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720fa25 | 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 | import Mathlib
namespace OAI
noncomputable section
open scoped BigOperators ContDiff
namespace ArnoldCounterexample
abbrev CV := Fin 5 → ℂ
def quadricPolynomial (z : CV) : ℂ := ∑ j, z j ^ 2
def normSquared (z : CV) : ℝ := ∑ j, Complex.normSq (z j)
def IsUnitQuadric (z : CV) : Prop :=
normSquared z = 1 ∧ quadricPolynomial z = 0
abbrev UnitQuadric := {z : CV // IsUnitQuadric z}
lemma unitQuadric_ne_zero (z : UnitQuadric) : (z : CV) ≠ 0 := by
intro h
have hz := z.property.1
simp [normSquared, h] at hz
/-- The literal projective quadric sum z_j^2 = 0 in CP^4. -/
def Q3 := {q : Projectivization ℂ CV //
∃ (z : CV) (hz : z ≠ 0), Projectivization.mk ℂ z hz = q ∧
quadricPolynomial z = 0}
def project (z : UnitQuadric) : Q3 := by
have hz : (z : CV) ≠ 0 := by
intro h
have hunit := z.property.1
simp [normSquared, h] at hunit
exact ⟨Projectivization.mk ℂ z hz, z, hz, rfl, z.property.2⟩
/-- Horizontal lifts of the real tangent of Q3 at [z]. -/
def Horizontal (z v : CV) : Prop :=
(∑ j, star (z j) * v j) = 0 ∧ (∑ j, z j * v j) = 0
/-- The restricted Fubini--Study form, evaluated on horizontal lifts. -/
def omega (v w : CV) : ℝ := 2 * (∑ j, star (v j) * w j).im
/-- A local smooth extension of the pullback of h to the unit isotropic cone. -/
def LocalFunctionLift (h : Q3 → ℝ) (z : UnitQuadric)
(U : Set CV) (g : CV → ℝ) : Prop :=
IsOpen U ∧ (z : CV) ∈ U ∧ ContDiffOn ℝ ∞ g U ∧
∀ y : UnitQuadric, (y : CV) ∈ U → g y = h (project y)
def IsSmoothFunction (h : Q3 → ℝ) : Prop :=
∀ z : UnitQuadric, ∃ (U : Set CV) (g : CV → ℝ), LocalFunctionLift h z U g
structure SmoothFunction where
toFun : Q3 → ℝ
smooth : IsSmoothFunction toFun
/-- dh_q = 0, expressed using all local lifts, independent of normal extensions. -/
def IsCritical (h : Q3 → ℝ) (q : Q3) : Prop :=
∀ z : UnitQuadric, project z = q →
∀ (U : Set CV) (g : CV → ℝ), LocalFunctionLift h z U g →
∀ v : CV, Horizontal z v → fderiv ℝ g z v = 0
def criticalSet (h : SmoothFunction) : Set Q3 := {q | IsCritical h.toFun q}
/-- The minimum critical number, including degenerate functions (ENat allows infinity). -/
def criticalNumber : ℕ∞ := ⨅ h : SmoothFunction, (criticalSet h).encard
/-- An explicit smooth lift of a Hamiltonian isotopy, with inverse and generator.
The Hamiltonian equation uses i_X omega = dH. -/
structure HamiltonianPath (φ : Q3 ≃ Q3) where
forward : ℝ × CV → CV
backward : ℝ × CV → CV
hamiltonian : ℝ × CV → ℝ
forward_smooth : ContDiff ℝ ∞ forward
backward_smooth : ContDiff ℝ ∞ backward
hamiltonian_smooth : ContDiff ℝ ∞ hamiltonian
forward_preserves : ∀ t ∈ Set.Icc (0 : ℝ) 1, ∀ z : UnitQuadric,
IsUnitQuadric (forward (t, z))
backward_preserves : ∀ t ∈ Set.Icc (0 : ℝ) 1, ∀ z : UnitQuadric,
IsUnitQuadric (backward (t, z))
forward_phase : ∀ t ∈ Set.Icc (0 : ℝ) 1, ∀ z : UnitQuadric,
∀ a : ℂ, Complex.normSq a = 1 → forward (t, a • (z : CV)) = a • forward (t, z)
backward_phase : ∀ t ∈ Set.Icc (0 : ℝ) 1, ∀ z : UnitQuadric,
∀ a : ℂ, Complex.normSq a = 1 → backward (t, a • (z : CV)) = a • backward (t, z)
hamiltonian_phase : ∀ t ∈ Set.Icc (0 : ℝ) 1, ∀ z : UnitQuadric,
∀ a : ℂ, Complex.normSq a = 1 → hamiltonian (t, a • (z : CV)) = hamiltonian (t, z)
left_inverse : ∀ t ∈ Set.Icc (0 : ℝ) 1, ∀ z : UnitQuadric,
backward (t, forward (t, z)) = z
right_inverse : ∀ t ∈ Set.Icc (0 : ℝ) 1, ∀ z : UnitQuadric,
forward (t, backward (t, z)) = z
start : ∀ z : UnitQuadric, forward (0, z) = z
finish : ∀ z : UnitQuadric,
project ⟨forward (1, z), forward_preserves 1 (by norm_num) z⟩ = φ (project z)
equation : ∀ t ∈ Set.Icc (0 : ℝ) 1, ∀ z : UnitQuadric, ∀ v : CV,
Horizontal (forward (t, z)) v →
omega (deriv (fun s : ℝ => forward (s, z)) t) v =
fderiv ℝ (fun w : CV => hamiltonian (t, w)) (forward (t, z)) v
def IsHamiltonian (φ : Q3 ≃ Q3) : Prop := Nonempty (HamiltonianPath φ)
def fixedSet (φ : Q3 ≃ Q3) : Set Q3 := {q | φ q = q}
/-- Eigenvalue 1 in the derivative on the quotient tangent, not an ambient
spurious vertical eigenvector. v is nonzero and horizontal. -/
def HasDegenerateFixedPoint (φ : Q3 ≃ Q3) : Prop :=
∃ (z : UnitQuadric) (v : CV), φ (project z) = project z ∧ v ≠ 0 ∧
Horizontal z v ∧ ∃ (U : Set CV) (G : CV → CV),
IsOpen U ∧ (z : CV) ∈ U ∧ ContDiffOn ℝ ∞ G U ∧ G z = z ∧
(∀ y : UnitQuadric, (y : CV) ∈ U →
∃ hy : IsUnitQuadric (G y), project ⟨G y, hy⟩ = φ (project y)) ∧
∃ a : ℝ, fderiv ℝ G z v - v = a • (Complex.I • (z : CV))
end ArnoldCounterexample
end
end OAI
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