ≤ M · ‖Z([A])‖, then the bound extends to any object E with narrow HN width:
‖U([E])‖ ≤ (M / cos(πη/2)) · ‖Z([E])‖, where η bounds the HN phase width.
The proof decomposes E via its HN filtration (a PostnikovTower with phase data),
applies K₀ additivity, the pointwise seminorm bound on factors, and the
sector estimate norm_sum_exp_ge_cos_mul_sum.
Sector bound using intrinsic phase width phiPlus - phiMinus.
Node 6.2d: For a τ-semistable nonzero object E of phase φ with
d(σ, τ) < ε < 1/2, the σ-HN width of E is less than 2ε, so the sector
bound applies. Combined with the seminorm bound on W - Z, this controls
‖Z([E])‖ by ‖W([E])‖ (where W = τ.Z and Z = σ.Z).
Seminorm comparison (Lemma 6.2 consequence) #
For σ-semistable nonzero A, the ratio ‖U(A)‖/‖Z(A)‖ is bounded by the seminorm.
Seminorm comparison for same central charge (Lemma 6.2 consequence).
If σ and τ have the same central charge and d(P, Q) < ε < 1/2, then
‖U‖_τ < ⊤ whenever ‖U‖_σ < ⊤. This shows V(σ) ⊆ V(τ), and by symmetry
V(σ) = V(τ) for nearby stability conditions with the same central charge.
Proposition 6.3 (same Z case). If σ and τ have the same central charge
and d(P, Q) < ε < 1/2, then the finite seminorm subgroups agree: V(σ) = V(τ).
This is the key ingredient for showing V(Σ) is well-defined on connected components.
General Lemma 6.2 (explicit seminorm bound). If d(P, Q) < ε < 1/2
and ‖τ.Z - σ.Z‖_σ < cos(πε), then for any U with finite σ-seminorm, the
τ-seminorm is bounded: ‖U‖_τ ≤ ‖U‖_σ / (cos(πε) - ‖τ.Z - σ.Z‖_σ).
This is the quantitative core of the seminorm comparison, used for both
stabilitySeminorm_lt_top_of_near and the reverse direction of Proposition 6.3.
General Lemma 6.2 (seminorm finiteness comparison). If d(P, Q) < ε < 1/2
and ‖τ.Z - σ.Z‖_σ < cos(πε), then V(σ) ⊆ V(τ).
Proposition 6.3 (V(Σ) equality for nearby stability conditions). If
τ ∈ B_ε(σ) with ε < 1/8, then V(σ) = V(τ). This shows the subgroup V
is locally constant on Stab(D), hence constant on connected components.
The forward direction uses stabilitySeminorm_le_of_near. The reverse direction uses
the explicit bound to show ‖σ.Z - τ.Z‖_τ < cos(πε), then applies the
comparison with σ and τ swapped. The key inequality is
sin(πε) · (1 + cos(πε)) < cos²(πε) for ε < 1/8, proved via
sin(x) < x and x² + 2x < 1 for x = πε < π/8 < √2 - 1.