| import Mathlib | |
| 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 | |