| import OAI.Geometry.KahlerSplitting.PolydiskAxes |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open Set Filter Topology |
| namespace ProductBoxTopology |
| variable {A B X C D : Type*} [TopologicalSpace A] [TopologicalSpace B] |
| [TopologicalSpace X] [TopologicalSpace C] [TopologicalSpace D] |
|
|
| def IsHorizontal (atlas : Set (OpenPartialHomeomorph X (C × D))) (f : A → X) : Prop := |
| Continuous f ∧ ∀ e ∈ atlas, |
| IsLocallyConstant (fun a : {a : A // f a ∈ e.source} ↦ (e (f a)).2) |
|
|
| def IsVertical (atlas : Set (OpenPartialHomeomorph X (C × D))) (f : A → X) : Prop := |
| Continuous f ∧ ∀ e ∈ atlas, |
| IsLocallyConstant (fun a : {a : A // f a ∈ e.source} ↦ (e (f a)).1) |
|
|
| lemma IsBox.horizontal {atlas : Set (OpenPartialHomeomorph X (C × D))} |
| {f : A × B → X} (hf : IsBox atlas f) (b : B) : |
| IsHorizontal atlas (fun a ↦ f (a,b)) := |
| ⟨hf.1.comp (continuous_id.prodMk continuous_const),fun e he ↦ (hf.2 e he).2 b⟩ |
|
|
| lemma IsBox.vertical {atlas : Set (OpenPartialHomeomorph X (C × D))} |
| {f : A × B → X} (hf : IsBox atlas f) (a : A) : |
| IsVertical atlas (fun b ↦ f (a,b)) := |
| ⟨hf.1.comp (continuous_const.prodMk continuous_id),fun e he ↦ (hf.2 e he).1 a⟩ |
|
|
| lemma IsHorizontal.precomp {atlas : Set (OpenPartialHomeomorph X (C × D))} |
| {f : A → X} (hf : IsHorizontal atlas f) {a : B → A} (ha : Continuous a) : |
| IsHorizontal atlas (f ∘ a) := by |
| refine ⟨hf.1.comp ha,fun e he ↦ ?_⟩ |
| exact (hf.2 e he).comp_continuous |
| ((ha.comp continuous_subtype_val).subtype_mk fun x ↦ x.property) |
|
|
| lemma IsVertical.precomp {atlas : Set (OpenPartialHomeomorph X (C × D))} |
| {f : A → X} (hf : IsVertical atlas f) {a : B → A} (ha : Continuous a) : |
| IsVertical atlas (f ∘ a) := by |
| refine ⟨hf.1.comp ha,fun e he ↦ ?_⟩ |
| exact (hf.2 e he).comp_continuous |
| ((ha.comp continuous_subtype_val).subtype_mk fun x ↦ x.property) |
|
|
| end ProductBoxTopology |
|
|
| open Metric |
| open scoped Manifold |
| namespace UniversalCoverSplitting |
| variable {n r s : ℕ} {X : ComplexManifold n} {S : HolomorphicSplitting X r s} |
| variable {A : Type*} [TopologicalSpace A] |
|
|
| lemma horizontal_local_parameter (h₁ : Integrable S.projection) |
| (h₂ : Integrable (complementaryProjection S)) {α : A → X.Carrier} |
| (hα : ProductBoxTopology.IsHorizontal (splittingAtlas S) α) (x : A) : |
| ∃ U : Set A, IsOpen U ∧ x ∈ U ∧ ∃ f : Polydisk r → X.Carrier, |
| IsDiskHorizontal S f ∧ ∃ a : U → Polydisk r, Continuous a ∧ ∀ y, f (a y)=α y := by |
| obtain ⟨e,hxe,he0⟩ := exists_splittingChart S h₁ h₂ (α x) |
| let W := α ⁻¹' e.chart.source |
| have hW : IsOpen W := e.chart.open_source.preimage hα.1 |
| have hlc := (hα.2 e.chart ⟨e,rfl⟩).eventually_eq (⟨x,hxe⟩ : W) |
| obtain ⟨V,hV,hsub⟩ := (mem_nhds_subtype W ⟨x,hxe⟩ _).mp hlc |
| obtain ⟨U,hUV,hU,hxU⟩ := mem_nhds_iff.mp (inter_mem hV (hW.mem_nhds hxe)) |
| have heU (y : U) : α y ∈ e.chart.source := (hUV y.property).2 |
| have hcoord (y : U) : (e.chart (α y)).2=0 := by |
| have hh := hsub (show (⟨y,heU y⟩ : W) ∈ Subtype.val ⁻¹' V from (hUV y.property).1) |
| change (e.chart (α y)).2=(e.chart (α x)).2 at hh |
| rw [he0] at hh |
| exact hh |
| let a : U → Polydisk r := fun y ↦ ⟨(e.radius : ℂ)⁻¹ • (e.chart (α y)).1, |
| unscale_mem_ball e.radius_pos (by |
| have hh := e.chart.map_source (heU y) |
| rw [e.product_range] at hh |
| exact hh.1)⟩ |
| have ha : Continuous a := by |
| apply Continuous.subtype_mk |
| exact ((e.chart.continuousOn.comp_continuous (hα.1.comp continuous_subtype_val) |
| heU).fst.const_smul ((e.radius : ℂ)⁻¹)) |
| let q : Polydisk s := ⟨0,mem_ball_self zero_lt_one⟩ |
| refine ⟨U,hU,hxU,fun p ↦ e.unitMap (p,q), |
| ⟨fun p ↦ e.unitMap (p,0),e.unitMap_block.horizontal_slice isOpen_ball isOpen_ball q.property, |
| fun _ ↦ rfl⟩,a,ha,?_⟩ |
| intro y |
| have hh := e.unitMap_coordinates (heU y) |
| simpa only [hcoord y,smul_zero] using hh |
|
|
| lemma vertical_local_parameter (h₁ : Integrable S.projection) |
| (h₂ : Integrable (complementaryProjection S)) {β : A → X.Carrier} |
| (hβ : ProductBoxTopology.IsVertical (splittingAtlas S) β) (x : A) : |
| ∃ U : Set A, IsOpen U ∧ x ∈ U ∧ ∃ f : Polydisk s → X.Carrier, |
| IsDiskVertical S f ∧ ∃ b : U → Polydisk s, Continuous b ∧ ∀ y, f (b y)=β y := by |
| obtain ⟨e,hxe,he0⟩ := exists_splittingChart S h₁ h₂ (β x) |
| let W := β ⁻¹' e.chart.source |
| have hW : IsOpen W := e.chart.open_source.preimage hβ.1 |
| have hlc := (hβ.2 e.chart ⟨e,rfl⟩).eventually_eq (⟨x,hxe⟩ : W) |
| obtain ⟨V,hV,hsub⟩ := (mem_nhds_subtype W ⟨x,hxe⟩ _).mp hlc |
| obtain ⟨U,hUV,hU,hxU⟩ := mem_nhds_iff.mp (inter_mem hV (hW.mem_nhds hxe)) |
| have heU (y : U) : β y ∈ e.chart.source := (hUV y.property).2 |
| have hcoord (y : U) : (e.chart (β y)).1=0 := by |
| have hh := hsub (show (⟨y,heU y⟩ : W) ∈ Subtype.val ⁻¹' V from (hUV y.property).1) |
| change (e.chart (β y)).1=(e.chart (β x)).1 at hh |
| rw [he0] at hh |
| exact hh |
| let b : U → Polydisk s := fun y ↦ ⟨(e.radius : ℂ)⁻¹ • (e.chart (β y)).2, |
| unscale_mem_ball e.radius_pos (by |
| have hh := e.chart.map_source (heU y) |
| rw [e.product_range] at hh |
| exact hh.2)⟩ |
| have hb : Continuous b := by |
| apply Continuous.subtype_mk |
| exact ((e.chart.continuousOn.comp_continuous (hβ.1.comp continuous_subtype_val) |
| heU).snd.const_smul ((e.radius : ℂ)⁻¹)) |
| let p : Polydisk r := ⟨0,mem_ball_self zero_lt_one⟩ |
| refine ⟨U,hU,hxU,fun q ↦ e.unitMap (p,q), |
| ⟨fun q ↦ e.unitMap (0,q),e.unitMap_block.vertical_slice isOpen_ball isOpen_ball p.property, |
| fun _ ↦ rfl⟩,b,hb,?_⟩ |
| intro y |
| have hh := e.unitMap_coordinates (heU y) |
| simpa only [hcoord y,smul_zero] using hh |
|
|
| end UniversalCoverSplitting |
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Set Filter Topology |
|
|
| namespace ClosedCoverGlue |
| variable {A X ι : Type*} [TopologicalSpace A] [TopologicalSpace X] [Finite ι] |
|
|
| theorem exists_glue (U : ι → Set A) (hU : ∀ i, IsClosed (U i)) |
| (cover : ∀ x, ∃ i, x ∈ U i) (f : ∀ i, U i → X) (hf : ∀ i, Continuous (f i)) |
| (compat : ∀ i j x (hi : x ∈ U i) (hj : x ∈ U j), f i ⟨x,hi⟩=f j ⟨x,hj⟩) : |
| ∃ F : A → X, Continuous F ∧ ∀ i (x : U i), F x=f i x := by |
| classical |
| let F : A → X := fun x ↦ f (cover x).choose ⟨x,(cover x).choose_spec⟩ |
| have herr (i : ι) (x : U i) : F x=f i x := compat _ _ _ _ _ |
| refine ⟨F,?_,herr⟩ |
| apply (locallyFinite_of_finite U).continuous (iUnion_eq_univ_iff.mpr cover) hU |
| intro i |
| rw [continuousOn_iff_continuous_domRestrict] |
| have he : (U i).domRestrict F=f i := funext fun x ↦ herr i x |
| rw [he] |
| exact hf i |
|
|
| end ClosedCoverGlue |
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Set Filter Topology |
| open scoped unitInterval |
| namespace ProductBoxTopology |
| open IntervalChain |
| variable {X C D : Type*} [TopologicalSpace X] [TopologicalSpace C] [TopologicalSpace D] |
|
|
| theorem grid_glue (atlas : Set (OpenPartialHomeomorph X (C × D))) |
| (t u : ℕ → I) (ht0 : t 0=0) (hu0 : u 0=0) (ht : Monotone t) (hu : Monotone u) |
| (N M : ℕ) (hN : t N=1) (hM : u M=1) |
| (R : ∀ i j, cell t i × cell u j → X) (hR : ∀ i j, IsBox atlas (R i j)) |
| (seamA : ∀ i j (y : cell u j), |
| R i j (⟨t (i+1),ht (Nat.le_succ i),le_rfl⟩,y)= |
| R (i+1) j (⟨t (i+1),le_rfl,ht (Nat.le_succ (i+1))⟩,y)) |
| (seamB : ∀ i j (x : cell t i), |
| R i j (x,⟨u (j+1),hu (Nat.le_succ j),le_rfl⟩)= |
| R i (j+1) (x,⟨u (j+1),le_rfl,hu (Nat.le_succ (j+1))⟩)) : |
| ∃ F : I × I → X, IsBox atlas F ∧ |
| ∀ i j (x : cell t i) (y : cell u j), F (x,y)=R i j (x,y) := by |
| classical |
| have compat (i j k l : ℕ) (p : I × I) (hi : p.1 ∈ cell t i) (hj : p.2 ∈ cell u j) |
| (hk : p.1 ∈ cell t k) (hl : p.2 ∈ cell u l) : |
| R i j (⟨p.1,hi⟩,⟨p.2,hj⟩)=R k l (⟨p.1,hk⟩,⟨p.2,hl⟩) := by |
| have hA := overlap t ht (fun i x ↦ R i j (x,⟨p.2,hj⟩)) |
| (fun i ↦ seamA i j ⟨p.2,hj⟩) i k p.1 hi hk |
| have hB := overlap u hu (fun j y ↦ R k j (⟨p.1,hk⟩,y)) |
| (fun j ↦ seamB k j ⟨p.1,hk⟩) j l p.2 hj hl |
| exact hA.trans hB |
| let K := Fin (N+1) × Fin (M+1) |
| let U : K → Set I := fun k ↦ cell t k.1 |
| let V : K → Set I := fun k ↦ cell u k.2 |
| have cover (p : I × I) : ∃ k : K, p ∈ U k ×ˢ V k := by |
| obtain ⟨i,hi,hpi⟩ := covers t ht0 ht N hN p.1 |
| obtain ⟨j,hj,hpj⟩ := covers u hu0 hu M hM p.2 |
| exact ⟨(⟨i,by omega⟩,⟨j,by omega⟩),hpi,hpj⟩ |
| let f : ∀ k : K, (U k ×ˢ V k) → X := fun k p ↦ |
| R k.1 k.2 (⟨p.val.1,p.property.1⟩,⟨p.val.2,p.property.2⟩) |
| have hf (k : K) : Continuous (f k) := (hR k.1 k.2).1.comp |
| (((continuous_fst.comp continuous_subtype_val).subtype_mk _).prodMk |
| ((continuous_snd.comp continuous_subtype_val).subtype_mk _)) |
| obtain ⟨F,hF,hFr⟩ := ClosedCoverGlue.exists_glue (fun k ↦ U k ×ˢ V k) |
| (fun _ ↦ isClosed_Icc.prod isClosed_Icc) cover f hf |
| (fun a b p ha hb ↦ compat a.1 a.2 b.1 b.2 p ha.1 ha.2 hb.1 hb.2) |
| have he (i j : ℕ) (x : cell t i) (y : cell u j) : F (x,y)=R i j (x,y) := by |
| obtain ⟨k,hk⟩ := cover (x,y) |
| exact (hFr k ⟨(x,y),hk⟩).trans (compat k.1 k.2 i j (x,y) hk.1 hk.2 x.property y.property) |
| refine ⟨F,?_,he⟩ |
| apply IsBox.of_closed_rectangles hF U V (fun _ ↦ isClosed_Icc) (fun _ ↦ isClosed_Icc) cover |
| intro k |
| have hh : (fun p : U k × V k ↦ F (p.1,p.2))=R k.1 k.2 := |
| funext fun p ↦ he k.1 k.2 p.1 p.2 |
| rw [hh] |
| exact hR k.1 k.2 |
|
|
| end ProductBoxTopology |
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Set Metric Filter Topology |
| open scoped Manifold unitInterval |
| namespace UniversalCoverSplitting |
| open IntervalChain |
| variable {n r s : ℕ} {X : ComplexManifold n} {S : HolomorphicSplitting X r s} |
|
|
| theorem path_axes [CompactSpace X.Carrier] (g : KahlerMetric X) |
| (h₁ : Integrable S.projection) (h₂ : Integrable (complementaryProjection S)) |
| {α β : I → X.Carrier} (hα : ProductBoxTopology.IsHorizontal (splittingAtlas S) α) |
| (hβ : ProductBoxTopology.IsVertical (splittingAtlas S) β) (hbase : α 0=β 0) : |
| ∃ R : I × I → X.Carrier, ProductBoxTopology.IsBox (splittingAtlas S) R ∧ |
| (∀ x, R (x,0)=α x) ∧ (∀ y, R (0,y)=β y) := by |
| obtain ⟨t,ht0,ht,⟨N,hN⟩,A,hA,a,ha,eA⟩ := |
| PathSubdivision.parameterize α (IsDiskHorizontal S) (horizontal_local_parameter h₁ h₂ hα) |
| obtain ⟨u,hu0,hu,⟨M,hM⟩,B,hB,b,hb,eB⟩ := |
| PathSubdivision.parameterize β (IsDiskVertical S) (vertical_local_parameter h₁ h₂ hβ) |
| let lo : ℕ → Polydisk r := fun i ↦ a i ⟨t i,le_rfl,ht (Nat.le_succ i)⟩ |
| let hi : ℕ → Polydisk r := fun i ↦ a i ⟨t (i+1),ht (Nat.le_succ i),le_rfl⟩ |
| let bot : ℕ → Polydisk s := fun j ↦ b j ⟨u j,le_rfl,hu (Nat.le_succ j)⟩ |
| let top : ℕ → Polydisk s := fun j ↦ b j ⟨u (j+1),hu (Nat.le_succ j),le_rfl⟩ |
| have joinA (i : ℕ) : A i (hi i)=A (i+1) (lo (i+1)) := |
| (eA i ⟨t (i+1),ht (Nat.le_succ i),le_rfl⟩).trans |
| (eA (i+1) ⟨t (i+1),le_rfl,ht (Nat.le_succ (i+1))⟩).symm |
| have joinB (j : ℕ) : B j (top j)=B (j+1) (bot (j+1)) := |
| (eB j ⟨u (j+1),hu (Nat.le_succ j),le_rfl⟩).trans |
| (eB (j+1) ⟨u (j+1),le_rfl,hu (Nat.le_succ (j+1))⟩).symm |
| have corner : A 0 (lo 0)=B 0 (bot 0) := by |
| change A 0 (a 0 _)=B 0 (b 0 _) |
| rw [eA,eB] |
| change α (t 0)=β (u 0) |
| simpa only [ht0,hu0] using hbase |
| obtain ⟨F,hF,hFa,hFb,hFh,hFv⟩ := |
| holomorphic_grid g h₁ h₂ A hA lo hi joinA B hB bot top joinB corner |
| let tile (i j : ℕ) : cell t i × cell u j → X.Carrier := |
| fun p ↦ F i j (a i p.1,b j p.2) |
| have htile (i j : ℕ) : ProductBoxTopology.IsBox (splittingAtlas S) (tile i j) := |
| (hF i j).plaqueBox.precomp (ha i) (hb j) |
| obtain ⟨R,hR,hRt⟩ := ProductBoxTopology.grid_glue (splittingAtlas S) t u ht0 hu0 ht hu |
| N M (hN N le_rfl) (hM M le_rfl) tile htile |
| (fun i j y ↦ hFh i j (b j y)) (fun i j x ↦ hFv i j (a i x)) |
| refine ⟨R,hR,?_,?_⟩ |
| · intro x |
| obtain ⟨i,_,hxi⟩ := covers t ht0 ht N (hN N le_rfl) x |
| let xx : cell t i := ⟨x,hxi⟩ |
| let yy : cell u 0 := ⟨u 0,le_rfl,hu (Nat.le_succ 0)⟩ |
| calc R (x,0) = tile i 0 (xx,yy) := by simpa only [yy,hu0] using hRt i 0 xx yy |
| _ = A i (a i xx) := hFa i (a i xx) |
| _ = α x := eA i xx |
| · intro y |
| obtain ⟨j,_,hyj⟩ := covers u hu0 hu M (hM M le_rfl) y |
| let xx : cell t 0 := ⟨t 0,le_rfl,ht (Nat.le_succ 0)⟩ |
| let yy : cell u j := ⟨y,hyj⟩ |
| calc R (0,y) = tile 0 j (xx,yy) := by simpa only [xx,ht0] using hRt 0 j xx yy |
| _ = B j (b j yy) := hFb j (b j yy) |
| _ = β y := eB j yy |
|
|
| end UniversalCoverSplitting |
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Set Metric Filter Topology |
| open scoped Manifold unitInterval |
| namespace UniversalCoverSplitting |
| open IntervalChain |
| variable {n r s : ℕ} {X : ComplexManifold n} {S : HolomorphicSplitting X r s} |
| variable {A : Type*} [TopologicalSpace A] |
|
|
| lemma continuous_local_parameter (h₁ : Integrable S.projection) |
| (h₂ : Integrable (complementaryProjection S)) {γ : A → X.Carrier} |
| (hγ : Continuous γ) (x : A) : |
| ∃ U : Set A, IsOpen U ∧ x ∈ U ∧ ∃ f : Polydisk r × Polydisk s → X.Carrier, |
| IsDiskTile S f ∧ ∃ a : U → Polydisk r × Polydisk s, Continuous a ∧ |
| ∀ y, f (a y)=γ y := by |
| obtain ⟨e,hxe,_⟩ := exists_splittingChart S h₁ h₂ (γ x) |
| let U := γ ⁻¹' e.chart.source |
| have heU (y : U) : γ y ∈ e.chart.source := y.property |
| have hcoord (y : U) : e.chart (γ y) ∈ ball 0 e.radius ×ˢ ball 0 e.radius := by |
| simpa only [←e.product_range] using e.chart.map_source (heU y) |
| let a : U → Polydisk r × Polydisk s := fun y ↦ |
| (⟨(e.radius : ℂ)⁻¹ • (e.chart (γ y)).1,unscale_mem_ball e.radius_pos (hcoord y).1⟩, |
| ⟨(e.radius : ℂ)⁻¹ • (e.chart (γ y)).2,unscale_mem_ball e.radius_pos (hcoord y).2⟩) |
| have hc := e.chart.continuousOn.comp_continuous (hγ.comp continuous_subtype_val) heU |
| have ha : Continuous a := |
| ((hc.fst.const_smul ((e.radius : ℂ)⁻¹)).subtype_mk _).prodMk |
| ((hc.snd.const_smul ((e.radius : ℂ)⁻¹)).subtype_mk _) |
| exact ⟨U,e.chart.open_source.preimage hγ,hxe,fun p ↦ e.unitMap (p.1,p.2), |
| ⟨e.unitMap,e.unitMap_block,fun _ ↦ rfl⟩,a,ha,fun y ↦ e.unitMap_coordinates (heU y)⟩ |
| |
| theorem path_diagonal [CompactSpace X.Carrier] (g : KahlerMetric X) |
| (h₁ : Integrable S.projection) (h₂ : Integrable (complementaryProjection S)) |
| {γ : I → X.Carrier} (hγ : Continuous γ) : |
| ∃ R : I × I → X.Carrier, ProductBoxTopology.IsBox (splittingAtlas S) R ∧ |
| ∀ t, R (t,t)=γ t := by |
| obtain ⟨t,ht0,ht,⟨N,hN⟩,D,hD,a,ha,eD⟩ := |
| PathSubdivision.parameterize γ (IsDiskTile (d:=r) (e:=s) S) |
| (continuous_local_parameter h₁ h₂ hγ) |
| let lo : ℕ → Polydisk r := fun i ↦ (a i ⟨t i,le_rfl,ht (Nat.le_succ i)⟩).1 |
| let hi : ℕ → Polydisk r := fun i ↦ (a i ⟨t (i+1),ht (Nat.le_succ i),le_rfl⟩).1 |
| let bot : ℕ → Polydisk s := fun i ↦ (a i ⟨t i,le_rfl,ht (Nat.le_succ i)⟩).2 |
| let top : ℕ → Polydisk s := fun i ↦ (a i ⟨t (i+1),ht (Nat.le_succ i),le_rfl⟩).2 |
| have join (i : ℕ) : D i (hi i,top i)=D (i+1) (lo (i+1),bot (i+1)) := |
| (eD i ⟨t (i+1),ht (Nat.le_succ i),le_rfl⟩).trans |
| (eD (i+1) ⟨t (i+1),le_rfl,ht (Nat.le_succ (i+1))⟩).symm |
| obtain ⟨F,hF,hFD,hFh,hFv⟩ := RectangleGrid.diagonal_grid |
| {a | IsDiskHorizontal S a} {b | IsDiskVertical S b} {f | IsDiskTile S f} |
| (fun _ h q ↦ h.horizontal q) (fun _ h p ↦ h.vertical p) |
| (fun _ hA _ hB p q hc ↦ disk_fill g h₁ h₂ hA hB p q hc) |
| D hD lo hi bot top join |
| let tile (i j : ℕ) : cell t i × cell t j → X.Carrier := |
| fun p ↦ F i j ((a i p.1).1,(a j p.2).2) |
| have htile (i j : ℕ) : ProductBoxTopology.IsBox (splittingAtlas S) (tile i j) := |
| (hF i j).plaqueBox.precomp (ha i).fst (ha j).snd |
| obtain ⟨R,hR,hRt⟩ := ProductBoxTopology.grid_glue (splittingAtlas S) t t ht0 ht0 ht ht |
| N N (hN N le_rfl) (hN N le_rfl) tile htile |
| (fun i j y ↦ hFh i j (a j y).2) (fun i j x ↦ hFv i j (a i x).1) |
| refine ⟨R,hR,fun x ↦ ?_⟩ |
| obtain ⟨i,_,hxi⟩ := covers t ht0 ht N (hN N le_rfl) x |
| let xx : cell t i := ⟨x,hxi⟩ |
| calc R (x,x)=tile i i (xx,xx) := hRt i i xx xx |
| _ = D i (a i xx) := congrFun (hFD i) (a i xx) |
| _ = γ x := eD i xx |
| |
| end UniversalCoverSplitting |
| end |
| end |
| |
| end OAI |
| |
