lean/OAI/Geometry/KahlerSplitting/PolydiskAxes.lean

import OAI.Geometry.KahlerSplitting.ParameterGrid

namespace OAI

section

noncomputable section
open Set Metric
namespace UnitPolydisk

def discShift (a z : ℂ) : ℂ := (a+z)/(1+star a*z)

lemma denominator_ne_zero {a z : ℂ} (ha : ‖a‖ < 1) (hz : ‖z‖ < 1) : 1+star a*z ≠ 0 := by
  intro he
  have hn : ‖star a*z‖ < 1 := by
    rw [norm_mul,norm_star]
    nlinarith [norm_nonneg a,norm_nonneg z]
  have hh : star a*z = -1 := by linear_combination he
  rw [hh,norm_neg,norm_one] at hn
  exact (lt_irrefl 1) hn

lemma normSq_difference (a z : ℂ) :
    Complex.normSq (1+star a*z) - Complex.normSq (a+z) =
      (1-Complex.normSq a)*(1-Complex.normSq z) := by
  simp only [Complex.star_def,Complex.normSq_apply,Complex.add_re,Complex.add_im,Complex.mul_re,
    Complex.mul_im,Complex.conj_re,Complex.conj_im,Complex.one_re,Complex.one_im]
  ring

lemma discShift_mem {a z : ℂ} (ha : ‖a‖ < 1) (hz : ‖z‖ < 1) : ‖discShift a z‖ < 1 := by
  have hden := denominator_ne_zero ha hz
  have ha2 : Complex.normSq a < 1 := by rw [Complex.normSq_eq_norm_sq]; nlinarith [norm_nonneg a]
  have hz2 : Complex.normSq z < 1 := by rw [Complex.normSq_eq_norm_sq]; nlinarith [norm_nonneg z]
  have hn : Complex.normSq (a+z) < Complex.normSq (1+star a*z) := by
    have hh := normSq_difference a z
    have hp := mul_pos (sub_pos.mpr ha2) (sub_pos.mpr hz2)
    linarith
  rw [Complex.normSq_eq_norm_sq,Complex.normSq_eq_norm_sq] at hn
  rw [discShift,norm_div,div_lt_one (norm_pos_iff.mpr hden)]
  nlinarith [norm_nonneg (a+z),norm_nonneg (1+star a*z)]

lemma discShift_zero (a : ℂ) : discShift a 0=a := by simp [discShift]

lemma discShift_neg_self {a : ℂ}  : discShift (-a) a=0 := by
  simp [discShift]

lemma discShift_inverse {a z : ℂ} (ha : ‖a‖ < 1) (hz : ‖z‖ < 1) :
    discShift (-a) (discShift a z)=z := by
  have hden := denominator_ne_zero ha hz
  have ha' : ‖-a‖ < 1 := by simpa using ha
  have hden' := denominator_ne_zero ha' (discShift_mem ha hz)
  change (-a+discShift a z)/(1+star (-a)*discShift a z)=z
  apply (div_eq_iff hden').mpr
  simp only [discShift,star_neg,div_eq_mul_inv]
  have hcancel : (1+star a*z)*(1+star a*z)⁻¹=1 := mul_inv_cancel₀ hden
  linear_combination (a+z)*hcancel

lemma differentiableOn_discShift {a : ℂ} (ha : ‖a‖ < 1) :
    DifferentiableOn ℂ (discShift a) (ball (0:ℂ) 1) := by
  exact ((differentiable_const a).add differentiable_id).differentiableOn.div
    ((differentiable_const 1).add ((differentiable_const (star a)).mul differentiable_id)).differentiableOn
    (fun z hz ↦ denominator_ne_zero ha (mem_ball_zero_iff.mp hz))

def shift {d : ℕ} (a z : Fin d → ℂ) : Fin d → ℂ := fun i ↦ discShift (a i) (z i)

lemma norm_coord_lt_one {d : ℕ} {a : Fin d → ℂ} (ha : a ∈ ball 0 1) (i : Fin d) : ‖a i‖ < 1 :=
  (norm_le_pi_norm a i).trans_lt (mem_ball_zero_iff.mp ha)

lemma shift_mem {d : ℕ} {a z : Fin d → ℂ} (ha : a ∈ ball 0 1) (hz : z ∈ ball 0 1) :
    shift a z ∈ ball 0 1 := by
  rw [mem_ball_zero_iff,pi_norm_lt_iff (by norm_num : (0:ℝ)<1)]
  exact fun i ↦ discShift_mem (norm_coord_lt_one ha i) (norm_coord_lt_one hz i)

lemma shift_zero {d : ℕ} (a : Fin d → ℂ) : shift a 0=a := by ext i; exact discShift_zero (a i)

lemma shift_inverse {d : ℕ} {a z : Fin d → ℂ} (ha : a ∈ ball 0 1) (hz : z ∈ ball 0 1) :
    shift (-a) (shift a z)=z := by
  ext i
  exact discShift_inverse (norm_coord_lt_one ha i) (norm_coord_lt_one hz i)

lemma differentiableOn_shift {d : ℕ} {a : Fin d → ℂ} (ha : a ∈ ball 0 1) :
    DifferentiableOn ℂ (shift a) (ball 0 1) := by
  apply differentiableOn_pi.mpr
  intro i
  exact (differentiableOn_discShift (norm_coord_lt_one ha i)).comp
    (differentiable_apply i).differentiableOn
    (fun z hz ↦ mem_ball_zero_iff.mpr (norm_coord_lt_one hz i))

end UnitPolydisk
end
end

section

noncomputable section
open Set Metric Filter Topology
open scoped Manifold
namespace UniversalCoverSplitting
variable {n r s d e : ℕ} {X : ComplexManifold n} {S : HolomorphicSplitting X r s}

theorem polydisk_axes [CompactSpace X.Carrier] (g : KahlerMetric X)
    (h₁ : Integrable S.projection) (h₂ : Integrable (complementaryProjection S))
    {α : Model d → X.Carrier} {β : Model e → X.Carrier}
    (hα : IsHorizontalMap S (ball 0 1) α) (hβ : IsVerticalMap S (ball 0 1) β)
    {p₀ : Model d} {q₀ : Model e} (hp₀ : p₀ ∈ ball 0 1) (hq₀ : q₀ ∈ ball 0 1)
    (hbase : α p₀=β q₀) :
    ∃ H : Model d × Model e → X.Carrier, IsBlockBox S (ball 0 1 ×ˢ ball 0 1) H ∧
      (∀ p ∈ ball 0 1, H (p,q₀)=α p) ∧ (∀ q ∈ ball 0 1, H (p₀,q)=β q) := by
  have hpN : -p₀ ∈ ball (0:Model d) 1 := by simpa using hp₀
  have hqN : -q₀ ∈ ball (0:Model e) 1 := by simpa using hq₀
  have hα' := hα.comp isOpen_ball isOpen_ball (UnitPolydisk.differentiableOn_shift hp₀)
    (fun _ h ↦ UnitPolydisk.shift_mem hp₀ h)
  have hβ' := hβ.comp isOpen_ball isOpen_ball (UnitPolydisk.differentiableOn_shift hq₀)
    (fun _ h ↦ UnitPolydisk.shift_mem hq₀ h)
  obtain ⟨F,hF,hFα,hFβ⟩ := ball_axes_zero g h₁ h₂ hα' hβ'
    (by simpa only [Function.comp_apply,UnitPolydisk.shift_zero] using hbase)
  let a := UnitPolydisk.shift (-p₀)
  let b := UnitPolydisk.shift (-q₀)
  have hap₀ : a p₀=0 := by ext i; exact UnitPolydisk.discShift_neg_self
  have hbq₀ : b q₀=0 := by ext i; exact UnitPolydisk.discShift_neg_self
  refine ⟨F ∘ Prod.map a b,hF.comp_parameters isOpen_ball isOpen_ball isOpen_ball isOpen_ball
    (UnitPolydisk.differentiableOn_shift hpN) (UnitPolydisk.differentiableOn_shift hqN)
    (fun _ h ↦ UnitPolydisk.shift_mem hpN h) (fun _ h ↦ UnitPolydisk.shift_mem hqN h),?_,?_⟩
  · intro p hp
    change F (a p,b q₀)=α p
    rw [hbq₀,hFα (a p) (UnitPolydisk.shift_mem hpN hp)]
    have hh := UnitPolydisk.shift_inverse hpN hp
    simpa only [neg_neg,Function.comp_apply] using congrArg α hh
  · intro q hq
    change F (a p₀,b q)=β q
    rw [hap₀,hFβ (b q) (UnitPolydisk.shift_mem hqN hq)]
    have hh := UnitPolydisk.shift_inverse hqN hq
    simpa only [neg_neg,Function.comp_apply] using congrArg β hh

lemma IsBlockBox.unique_polydisk {F G : Model d × Model e → X.Carrier}
    (h₁ : Integrable S.projection) (h₂ : Integrable (complementaryProjection S))
    (hF : IsBlockBox S (ball 0 1 ×ˢ ball 0 1) F) (hG : IsBlockBox S (ball 0 1 ×ˢ ball 0 1) G)
    {p₀ : Model d} {q₀ : Model e} (hp₀ : p₀ ∈ ball 0 1) (hq₀ : q₀ ∈ ball 0 1)
    (hb : ∀ p ∈ ball 0 1, F (p,q₀)=G (p,q₀))
    (hl : ∀ q ∈ ball 0 1, F (p₀,q)=G (p₀,q)) : EqOn F G (ball 0 1 ×ˢ ball 0 1) := by
  let : PathConnectedSpace (ball (0:Model d) 1) :=
    isPathConnected_iff_pathConnectedSpace.mp ((convex_ball (0:Model d) 1).isPathConnected ⟨p₀,hp₀⟩)
  let : PathConnectedSpace (ball (0:Model e) 1) :=
    isPathConnected_iff_pathConnectedSpace.mp ((convex_ball (0:Model e) 1).isPathConnected ⟨q₀,hq₀⟩)
  have hcover : ∀ x : X.Carrier, ∃ c ∈ splittingAtlas S, x ∈ c.source := by
    intro x
    obtain ⟨c,hc,_⟩ := exists_splittingChart S h₁ h₂ x
    exact ⟨c.chart,⟨c,rfl⟩,hc⟩
  have he := ProductBoxTopology.unique_connected (splittingAtlas S) hcover
    (hF.plaqueBox isOpen_ball isOpen_ball) (hG.plaqueBox isOpen_ball isOpen_ball)
    (⟨p₀,hp₀⟩ : ball (0:Model d) 1) (⟨q₀,hq₀⟩ : ball (0:Model e) 1)
    (fun p ↦ hb p p.property) (fun q ↦ hl q q.property)
  intro z hz
  exact congrFun he (⟨z.1,hz.1⟩,⟨z.2,hz.2⟩)

end UniversalCoverSplitting
end
end

section

noncomputable section
open Set Metric Filter Topology
open scoped Manifold
namespace UniversalCoverSplitting
variable {n r s d e : ℕ} {X : ComplexManifold n} {S : HolomorphicSplitting X r s}

def IsDiskHorizontal (S : HolomorphicSplitting X r s) (f : Polydisk d → X.Carrier) : Prop :=
  ∃ g : Model d → X.Carrier, IsHorizontalMap S (Polydisk d) g ∧ ∀ z : Polydisk d, g z=f z

def IsDiskVertical (S : HolomorphicSplitting X r s) (f : Polydisk e → X.Carrier) : Prop :=
  ∃ g : Model e → X.Carrier, IsVerticalMap S (Polydisk e) g ∧ ∀ z : Polydisk e, g z=f z

def IsDiskTile (S : HolomorphicSplitting X r s) (f : Polydisk d × Polydisk e → X.Carrier) : Prop :=
  ∃ g : Model d × Model e → X.Carrier, IsBlockBox S (Polydisk d ×ˢ Polydisk e) g ∧
    ∀ z : Polydisk d × Polydisk e, g (z.1,z.2)=f z

lemma IsDiskTile.horizontal {f : Polydisk d × Polydisk e → X.Carrier}
    (hf : IsDiskTile S f) (q : Polydisk e) : IsDiskHorizontal S (fun p ↦ f (p,q)) := by
  obtain ⟨g,hg,he⟩ := hf
  exact ⟨fun p ↦ g (p,q),hg.horizontal_slice isOpen_ball isOpen_ball q.property,
    fun p ↦ he (p,q)⟩

lemma IsDiskTile.vertical {f : Polydisk d × Polydisk e → X.Carrier}
    (hf : IsDiskTile S f) (p : Polydisk d) : IsDiskVertical S (fun q ↦ f (p,q)) := by
  obtain ⟨g,hg,he⟩ := hf
  exact ⟨fun q ↦ g (p,q),hg.vertical_slice isOpen_ball isOpen_ball p.property,
    fun q ↦ he (p,q)⟩

lemma IsDiskTile.plaqueBox {f : Polydisk d × Polydisk e → X.Carrier}
    (hf : IsDiskTile S f) : ProductBoxTopology.IsBox (splittingAtlas S) f := by
  obtain ⟨g,hg,he⟩ := hf
  have heq : f=(fun z : Polydisk d × Polydisk e ↦ g (z.1,z.2)) := funext fun z ↦ (he z).symm
  rw [heq]
  exact hg.plaqueBox isOpen_ball isOpen_ball

lemma disk_fill [CompactSpace X.Carrier] (g : KahlerMetric X)
    (h₁ : Integrable S.projection) (h₂ : Integrable (complementaryProjection S))
    {a : Polydisk d → X.Carrier} {b : Polydisk e → X.Carrier}
    (ha : IsDiskHorizontal S a) (hb : IsDiskVertical S b)
    (p : Polydisk d) (q : Polydisk e) (hc : a p=b q) :
    ∃ f, IsDiskTile S f ∧ (∀ x, f (x,q)=a x) ∧ (∀ y, f (p,y)=b y) := by
  obtain ⟨A,hA,eA⟩ := ha
  obtain ⟨B,hB,eB⟩ := hb
  obtain ⟨F,hF,hFa,hFb⟩ := polydisk_axes g h₁ h₂ hA hB p.property q.property
    (by rw [eA p,eB q]; exact hc)
  refine ⟨fun z ↦ F (z.1,z.2),⟨F,hF,fun _ ↦ rfl⟩,?_,?_⟩
  · intro x; exact (hFa x x.property).trans (eA x)
  · intro y; exact (hFb y y.property).trans (eB y)

theorem holomorphic_grid [CompactSpace X.Carrier] (g : KahlerMetric X)
    (h₁ : Integrable S.projection) (h₂ : Integrable (complementaryProjection S))
    (a : ℕ → Polydisk d → X.Carrier) (ha : ∀ i, IsDiskHorizontal S (a i))
    (lo hi : ℕ → Polydisk d) (joinA : ∀ i, a i (hi i)=a (i+1) (lo (i+1)))
    (b : ℕ → Polydisk e → X.Carrier) (hb : ∀ j, IsDiskVertical S (b j))
    (bot top : ℕ → Polydisk e) (joinB : ∀ j, b j (top j)=b (j+1) (bot (j+1)))
    (corner : a 0 (lo 0)=b 0 (bot 0)) :
    ∃ f : ℕ → ℕ → Polydisk d × Polydisk e → X.Carrier,
      (∀ i j, IsDiskTile S (f i j)) ∧
      (∀ i x, f i 0 (x,bot 0)=a i x) ∧ (∀ j y, f 0 j (lo 0,y)=b j y) ∧
      (∀ i j y, f i j (hi i,y)=f (i+1) j (lo (i+1),y)) ∧
      (∀ i j x, f i j (x,top j)=f i (j+1) (x,bot (j+1))) := by
  exact RectangleGrid.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)
    a ha lo hi joinA b hb bot top joinB corner

end UniversalCoverSplitting
end
end

end OAI