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namespace OAI
noncomputable section
open scoped Manifold ContDiff
open Set
namespace UniversalCoverSplitting
universe u v
abbrev Model (n : ℕ) := Fin n → ℂ
abbrev model (n : ℕ) := 𝓘(ℂ, Model n)
structure ComplexManifold (n : ℕ) where
Carrier : Type u
[topology : TopologicalSpace Carrier]
[charted : ChartedSpace (Model n) Carrier]
[hausdorff : T2Space Carrier]
[secondCountable : SecondCountableTopology Carrier]
[complexSmooth : IsManifold (model n) ∞ Carrier]
attribute [instance] ComplexManifold.topology ComplexManifold.charted
ComplexManifold.hausdorff ComplexManifold.secondCountable ComplexManifold.complexSmooth
variable {n r₁ r₂ : ℕ}
abbrev Tangent (X : ComplexManifold.{u} n) (x : X.Carrier) := TangentSpace (model n) x
abbrev RealMetricField (X : ComplexManifold.{u} n) :=
∀ x : X.Carrier, Tangent X x →ₗ[ℝ] Tangent X x →ₗ[ℝ] ℝ
def metricInChart {X : ComplexManifold.{u} n} (g : RealMetricField X)
(c : X.Carrier) (z u v : Model n) : ℝ :=
let e := extChartAt (model n) c
let q := mfderiv (model n) (model n) e.symm z
g (e.symm z) (q u) (q v)
def fundamentalInChart {X : ComplexManifold.{u} n} (g : RealMetricField X)
(c : X.Carrier) (z u v : Model n) : ℝ :=
metricInChart g c z (Complex.I • u) v
structure KahlerMetric (X : ComplexManifold.{u} n) where
metric : RealMetricField X
symmetric : ∀ x u v, metric x u v = metric x v u
positive : ∀ x u, u ≠ 0 → 0 < metric x u u
hermitian : ∀ x u v, metric x (Complex.I • u) (Complex.I • v) = metric x u v
smooth : ∀ c u v, ContDiffOn ℝ ∞ (fun z ↦ metricInChart metric c z u v)
(extChartAt (model n) c).target
closed : ∀ c z, z ∈ (extChartAt (model n) c).target → ∀ u v w : Model n,
fderiv ℝ (fun y ↦ fundamentalInChart metric c y v w) z u +
fderiv ℝ (fun y ↦ fundamentalInChart metric c y w u) z v +
fderiv ℝ (fun y ↦ fundamentalInChart metric c y u v) z w = 0
abbrev ProjectionField (X : ComplexManifold.{u} n) :=
∀ x : X.Carrier, Tangent X x →L[ℂ] Tangent X x
def projectionInChart {X : ComplexManifold.{u} n} (P : ProjectionField X)
(c : X.Carrier) (z : Model n) : Model n →L[ℂ] Model n :=
let e := extChartAt (model n) c
(mfderiv (model n) (model n) e (e.symm z)).comp
((P (e.symm z)).comp (mfderiv (model n) (model n) e.symm z))
def HolomorphicProjection {X : ComplexManifold.{u} n} (P : ProjectionField X) : Prop :=
∀ c, DifferentiableOn ℂ (projectionInChart P c) (extChartAt (model n) c).target
def Integrable {X : ComplexManifold.{u} n} (P : ProjectionField X) : Prop :=
∀ c U, IsOpen U → U ⊆ (extChartAt (model n) c).target →
∀ V W : Model n → Model n,
DifferentiableOn ℂ V U → DifferentiableOn ℂ W U →
(∀ z ∈ U, projectionInChart P c z (V z) = V z) →
(∀ z ∈ U, projectionInChart P c z (W z) = W z) →
∀ z ∈ U, projectionInChart P c z (VectorField.lieBracket ℂ V W z) =
VectorField.lieBracket ℂ V W z
structure HolomorphicSplitting (X : ComplexManifold.{u} n) (r₁ r₂ : ℕ) where
projection : ProjectionField X
idempotent : ∀ x v, projection x (projection x v) = projection x v
holomorphic : HolomorphicProjection projection
first_rank : ∀ x, Module.finrank ℂ (LinearMap.range (projection x).toLinearMap) = r₁
second_rank : ∀ x, Module.finrank ℂ (LinearMap.ker (projection x).toLinearMap) = r₂
def complementaryProjection {X : ComplexManifold.{u} n}
(S : HolomorphicSplitting X r₁ r₂) : ProjectionField X :=
fun x ↦ ContinuousLinearMap.id ℂ (Tangent X x) - S.projection x
def LocalBiholomorphism {X : ComplexManifold.{u} n} {Z : ComplexManifold.{v} n}
(π : Z.Carrier → X.Carrier) : Prop :=
∀ z, ∃ e : OpenPartialHomeomorph Z.Carrier X.Carrier,
z ∈ e.source ∧ (∀ w ∈ e.source, e w = π w) ∧
MDifferentiableOn (model n) (model n) e e.source ∧
MDifferentiableOn (model n) (model n) e.symm e.target
structure OrdinaryUniversalCover (X : ComplexManifold.{u} n)
(Z : ComplexManifold.{v} n) where
projection : Z.Carrier → X.Carrier
covering : IsCoveringMap projection
[connected : ConnectedSpace Z.Carrier]
[simplyConnected : SimplyConnectedSpace Z.Carrier]
liftedComplexStructure : LocalBiholomorphism projection
def CompatibleProduct {X : ComplexManifold.{u} n} {Z : ComplexManifold.{v} n}
(S : HolomorphicSplitting X r₁ r₂) (π : OrdinaryUniversalCover X Z)
(Y₁ : ComplexManifold.{v} r₁) (Y₂ : ComplexManifold.{v} r₂) : Prop :=
∃ Φ : Z.Carrier ≃ₜ (Y₁.Carrier × Y₂.Carrier),
MDifferentiable (model n) ((model r₁).prod (model r₂)) Φ ∧
MDifferentiable ((model r₁).prod (model r₂)) (model n) Φ.symm ∧
(∀ z, Submodule.map
(mfderiv (model n) ((model r₁).prod (model r₂)) Φ z).toLinearMap
(Submodule.comap (mfderiv (model n) (model n) π.projection z).toLinearMap
(LinearMap.range (S.projection (π.projection z)).toLinearMap)) =
LinearMap.ker (ContinuousLinearMap.snd ℂ (Model r₁) (Model r₂)).toLinearMap) ∧
(∀ z, Submodule.map
(mfderiv (model n) ((model r₁).prod (model r₂)) Φ z).toLinearMap
(Submodule.comap (mfderiv (model n) (model n) π.projection z).toLinearMap
(LinearMap.ker (S.projection (π.projection z)).toLinearMap)) =
LinearMap.ker (ContinuousLinearMap.fst ℂ (Model r₁) (Model r₂)).toLinearMap)
open Set
open scoped Manifold ContDiff
variable {n r₁ r₂ : ℕ}
theorem main (X : ComplexManifold.{u} n) [CompactSpace X.Carrier]
[ConnectedSpace X.Carrier] (hn : 2 ≤ n) (hr₁ : 0 < r₁) (hr₂ : 0 < r₂)
(g : KahlerMetric X) (S : HolomorphicSplitting X r₁ r₂)
(h₁ : Integrable S.projection) (h₂ : Integrable (complementaryProjection S))
(Z : ComplexManifold.{v} n) (π : OrdinaryUniversalCover X Z) :
∃ (Y₁ : ComplexManifold.{v} r₁) (Y₂ : ComplexManifold.{v} r₂),
ConnectedSpace Y₁.Carrier ∧ ConnectedSpace Y₂.Carrier ∧
SimplyConnectedSpace Y₁.Carrier ∧ SimplyConnectedSpace Y₂.Carrier ∧
CompatibleProduct S π Y₁ Y₂ := by
sorry
end UniversalCoverSplitting
end
end OAI
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