File size: 4,991 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 | import OAI.Geometry.Arnold.Model
namespace OAI
noncomputable section
open scoped BigOperators ContDiff
namespace ArnoldCounterexample
namespace Quadric
/-- The linear involution fixing the first coordinate. -/
def involutionLinear : CV ≃ₗ[ℂ] CV where
toFun z j := if j = 0 then z j else -z j
invFun z j := if j = 0 then z j else -z j
left_inv z := by funext j; by_cases hj : j = 0 <;> simp [hj]
right_inv z := by funext j; by_cases hj : j = 0 <;> simp [hj]
map_add' z w := by funext j; by_cases hj : j = 0 <;> simp [hj, add_comm]
map_smul' a z := by funext j; by_cases hj : j = 0 <;> simp [hj]
@[simp] lemma involutionLinear_apply (z : CV) (j : Fin 5) :
involutionLinear z j = if j = 0 then z j else -z j := rfl
@[simp] lemma involutionLinear_twice (z : CV) :
involutionLinear (involutionLinear z) = z := involutionLinear.left_inv z
@[simp] lemma involutionLinear_quadricPolynomial (z : CV) :
quadricPolynomial (involutionLinear z) = quadricPolynomial z := by
apply Finset.sum_congr rfl
intro j _
by_cases hj : j = 0 <;> simp [hj]
@[simp] lemma involutionLinear_normSquared (z : CV) :
normSquared (involutionLinear z) = normSquared z := by
apply Finset.sum_congr rfl
intro j _
by_cases hj : j = 0 <;> simp [hj]
lemma involutionLinear_unit (z : UnitQuadric) : IsUnitQuadric (involutionLinear z) := by
simpa only [IsUnitQuadric, involutionLinear_quadricPolynomial,
involutionLinear_normSquared] using z.property
/-- An injective complex-linear symmetry of the cone acts on the actual
projective quadric, not on a substitute abstract model. -/
lemma linearEquiv_ne_zero (D : CV ≃ₗ[ℂ] CV) {z : CV} (hz : z ≠ 0) : D z ≠ 0 := by
intro h
exact hz (D.injective (by simpa using h))
def map (D : CV ≃ₗ[ℂ] CV)
(hD : ∀ z, quadricPolynomial z = 0 → quadricPolynomial (D z) = 0) : Q3 → Q3 :=
fun q ↦ ⟨Projectivization.map D.toLinearMap D.injective q.val, by
obtain ⟨z, hz, hq, hpoly⟩ := q.property
refine ⟨D z, linearEquiv_ne_zero D hz, ?_, hD z hpoly⟩
rw [← hq]
rfl⟩
lemma map_mk (D : CV ≃ₗ[ℂ] CV)
(hD : ∀ z, quadricPolynomial z = 0 → quadricPolynomial (D z) = 0)
(z : CV) (hz : z ≠ 0) (hp : quadricPolynomial z = 0) :
map D hD ⟨Projectivization.mk ℂ z hz, z, hz, rfl, hp⟩ =
⟨Projectivization.mk ℂ (D z) (linearEquiv_ne_zero D hz),
D z, linearEquiv_ne_zero D hz, rfl, hD z hp⟩ := rfl
/-- The Hamiltonian involution's underlying projective equivalence. -/
def involution : Q3 ≃ Q3 where
toFun := map involutionLinear (fun z hz ↦ by simpa using hz)
invFun := map involutionLinear (fun z hz ↦ by simpa using hz)
left_inv q := by
apply Subtype.ext
obtain ⟨z, hz, hq, hp⟩ := q.property
change Projectivization.map involutionLinear.toLinearMap involutionLinear.injective
(Projectivization.map involutionLinear.toLinearMap involutionLinear.injective q.val) = q.val
rw [← hq]
simp only [Projectivization.map_mk, LinearEquiv.coe_coe, involutionLinear_twice]
right_inv q := by
apply Subtype.ext
obtain ⟨z, hz, hq, hp⟩ := q.property
change Projectivization.map involutionLinear.toLinearMap involutionLinear.injective
(Projectivization.map involutionLinear.toLinearMap involutionLinear.injective q.val) = q.val
rw [← hq]
simp only [Projectivization.map_mk, LinearEquiv.coe_coe, involutionLinear_twice]
@[simp] theorem involution_sq (q : Q3) : involution (involution q) = q :=
involution.left_inv q
lemma involution_project (z : UnitQuadric) :
involution (project z) = project ⟨involutionLinear z, involutionLinear_unit z⟩ := rfl
/-- The +1 eigenline is not on the quadric. All projectively fixed lines
therefore lie in precisely the hyperplane z0=0. -/
theorem involution_fixed_project_iff (z : UnitQuadric) :
involution (project z) = project z ↔ (z : CV) 0 = 0 := by
rw [involution_project]
change (Subtype.mk _ _ : Q3) = Subtype.mk _ _ ↔ _
rw [Subtype.mk.injEq, Projectivization.mk_eq_mk_iff']
constructor
· rintro ⟨a, ha⟩
by_contra hz0
have hzero := congrFun ha 0
simp only [Pi.smul_apply, smul_eq_mul, involutionLinear_apply] at hzero
have ha1 : a = 1 := mul_right_cancel₀ hz0 (by simpa using hzero)
have hj0 : ∀ j : Fin 5, j ≠ 0 → (z : CV) j = 0 := by
intro j hj
have hh := congrFun ha j
simp only [Pi.smul_apply, smul_eq_mul, ha1, one_mul, involutionLinear_apply,
ite_eq_right hj] at hh
linear_combination (1 / 2 : ℂ) * hh
have hp : quadricPolynomial z = (z : CV) 0 ^ 2 := by
apply Finset.sum_eq_single 0
· intro j _ hj
simp [hj0 j hj]
· simp
have hsq : (z : CV) 0 ^ 2 = 0 := hp.symm.trans z.property.2
exact hz0 (sq_eq_zero_iff.mp hsq)
· intro hz0
refine ⟨-1, ?_⟩
funext j
by_cases hj : j = 0
· subst j
simp [hz0]
· simp [hj]
end Quadric
end ArnoldCounterexample
end
end OAI
|