| import OAI.Geometry.KahlerSplitting.AxisContinuation |
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| namespace OAI |
|
|
| noncomputable section |
| open Set Topology Filter Metric |
| open Set MeasureTheory Metric Filter Topology |
| open scoped ENNReal NNReal |
| open Complex |
| open Set Metric Filter Topology Complex |
| open Set Metric Filter Topology MeasureTheory Complex |
| open Set Metric Filter Topology Asymptotics |
| open scoped Interval |
| open Set Filter Topology Metric |
| open Set Filter Topology Metric Complex |
| open scoped Real |
| open Set Filter Topology |
| open scoped Manifold ContDiff |
| open Set MeasureTheory |
| open scoped ContDiff |
| open scoped Manifold ContDiff ENNReal |
| open scoped Manifold ENNReal NNReal |
| open Set Metric |
| open Set MeasureTheory Filter Topology |
| open scoped ENNReal |
| open Set |
| open Set Metric Filter Topology MeasureTheory |
| open scoped Manifold ENNReal |
| open Set Metric Filter Topology |
| open scoped Manifold |
|
|
| namespace UniversalCoverSplitting |
| variable {n r s : ℕ} {X : ComplexManifold n} {S : HolomorphicSplitting X r s} |
|
|
| lemma superDomain_axes_of_eventually {m : ℕ} (hm : 0 < m) {a : ℝ} (ha : 0 < a) (ha1 : a ≤ 1) |
| {f : ℂ × ℂ → X.Carrier} (hf : IsBox S (superDomain m a) f) |
| {α β : ℂ → X.Carrier} (hα : IsHorizontalCurve S (ball (0:ℂ) 1) α) |
| (hβ : IsVerticalCurve S (ball (0:ℂ) 1) β) |
| (he₁ : (fun z : ℂ ↦ f (z,0)) =ᶠ[𝓝 0] α) |
| (he₂ : (fun z : ℂ ↦ f (0,z)) =ᶠ[𝓝 0] β) : |
| (∀ z, (z,0) ∈ superDomain m a → f (z,0)=α z) ∧ |
| (∀ z, (0,z) ∈ superDomain m a → f (0,z)=β z) := by |
| let L₁ : ℂ →L[ℝ] ℂ × ℂ := (ContinuousLinearMap.id ℝ ℂ).prod 0 |
| let L₂ : ℂ →L[ℝ] ℂ × ℂ := (0:ℂ →L[ℝ] ℂ).prod (ContinuousLinearMap.id ℝ ℂ) |
| have hD₁ : IsOpen {z : ℂ | (z,0) ∈ superDomain m a} := (superDomain_open m a).preimage L₁.continuous |
| have hD₂ : IsOpen {z : ℂ | (0,z) ∈ superDomain m a} := (superDomain_open m a).preimage L₂.continuous |
| have hC₁ : Convex ℝ {z : ℂ | (z,0) ∈ superDomain m a} := (superDomain_convex m a).linear_preimage L₁.toLinearMap |
| have hC₂ : Convex ℝ {z : ℂ | (0,z) ∈ superDomain m a} := (superDomain_convex m a).linear_preimage L₂.toLinearMap |
| constructor |
| · exact holomorphic_manifold_eqOn_of_preconnected_of_eventuallyEq hD₁ hC₁.isPreconnected |
| (hf.horizontal_slice (superDomain_open m a) 0 (fun _ hz ↦ hz)).1 |
| (hα.1.mono (fun _ hz ↦ (superDomain_subset_bidisk hm ha1 hz).1)) |
| (show (0:ℂ) ∈ {z : ℂ | (z,0) ∈ superDomain m a} from superDomain_zero hm ha) he₁ |
| · exact holomorphic_manifold_eqOn_of_preconnected_of_eventuallyEq hD₂ hC₂.isPreconnected |
| (hf.vertical_slice (superDomain_open m a) 0 (fun _ hz ↦ hz)).1 |
| (hβ.1.mono (fun _ hz ↦ (superDomain_subset_bidisk hm ha1 hz).2)) |
| (show (0:ℂ) ∈ {z : ℂ | (0,z) ∈ superDomain m a} from superDomain_zero hm ha) he₂ |
|
|
| lemma superDomain_full_boundary [CompactSpace X.Carrier] |
| (g : KahlerMetric X) (h₁ : Integrable S.projection) |
| (h₂ : Integrable (complementaryProjection S)) {m : ℕ} (hm : 0 < m) |
| {a : ℝ} (ha : 0 < a) (ha1 : a < 1) |
| {f : ℂ × ℂ → X.Carrier} (hf : IsBox S (superDomain m a) f) |
| {α β : ℂ → X.Carrier} (hα : IsHorizontalCurve S (ball (0:ℂ) 1) α) |
| (hβ : IsVerticalCurve S (ball (0:ℂ) 1) β) |
| (he₁ : ∀ z, (z,0) ∈ superDomain m a → f (z,0)=α z) |
| (he₂ : ∀ z, (0,z) ∈ superDomain m a → f (0,z)=β z) |
| {p : ℂ × ℂ} (hp : p ∈ frontier (superDomain m a)) : |
| ∃ V : Set (ℂ × ℂ), IsOpen V ∧ p ∈ V ∧ |
| ∃ F : ℂ × ℂ → X.Carrier, IsBox S V F ∧ EqOn F f (V ∩ superDomain m a) := by |
| have hpa : superPotential m p = a := by rwa [superDomain_frontier hm ha] at hp |
| have hpD := superSublevel_subset_bidisk hm ha1 (show superPotential m p ≤ a from hpa.le) |
| rcases p with ⟨u,v⟩ |
| by_cases hv : v = 0 |
| · subst v |
| exact horizontal_axis_boundary_extension h₁ h₂ (superDomain_open m a) (superDomain_convex m a) hf |
| isOpen_ball hα hpD.1 (superDomain_axis_closure hm ha hpa.le) (fun z _ hz ↦ he₁ z hz) |
| by_cases hu : u = 0 |
| · subst u |
| exact vertical_axis_boundary_extension h₁ h₂ (superDomain_open m a) (superDomain_convex m a) hf |
| isOpen_ball hβ hpD.2 (superDomain_vertical_axis_closure hm ha hpa.le) (fun z _ hz ↦ he₂ z hz) |
| exact superDomain_mixed_boundary g h₁ h₂ hm ha hf hpa hu hv |
|
|
| end UniversalCoverSplitting |
|
|
| open Set Metric Filter Topology |
| open scoped Manifold |
| namespace UniversalCoverSplitting |
| variable {n r s : ℕ} {X : ComplexManifold n} {S : HolomorphicSplitting X r s} |
|
|
| lemma super_box_maximal {m : ℕ} (hm : 0 < m) (f₀ : ℂ × ℂ → X.Carrier) |
| (hini : ∃ a : ℝ, 0 < a ∧ ∃ f : ℂ × ℂ → X.Carrier, |
| IsBox S (superDomain m a) f ∧ f =ᶠ[𝓝 0] f₀) |
| (hforward : ∀ a : ℝ, 0 < a → a < 1 → ∀ f : ℂ × ℂ → X.Carrier, |
| IsBox S (superDomain m a) f → f =ᶠ[𝓝 0] f₀ → |
| ∃ b : ℝ, a < b ∧ ∃ F : ℂ × ℂ → X.Carrier, |
| IsBox S (superDomain m b) F ∧ F =ᶠ[𝓝 0] f₀) : |
| ∃ f : ℂ × ℂ → X.Carrier, IsBox S (superDomain m 1) f ∧ f =ᶠ[𝓝 0] f₀ := by |
| classical |
| let T : Set ℝ := {a | 0 < a ∧ a ≤ 1 ∧ ∃ f : ℂ × ℂ → X.Carrier, |
| IsBox S (superDomain m a) f ∧ f =ᶠ[𝓝 0] f₀} |
| obtain ⟨a₀,ha₀,f,hf,hfg⟩ := hini |
| have hTne : T.Nonempty := ⟨min a₀ 1,lt_min ha₀ (by norm_num),min_le_right _ _,f, |
| hf.mono (superDomain_mono m (min_le_left _ _)),hfg⟩ |
| have hTbdd : BddAbove T := ⟨1,fun _ h ↦ h.2.1⟩ |
| let A := sSup T |
| have hA : 0 < A := lt_csSup_of_lt hTbdd hTne.choose_spec hTne.choose_spec.1 |
| have hA1 : A ≤ 1 := csSup_le hTne (fun _ h ↦ h.2.1) |
| have hfchoose (a : T) : ∃ f : ℂ × ℂ → X.Carrier, |
| IsBox S (superDomain m a) f ∧ f =ᶠ[𝓝 0] f₀ := a.property.2.2 |
| choose F hF hF₀ using hfchoose |
| have hcompat (a b : T) : EqOn (F a) (F b) (superDomain m a ∩ superDomain m b) := by |
| apply holomorphic_manifold_eqOn_of_preconnected_of_eventuallyEq |
| ((superDomain_open m a).inter (superDomain_open m b)) |
| (((superDomain_convex m a).inter (superDomain_convex m b)).isPreconnected) |
| ((hF a).1.mono inter_subset_left) ((hF b).1.mono inter_subset_right) |
| (show (0:ℂ × ℂ) ∈ superDomain m a ∩ superDomain m b from |
| ⟨superDomain_zero hm a.property.1,superDomain_zero hm b.property.1⟩) |
| ((hF₀ a).trans (hF₀ b).symm) |
| obtain ⟨G,hG,hGF⟩ := glue_boxes (fun a : T ↦ superDomain m a) |
| (fun a ↦ superDomain_open m a) F hF hcompat (f₀ 0) |
| have hUnion : (⋃ a : T, superDomain m a) = superDomain m A := by |
| ext z |
| constructor |
| · intro hz |
| obtain ⟨a,ha⟩ := mem_iUnion.mp hz |
| exact superDomain_mono m (le_csSup hTbdd a.property) ha |
| · intro hz |
| obtain ⟨a,ha,haP⟩ := exists_lt_of_lt_csSup hTne hz |
| exact mem_iUnion.mpr ⟨⟨a,ha⟩,haP⟩ |
| have hGb : IsBox S (superDomain m A) G := hUnion ▸ hG |
| let a : T := ⟨hTne.choose,hTne.choose_spec⟩ |
| have hGg : G =ᶠ[𝓝 0] f₀ := by |
| apply EventuallyEq.trans _ (hF₀ a) |
| exact mem_of_superset ((superDomain_open m a).mem_nhds (superDomain_zero hm a.property.1)) (hGF a) |
| have hAeq : A = 1 := by |
| by_contra hne |
| obtain ⟨b,hb,H,hH,hH₀⟩ := hforward A hA (lt_of_le_of_ne hA1 hne) G hGb hGg |
| have hmin : min b 1 ∈ T := ⟨lt_min (hA.trans hb) (by norm_num),min_le_right _ _,H, |
| hH.mono (superDomain_mono m (min_le_left _ _)),hH₀⟩ |
| have hle : min b 1 ≤ A := le_csSup hTbdd hmin |
| have hlt : A < min b 1 := lt_min hb (lt_of_le_of_ne hA1 hne) |
| exact not_lt_of_ge hle hlt |
| exact ⟨G,hAeq ▸ hGb,hGg⟩ |
|
|
| end UniversalCoverSplitting |
|
|
| open Set Metric Filter Topology |
| namespace UniversalCoverSplitting |
|
|
| lemma superSublevel_zero {m : ℕ} (hm : 0 < m) : |
| {z | superPotential m z ≤ 0} = {(0:ℂ × ℂ)} := by |
| ext z |
| constructor |
| · intro hz |
| have h₁ : Complex.normSq z.1 ^ m = 0 := by |
| have h1 := pow_nonneg (Complex.normSq_nonneg z.1) m |
| have h2 := pow_nonneg (Complex.normSq_nonneg z.2) m |
| change Complex.normSq z.1 ^ m + Complex.normSq z.2 ^ m ≤ 0 at hz |
| linarith |
| have h₂ : Complex.normSq z.2 ^ m = 0 := by |
| have h1 := pow_nonneg (Complex.normSq_nonneg z.1) m |
| have h2 := pow_nonneg (Complex.normSq_nonneg z.2) m |
| change Complex.normSq z.1 ^ m + Complex.normSq z.2 ^ m ≤ 0 at hz |
| linarith |
| exact Prod.ext (Complex.normSq_eq_zero.mp ((pow_eq_zero_iff hm.ne').mp h₁)) (Complex.normSq_eq_zero.mp ((pow_eq_zero_iff hm.ne').mp h₂)) |
| · intro hz |
| change superPotential m z ≤ 0 |
| rw [mem_singleton_iff.mp hz,superPotential_zero hm] |
|
|
| lemma superDomain_small {m : ℕ} (hm : 0 < m) {U : Set (ℂ × ℂ)} (hU : IsOpen U) (h0 : 0 ∈ U) : |
| ∃ a : ℝ, 0 < a ∧ superDomain m a ⊆ U := by |
| apply superSublevel_widen hm hU |
| rw [superSublevel_zero hm] |
| exact singleton_subset_iff.mpr h0 |
|
|
| lemma superDomain_exhaustion : |
| (⋃ m : {m : ℕ |
| apply Subset.antisymm |
| · intro z hz |
| obtain ⟨m,hm⟩ := mem_iUnion.mp hz |
| exact superDomain_subset_bidisk m.property (le_refl 1) hm |
| intro z hz |
| have hn₁ : Complex.normSq z.1 < 1 := by |
| have hh : ‖z.1‖ < 1 := by simpa only [mem_ball,dist_zero_right] using hz.1 |
| rw [Complex.normSq_eq_norm_sq] |
| nlinarith [norm_nonneg z.1] |
| have hn₂ : Complex.normSq z.2 < 1 := by |
| have hh : ‖z.2‖ < 1 := by simpa only [mem_ball,dist_zero_right] using hz.2 |
| rw [Complex.normSq_eq_norm_sq] |
| nlinarith [norm_nonneg z.2] |
| have ht := (tendsto_pow_atTop_nhds_zero_of_lt_one (Complex.normSq_nonneg z.1) hn₁).add |
| (tendsto_pow_atTop_nhds_zero_of_lt_one (Complex.normSq_nonneg z.2) hn₂) |
| have hev : ∀ᶠ m : ℕ in atTop, superPotential m z < 1 := |
| ht (Iio_mem_nhds (show (0:ℝ)+0 < 1 by norm_num)) |
| obtain ⟨m,hm,hPm⟩ := (eventually_gt_atTop (0:ℕ) |>.and hev).exists |
| exact mem_iUnion.mpr ⟨⟨m,hm⟩,hPm⟩ |
|
|
| end UniversalCoverSplitting |
|
|
| open Set Metric Filter Topology |
| open scoped Manifold |
| namespace UniversalCoverSplitting |
| variable {n r s : ℕ} {X : ComplexManifold n} {S : HolomorphicSplitting X r s} |
|
|
| theorem disk_axes [CompactSpace X.Carrier] (g : KahlerMetric X) |
| (h₁ : Integrable S.projection) (h₂ : Integrable (complementaryProjection S)) |
| {α β : ℂ → X.Carrier} (hα : IsHorizontalCurve S (ball (0:ℂ) 1) α) |
| (hβ : IsVerticalCurve S (ball (0:ℂ) 1) β) (hbase : α 0 = β 0) : |
| ∃ F : ℂ × ℂ → X.Carrier, IsBox S (ball (0:ℂ) 1 ×ˢ ball (0:ℂ) 1) F ∧ |
| (∀ z ∈ ball (0:ℂ) 1, F (z,0)=α z) ∧ (∀ z ∈ ball (0:ℂ) 1, F (0,z)=β z) := by |
| classical |
| obtain ⟨d,hd,_,_,f₀,hf₀,hfα,hfβ⟩ := exists_axis_box_germ h₁ h₂ isOpen_ball isOpen_ball |
| (mem_ball_self (show (0:ℝ)<1 by norm_num)) (mem_ball_self (show (0:ℝ)<1 by norm_num)) hα hβ hbase |
| have hgα : (fun z : ℂ ↦ f₀ (z,0)) =ᶠ[𝓝 0] α := mem_of_superset (ball_mem_nhds 0 hd) hfα |
| have hgβ : (fun z : ℂ ↦ f₀ (0,z)) =ᶠ[𝓝 0] β := mem_of_superset (ball_mem_nhds 0 hd) hfβ |
| have hcont₁ : Tendsto (fun z : ℂ ↦ (z,(0:ℂ))) (𝓝 0) (𝓝 0) := |
| (continuous_id.prodMk continuous_const).continuousAt |
| have hcont₂ : Tendsto (fun z : ℂ ↦ ((0:ℂ),z)) (𝓝 0) (𝓝 0) := |
| (continuous_const.prodMk continuous_id).continuousAt |
| have hex (m : ℕ) (hm : 0 < m) : |
| ∃ f : ℂ × ℂ → X.Carrier, IsBox S (superDomain m 1) f ∧ f =ᶠ[𝓝 0] f₀ := by |
| apply super_box_maximal hm f₀ |
| · obtain ⟨a,ha,haU⟩ := superDomain_small hm (isOpen_ball.prod isOpen_ball) |
| (show (0:ℂ × ℂ) ∈ ball (0:ℂ) d ×ˢ ball (0:ℂ) d from ⟨mem_ball_self hd,mem_ball_self hd⟩) |
| exact ⟨a,ha,f₀,hf₀.mono haU,EventuallyEq.rfl⟩ |
| intro a ha ha1 f hf hfg |
| have haxes := superDomain_axes_of_eventually hm ha ha1.le hf hα hβ |
| ((hfg.comp_tendsto hcont₁).trans hgα) ((hfg.comp_tendsto hcont₂).trans hgβ) |
| obtain ⟨W,hW,hWsub,H,hH,hHf⟩ := box_extend_closure (superDomain_open m a) (superDomain_convex m a) hf |
| (fun _ hp ↦ superDomain_full_boundary g h₁ h₂ hm ha ha1 hf hα hβ haxes.1 haxes.2 hp) |
| obtain ⟨b,hb,hbW⟩ := superSublevel_widen hm hW (by rwa [← superDomain_closure hm ha]) |
| refine ⟨b,hb,H,hH.mono hbW,?_⟩ |
| have hHfg : H =ᶠ[𝓝 0] f := mem_of_superset ((superDomain_open m a).mem_nhds (superDomain_zero hm ha)) hHf |
| exact hHfg.trans hfg |
| choose F hF hFg using (fun m : {m : ℕ |
| have hcompat (m k : {m : ℕ |
| EqOn (F m) (F k) (superDomain m 1 ∩ superDomain k 1) := by |
| exact holomorphic_manifold_eqOn_of_preconnected_of_eventuallyEq |
| ((superDomain_open m 1).inter (superDomain_open k 1)) |
| (((superDomain_convex m 1).inter (superDomain_convex k 1)).isPreconnected) |
| ((hF m).1.mono inter_subset_left) ((hF k).1.mono inter_subset_right) |
| (show (0:ℂ × ℂ) ∈ superDomain m 1 ∩ superDomain k 1 from |
| ⟨superDomain_zero m.property (by norm_num),superDomain_zero k.property (by norm_num)⟩) |
| ((hFg m).trans (hFg k).symm) |
| obtain ⟨G,hG,hGF⟩ := glue_boxes (fun m : {m : ℕ |
| (fun m ↦ superDomain_open m 1) F hF hcompat (α 0) |
| refine ⟨G,superDomain_exhaustion ▸ hG,?_,?_⟩ |
| · intro z hz |
| have hzU : (z,0) ∈ ⋃ m : {m : ℕ |
| rw [superDomain_exhaustion] |
| exact ⟨hz,mem_ball_self (by norm_num)⟩ |
| obtain ⟨m,hm⟩ := mem_iUnion.mp hzU |
| exact (hGF m hm).trans ((superDomain_axes_of_eventually m.property (by norm_num) (le_refl 1) |
| (hF m) hα hβ ((hFg m).comp_tendsto hcont₁ |>.trans hgα) |
| ((hFg m).comp_tendsto hcont₂ |>.trans hgβ)).1 z hm) |
| · intro z hz |
| have hzU : (0,z) ∈ ⋃ m : {m : ℕ |
| rw [superDomain_exhaustion] |
| exact ⟨mem_ball_self (by norm_num),hz⟩ |
| obtain ⟨m,hm⟩ := mem_iUnion.mp hzU |
| exact (hGF m hm).trans ((superDomain_axes_of_eventually m.property (by norm_num) (le_refl 1) |
| (hF m) hα hβ ((hFg m).comp_tendsto hcont₁ |>.trans hgα) |
| ((hFg m).comp_tendsto hcont₂ |>.trans hgβ)).2 z hm) |
|
|
| lemma disk_axes_unique (h₁ : Integrable S.projection) (h₂ : Integrable (complementaryProjection S)) |
| {F G : ℂ × ℂ → X.Carrier} (hF : IsBox S (ball (0:ℂ) 1 ×ˢ ball (0:ℂ) 1) F) |
| (hG : IsBox S (ball (0:ℂ) 1 ×ˢ ball (0:ℂ) 1) G) |
| (hα : ∀ z ∈ ball (0:ℂ) 1, F (z,0)=G (z,0)) |
| (hβ : ∀ z ∈ ball (0:ℂ) 1, F (0,z)=G (0,z)) : |
| EqOn F G (ball (0:ℂ) 1 ×ˢ ball (0:ℂ) 1) := by |
| have h0 : (0:ℂ) ∈ ball (0:ℂ) 1 := mem_ball_self (by norm_num) |
| exact holomorphic_manifold_eqOn_of_preconnected_of_eventuallyEq (isOpen_ball.prod isOpen_ball) |
| (((convex_ball (0:ℂ) 1).prod (convex_ball (0:ℂ) 1)).isPreconnected) |
| hF.1 hG.1 (show (0,0) ∈ ball (0:ℂ) 1 ×ˢ ball (0:ℂ) 1 from ⟨h0,h0⟩) |
| (box_axes_germ_unique h₁ h₂ isOpen_ball isOpen_ball hF hG 0 0 h0 h0 hα hβ) |
|
|
| end UniversalCoverSplitting |
|
|
| end |
|
|
| end OAI |
|
|
