| import OAI.Geometry.KahlerSplitting.ProductRetracts |
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| namespace OAI |
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| section |
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| noncomputable section |
| open Set Filter Topology |
| open scoped Manifold ContDiff |
| namespace UniversalCoverSplitting |
| variable {E F : Type} [NormedAddCommGroup E] [NormedSpace ℂ E] [FiniteDimensional ℂ E] |
| [NormedAddCommGroup F] [NormedSpace ℂ F] [FiniteDimensional ℂ F] |
| {M N : Type*} [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℂ,E) ∞ M] |
| [TopologicalSpace N] [ChartedSpace F N] [IsManifold 𝓘(ℂ,F) ∞ N] |
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| lemma holomorphic_local_chart_of_bijective {f : M → N} |
| (hf : MDifferentiable 𝓘(ℂ,E) 𝓘(ℂ,F) f) {p : M} |
| (hbij : Function.Bijective (mfderiv 𝓘(ℂ,E) 𝓘(ℂ,F) f p)) : |
| ∃ e : OpenPartialHomeomorph M N, p ∈ e.source ∧ EqOn e f e.source ∧ |
| e.MDifferentiable 𝓘(ℂ,E) 𝓘(ℂ,F) := by |
| let c := chartAt E p |
| let d := chartAt F (f p) |
| have hc : c.MDifferentiable 𝓘(ℂ,E) 𝓘(ℂ,E) := mdifferentiable_chart p |
| have hd : d.MDifferentiable 𝓘(ℂ,F) 𝓘(ℂ,F) := mdifferentiable_chart (f p) |
| have hpc : p ∈ c.source := mem_chart_source E p |
| have hpd : f p ∈ d.source := mem_chart_source F (f p) |
| let U : Set E := c.target ∩ (f ∘ c.symm) ⁻¹' d.source |
| have hUc : IsOpen U := (hf.continuous.comp_continuousOn c.symm.continuousOn).isOpen_inter_preimage |
| c.open_target d.open_source |
| have hpU : c p ∈ U := ⟨c.map_source hpc,by change f (c.symm (c p)) ∈ d.source; rw [c.left_inv hpc]; exact hpd⟩ |
| let H : E → F := d ∘ f ∘ c.symm |
| have hH : DifferentiableOn ℂ H U := mdifferentiableOn_iff_differentiableOn.mp |
| (hd.1.comp ((hf.mdifferentiableOn (s := univ)).comp (hc.2.mono inter_subset_left) (fun _ _ ↦ mem_univ _)) |
| (fun _ hz ↦ hz.2)) |
| have hD : Function.Bijective (fderiv ℂ H (c p)) := by |
| have hci := hc.mdifferentiableAt_symm (c.map_source hpc) |
| have hfp := hf p |
| have hdp := hd.mdifferentiableAt hpd |
| have hcomp := mfderiv_comp (I := 𝓘(ℂ,E)) (I' := 𝓘(ℂ,E)) (I'' := 𝓘(ℂ,F)) |
| (c p) (by simpa only [c.left_inv hpc] using hfp) hci |
| have hfp' : MDifferentiableAt 𝓘(ℂ,E) 𝓘(ℂ,F) f (c.symm (c p)) := by |
| simpa only [c.left_inv hpc] using hfp |
| have hfc : MDifferentiableAt 𝓘(ℂ,E) 𝓘(ℂ,F) (f ∘ c.symm) (c p) := hfp'.comp (c p) hci |
| have hdp' : MDifferentiableAt 𝓘(ℂ,F) 𝓘(ℂ,F) d ((f ∘ c.symm) (c p)) := by |
| simpa only [Function.comp_apply,c.left_inv hpc] using hdp |
| have hcomp2 := mfderiv_comp (I := 𝓘(ℂ,E)) (I' := 𝓘(ℂ,F)) (I'' := 𝓘(ℂ,F)) |
| (c p) hdp' hfc |
| rw [mfderiv_eq_fderiv] at hcomp2 |
| change fderiv ℂ H (c p)=_ at hcomp2 |
| rw [hcomp2,hcomp] |
| have hbij' : Function.Bijective (mfderiv 𝓘(ℂ,E) 𝓘(ℂ,F) f (c.symm (c p))) := by |
| rw [c.left_inv hpc] |
| exact hbij |
| have hpd' : f (c.symm (c p)) ∈ d.source := by rwa [c.left_inv hpc] |
| exact (hd.mfderiv_bijective hpd').comp (hbij'.comp (hc.symm.mfderiv_bijective (c.map_source hpc))) |
| let L : E ≃L[ℂ] F := (LinearEquiv.ofBijective (fderiv ℂ H (c p)).toLinearMap hD).toContinuousLinearEquiv |
| have hL : HasFDerivAt H (L : E →L[ℂ] F) (c p) := (hH.differentiableAt (hUc.mem_nhds hpU)).hasFDerivAt |
| let : CompleteSpace E := FiniteDimensional.complete ℂ E |
| obtain ⟨b,hb,hpb,hbU,hbd,hbdi⟩ := exists_holomorphic_ift_chart hUc |
| (finite_holomorphic_contDiffOn_one hUc hH) hpU L hL |
| have hbM : b.MDifferentiable 𝓘(ℂ,E) 𝓘(ℂ,F) := |
| ⟨mdifferentiableOn_iff_differentiableOn.mpr (hbd.differentiableOn (by norm_num)), |
| mdifferentiableOn_iff_differentiableOn.mpr (hbdi.differentiableOn (by norm_num))⟩ |
| let e := (c.trans b).trans d.symm |
| have heq (x : M) (hx : x ∈ e.source) : e x=f x := by |
| change d.symm (b (c x))=f x |
| rw [hb] |
| change d.symm (d (f (c.symm (c x))))=f x |
| rw [c.left_inv hx.1.1] |
| have hh : f x ∈ d.source := by |
| have h := (hbU hx.1.2).2 |
| change f (c.symm (c x)) ∈ d.source at h |
| rwa [c.left_inv hx.1.1] at h |
| exact d.left_inv hh |
| have hpe : p ∈ e.source := by |
| refine ⟨⟨hpc,hpb⟩,?_⟩ |
| change b (c p) ∈ d.target |
| rw [hb] |
| change d (f (c.symm (c p))) ∈ d.target |
| rw [c.left_inv hpc] |
| exact d.map_source hpd |
| exact ⟨e,hpe,heq,(hc.trans hbM).trans hd.symm⟩ |
|
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| end UniversalCoverSplitting |
| end |
| end |
|
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| section |
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| noncomputable section |
| open Set Filter Topology |
| open scoped Manifold ContDiff |
| namespace UniversalCoverSplitting |
| namespace ProductRetracts |
| variable {n r s : ℕ} {X Z : ComplexManifold n} {S : HolomorphicSplitting X r s} |
| {π : OrdinaryUniversalCover X Z} {o : Z.Carrier} |
| {h₁ : Integrable S.projection} {h₂ : Integrable (complementaryProjection S)} |
| [CompactSpace X.Carrier] (g : KahlerMetric X) (P : ProductRetracts π o h₁ h₂) |
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| lemma pair_derivative_bijective (z : Z.Carrier) : |
| Function.Bijective (mfderiv (model n) ((model r).prod (model s)) (P.pair g) z) := by |
| obtain ⟨e,hz,_⟩ := exists_coverProductChart π S h₁ h₂ z |
| have heDiff : e.chart.MDifferentiable (model n) 𝓘(ℂ,Model r × Model s) := |
| ⟨e.holomorphic,e.inverse_holomorphic⟩ |
| have hd := mfderiv_prodMk (P.q₁_holomorphic g z) (P.q₂_holomorphic g z) |
| change mfderiv (model n) ((model r).prod (model s)) (P.pair g) z = _ at hd |
| have hinj : Function.Injective (mfderiv (model n) ((model r).prod (model s)) (P.pair g) z) := by |
| apply (injective_iff_map_eq_zero _).mpr |
| intro v hv |
| rw [hd] at hv |
| have hv₁ := congrArg Prod.fst hv |
| have hv₂ := congrArg Prod.snd hv |
| have he₁ := (e.factor_kernel (P.q₁_holomorphic g) (P.q₁_factor g e) hz |
| (P.q₁_submersion g z) v).mp hv₁ |
| have he₂ := (e.swap.factor_kernel (P.q₂_holomorphic g) (P.q₂_factor g e) |
| hz (P.q₂_submersion g z) v).mp hv₂ |
| rw [e.swap_derivative hz v] at he₂ |
| have he : mfderiv (model n) 𝓘(ℂ,Model r × Model s) e.chart z v=0 := Prod.ext he₁ he₂ |
| exact heDiff.mfderiv_injective hz |
| (he.trans (map_zero _).symm) |
| refine ⟨hinj,?_⟩ |
| have hb := heDiff.mfderiv_bijective hz |
| have hfin : Module.finrank ℂ (Model n)=Module.finrank ℂ (Model r × Model s) := |
| LinearEquiv.finrank_eq (LinearEquiv.ofBijective |
| (show Model n →ₗ[ℂ] (Model r × Model s) from (mfderiv (model n) 𝓘(ℂ,Model r × Model s) e.chart z).toLinearMap) hb) |
| exact (LinearMap.injective_iff_surjective_of_finrank_eq_finrank hfin |
| (f := (show Model n →ₗ[ℂ] (Model r × Model s) from |
| (mfderiv (model n) ((model r).prod (model s)) (P.pair g) z).toLinearMap))).mp hinj |
|
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| lemma pair_local_chart (z : Z.Carrier) : |
| ∃ e : OpenPartialHomeomorph Z.Carrier (P.Y₁.Carrier × P.Y₂.Carrier), |
| z ∈ e.source ∧ EqOn e (P.pair g) e.source ∧ |
| e.MDifferentiable (model n) ((model r).prod (model s)) := by |
| let : ChartedSpace (Model r × Model s) (P.Y₁.Carrier × P.Y₂.Carrier) := |
| prodChartedSpace (Model r) P.Y₁.Carrier (Model s) P.Y₂.Carrier |
| have hmodel : (model r).prod (model s)=𝓘(ℂ,Model r × Model s) := modelWithCornersSelf_prod.symm |
| let : IsManifold 𝓘(ℂ,Model r × Model s) ∞ (P.Y₁.Carrier × P.Y₂.Carrier) := by |
| rw [←hmodel] |
| exact IsManifold.prod (I := model r) (I' := model s) P.Y₁.Carrier P.Y₂.Carrier |
| have hf : MDifferentiable (model n) 𝓘(ℂ,Model r × Model s) (P.pair g) := by |
| rw [←hmodel]; exact P.pair_holomorphic g |
| have hb : Function.Bijective (mfderiv (model n) 𝓘(ℂ,Model r × Model s) (P.pair g) z) := by |
| rw [←hmodel]; exact P.pair_derivative_bijective g z |
| obtain ⟨e,he,heq,hd⟩ := holomorphic_local_chart_of_bijective hf hb |
| exact ⟨e,he,heq,by rw [hmodel]; exact hd⟩ |
| |
| lemma pair_isLocalHomeomorph : IsLocalHomeomorph (P.pair g) := by |
| apply isLocalHomeomorph_iff_isOpenEmbedding_restrict.mpr |
| intro z |
| obtain ⟨e,he,heq,_⟩ := P.pair_local_chart g z |
| refine ⟨e.source,e.open_source.mem_nhds he,?_⟩ |
| have h : e.source.domRestrict (P.pair g)=e.source.domRestrict e := funext (fun x ↦ (heq x.property).symm) |
| rw [h] |
| exact e.isOpenEmbedding_restrict |
|
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| end ProductRetracts |
| end UniversalCoverSplitting |
| end |
| end |
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| end OAI |
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