| import OAI.Geometry.KahlerSplitting.Biholomorphism | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set | |
| open scoped Manifold ContDiff | |
| namespace UniversalCoverSplitting | |
| universe u v | |
| 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 | |
| have _ := hn | |
| have _ := hr₁ | |
| have _ := hr₂ | |
| let := π.connected | |
| let o : Z.Carrier := Classical.choice inferInstance | |
| obtain ⟨P⟩ := exists_productRetracts g h₁ h₂ π o | |
| let := P.simplyConnected_first g | |
| let := P.simplyConnected_second g | |
| refine ⟨P.Y₁,P.Y₂,inferInstance,inferInstance,inferInstance,inferInstance,?_⟩ | |
| obtain ⟨Φ,hΦ,hd,hdi⟩ := P.pair_biholomorph g | |
| refine ⟨Φ,hd,hdi,?_,?_⟩ | |
| · rw [hΦ] | |
| exact P.first_tangent g | |
| · rw [hΦ] | |
| exact P.second_tangent g | |
| end UniversalCoverSplitting | |
| end | |
| end | |
| end OAI | |