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465b10d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 | import Mathlib
namespace OAI
noncomputable section
open scoped Manifold ContDiff LinearAlgebra.Projectivization
open Set Bundle
universe u
namespace SplitTangent
structure ComplexManifold where
dim : ℕ
Carrier : Type u
[topology : TopologicalSpace Carrier]
[charts : ChartedSpace (Fin dim → ℂ) Carrier]
[holomorphicAtlas : IsManifold 𝓘(ℂ, Fin dim → ℂ) ω Carrier]
[hausdorff : T2Space Carrier]
[secondCountable : SecondCountableTopology Carrier]
attribute [instance] ComplexManifold.topology ComplexManifold.charts
ComplexManifold.holomorphicAtlas ComplexManifold.hausdorff
ComplexManifold.secondCountable
abbrev Model (X : ComplexManifold.{u}) := Fin X.dim → ℂ
abbrev ModelWithCorners (X : ComplexManifold.{u}) := 𝓘(ℂ, Model X)
abbrev Tangent (X : ComplexManifold.{u}) (x : X.Carrier) :=
TangentSpace (ModelWithCorners X) x
abbrev ProjectiveSpace (N : ℕ) := ℙ ℂ (Fin (N + 1) → ℂ)
def projectiveCoordinates {N : ℕ} (i : Fin (N + 1))
(p : ProjectiveSpace N) : Fin (N + 1) → ℂ :=
fun j => (p.rep i)⁻¹ * p.rep j
structure ProjectiveEmbedding (X : ComplexManifold.{u}) where
N : ℕ
map : X.Carrier → ProjectiveSpace N
injective : Function.Injective map
open_patch : ∀ i, IsOpen {x | (map x).rep i ≠ 0}
holomorphic : ∀ i, ContMDiffOn (ModelWithCorners X) 𝓘(ℂ, Fin (N + 1) → ℂ) ω
(fun x => projectiveCoordinates i (map x)) {x | (map x).rep i ≠ 0}
immersion : ∀ x i, (map x).rep i ≠ 0 → Function.Injective
(mfderiv (ModelWithCorners X) 𝓘(ℂ, Fin (N + 1) → ℂ)
(fun y => projectiveCoordinates i (map y)) x)
structure RationalCurve (X : ComplexManifold.{u}) where
finite : ℂ → X.Carrier
infinity : ℂ → X.Carrier
finite_holomorphic : ContMDiff 𝓘(ℂ, ℂ) (ModelWithCorners X) ω finite
infinity_holomorphic : ContMDiff 𝓘(ℂ, ℂ) (ModelWithCorners X) ω infinity
overlap : ∀ z : ℂ, z ≠ 0 → finite z = infinity z⁻¹
def RationalCurve.value {X : ComplexManifold.{u}} (g : RationalCurve X)
(z : OnePoint ℂ) : X.Carrier :=
z.elim (g.infinity 0) g.finite
abbrev PairPolynomial (N : ℕ) := MvPolynomial (Fin 2 × Fin (N + 1)) ℂ
def Bihomogeneous {N : ℕ} (p : PairPolynomial N) : Prop :=
∃ a b : ℕ, ∀ m ∈ p.support,
(∑ i : Fin (N + 1), m (0, i)) = a ∧
(∑ i : Fin (N + 1), m (1, i)) = b
def pairEval {X : ComplexManifold.{u}} (e : ProjectiveEmbedding X)
(p : PairPolynomial e.N) (xy : X.Carrier × X.Carrier) : ℂ :=
MvPolynomial.eval
(fun k : Fin 2 × Fin (e.N + 1) =>
if k.1 = 0 then (e.map xy.1).rep k.2 else (e.map xy.2).rep k.2) p
def IsZariskiOpenPair {X : ComplexManifold.{u}} (e : ProjectiveEmbedding X)
(U : Set (X.Carrier × X.Carrier)) : Prop :=
∃ equations : Set (PairPolynomial e.N),
(∀ p ∈ equations, Bihomogeneous p) ∧
U = {xy | ∃ p ∈ equations, pairEval e p xy ≠ 0}
def RationallyConnected {X : ComplexManifold.{u}} (e : ProjectiveEmbedding X) : Prop :=
∃ U : Set (X.Carrier × X.Carrier), IsZariskiOpenPair e U ∧ U.Nonempty ∧ Dense U ∧
∀ xy ∈ U, ∃ g : RationalCurve X, ∃ a b : OnePoint ℂ,
g.value a = xy.1 ∧ g.value b = xy.2
structure TangentSplitting (X : ComplexManifold.{u}) where
projector : ∀ x : X.Carrier, Tangent X x →L[ℂ] Tangent X x
idempotent : ∀ x v, projector x (projector x v) = projector x v
holomorphic : ContMDiff
((ModelWithCorners X).prod 𝓘(ℂ, Model X))
((ModelWithCorners X).prod 𝓘(ℂ, Model X)) ω
(fun v : TangentBundle (ModelWithCorners X) X.Carrier =>
(⟨v.proj, projector v.proj v.2⟩ : TangentBundle (ModelWithCorners X) X.Carrier))
first_positive : ∀ x, 0 < Module.finrank ℂ (LinearMap.range (projector x).toLinearMap)
second_positive : ∀ x, 0 < Module.finrank ℂ (LinearMap.ker (projector x).toLinearMap)
def TangentSplitting.first {X : ComplexManifold.{u}} (S : TangentSplitting X) (x : X.Carrier) :
Submodule ℂ (Tangent X x) := LinearMap.range (S.projector x).toLinearMap
def TangentSplitting.second {X : ComplexManifold.{u}} (S : TangentSplitting X) (x : X.Carrier) :
Submodule ℂ (Tangent X x) := LinearMap.ker (S.projector x).toLinearMap
def HolomorphicSectionOn (X : ComplexManifold.{u})
(V : ∀ x : X.Carrier, Tangent X x) (U : Set X.Carrier) : Prop :=
ContMDiffOn (ModelWithCorners X) ((ModelWithCorners X).prod 𝓘(ℂ, Model X)) ω
(fun x => (⟨x, V x⟩ : TangentBundle (ModelWithCorners X) X.Carrier)) U
def Integrable (X : ComplexManifold.{u}) (D : ∀ x : X.Carrier, Submodule ℂ (Tangent X x)) : Prop :=
∀ U : Set X.Carrier, IsOpen U → ∀ V W : ∀ x : X.Carrier, Tangent X x,
HolomorphicSectionOn X V U → HolomorphicSectionOn X W U →
(∀ x ∈ U, V x ∈ D x) → (∀ x ∈ U, W x ∈ D x) →
∀ x ∈ U, VectorField.mlieBracketWithin (ModelWithCorners X) V W U x ∈ D x
def IntegrabilityStatement : Prop :=
∀ (X : ComplexManifold.{u}), ConnectedSpace X.Carrier → CompactSpace X.Carrier →
2 ≤ X.dim → ∀ e : ProjectiveEmbedding X, RationallyConnected e →
∀ S : TangentSplitting X, Integrable X S.first ∧ Integrable X S.second
theorem integrability_main : IntegrabilityStatement.{u} := by
sorry
end SplitTangent
end
end OAI
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