10.Β StabilityCondition.Defs
Module BridgelandStability.StabilityCondition.Defs β 9 declarations (Structure, Definition)
10.1.Β CategoryTheory.Triangulated.PreStabilityCondition.WithClassMap
Structure | BridgelandStability.StabilityCondition.Defs | Source | Open Issue
/-- A Bridgeland prestability condition with respect to a class map
`v : Kβ(C) β Ξ`. The central charge lives on `Ξ`, and the ordinary ambient
charge is recovered by precomposition with `v`. -/
@[informal "Definition 5.1"]
structure WithClassMap (v : Kβ C β+ Ξ) where
/-- The underlying slicing. -/
slicing : Slicing C
/-- The central charge on the class lattice `Ξ`. -/
Z : Ξ β+ β
/-- Compatibility (raw). Use `Ο.compat` instead. -/
compat' : β (Ο : β) (E : C), slicing.P Ο E β Β¬IsZero E β
β (m : β), 0 < m β§
Z (v (Kβ.of C E)) = βm * Complex.exp (β(Real.pi * Ο) * Complex.I)
10.2.Β CategoryTheory.Triangulated.PreStabilityCondition.WithClassMap.charge
Definition | BridgelandStability.StabilityCondition.Defs | Source | Open Issue
/-- The central charge evaluated at the class of `E`. This is `Z(v[E])`. -/
noncomputable abbrev WithClassMap.charge {v : Kβ C β+ Ξ}
(Ο : WithClassMap C v) (E : C) : β :=
Ο.Z (cl C v E)
10.3.Β CategoryTheory.Triangulated.slicingDist
Definition | BridgelandStability.StabilityCondition.Defs | Source | Open Issue
/-- The Bridgeland generalized metric on slicings. For slicings `sβ` and `sβ`,
this is the supremum over all nonzero objects `E` of
`max(|ΟββΊ(E) - ΟββΊ(E)|, |Οββ»(E) - Οββ»(E)|)`. -/
def slicingDist (sβ sβ : Slicing C) : ββ₯0β :=
β¨ (E : C) (hE : Β¬IsZero E),
ENNReal.ofReal (max |sβ.phiPlus C E hE - sβ.phiPlus C E hE|
|sβ.phiMinus C E hE - sβ.phiMinus C E hE|)
10.4.Β CategoryTheory.Triangulated.StabilityCondition.WithClassMap
Structure | BridgelandStability.StabilityCondition.Defs | Source | Open Issue
/-- A Bridgeland stability condition with respect to a class map `v : Kβ(C) β Ξ`.
This is the locally-finite refinement of `PreStabilityCondition.WithClassMap`. -/
@[informal "Definition 5.7"]
structure WithClassMap (v : Kβ C β+ Ξ)
extends PreStabilityCondition.WithClassMap C v where
/-- The slicing is locally finite. -/
locallyFinite : slicing.IsLocallyFinite C
10.5.Β CategoryTheory.Triangulated.stabSeminorm
Definition | BridgelandStability.StabilityCondition.Defs | Source | Open Issue
/-- The Bridgeland seminorm `βUβ_Ο` on `Hom(Ξ, β)`. For a class-map stability
condition `Ο = (Z, P)` with class map `v : Kβ(D) β Ξ` and a group homomorphism
`U : Ξ β β`, this is `sup { |U(v[E])| / |Z(v[E])| : E is Ο-semistable and nonzero }`.
When `v = id` (i.e., `Ξ = Kβ(D)`), this recovers Bridgeland's original seminorm. -/
def stabSeminorm {v : Kβ C β+ Ξ} (Ο : StabilityCondition.WithClassMap C v)
(U : Ξ β+ β) : ββ₯0β :=
β¨ (E : C) (Ο : β) (_ : Ο.slicing.P Ο E) (_ : Β¬IsZero E),
ENNReal.ofReal (βU (cl C v E)β / βΟ.charge Eβ)
10.6.Β CategoryTheory.Triangulated.basisNhd
Definition | BridgelandStability.StabilityCondition.Defs | Source | Open Issue
/-- The basis neighborhood `B_Ξ΅(Ο)` for the Bridgeland topology on `Stab_v(D)`. -/
def basisNhd {v : Kβ C β+ Ξ} (Ο : StabilityCondition.WithClassMap C v) (Ξ΅ : β) :
Set (StabilityCondition.WithClassMap C v) :=
{Ο | stabSeminorm C Ο (Ο.Z - Ο.Z) < ENNReal.ofReal (Real.sin (Real.pi * Ξ΅)) β§
slicingDist C Ο.slicing Ο.slicing < ENNReal.ofReal Ξ΅}
10.7.Β CategoryTheory.Triangulated.StabilityCondition.WithClassMap.topologicalSpace
Definition | BridgelandStability.StabilityCondition.Defs | Source | Open Issue
/-- The Bridgeland topology on `Stab_v(D)`, generated by the basis neighborhoods
`B_Ξ΅(Ο)` for all stability conditions `Ο` and all `Ξ΅ β (0, 1/8)`.
This is the BLMNPS topology: the coarsest making both the charge map `Ο β¦ Ο.Z`
and the slicing map continuous. When `v = id`, this recovers Bridgeland's original
topology on `Stab(D)`. -/
instance StabilityCondition.WithClassMap.topologicalSpace {v : Kβ C β+ Ξ} :
TopologicalSpace (StabilityCondition.WithClassMap C v) :=
TopologicalSpace.generateFrom
{U | β (Ο : StabilityCondition.WithClassMap C v) (Ξ΅ : β), 0 < Ξ΅ β§ Ξ΅ < 1 / 8 β§
U = basisNhd C Ο Ξ΅}
10.8.Β CategoryTheory.Triangulated.StabilityCondition.WithClassMap.ComponentIndex
Definition | BridgelandStability.StabilityCondition.Defs | Source | Open Issue
/-- The connected-component index set for `Stab_v(D)`. -/
abbrev ComponentIndex (v : Kβ C β+ Ξ) :=
_root_.ConnectedComponents (StabilityCondition.WithClassMap C v)
10.9.Β CategoryTheory.Triangulated.StabilityCondition.WithClassMap.Component
Definition | BridgelandStability.StabilityCondition.Defs | Source | Open Issue
/-- The type of `v`-relative stability conditions in a fixed connected component. -/
abbrev Component (v : Kβ C β+ Ξ) (cc : StabilityCondition.WithClassMap.ComponentIndex C v) :=
{Ο : StabilityCondition.WithClassMap C v // _root_.ConnectedComponents.mk Ο = cc}