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8e4d533 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 | import OAI.Geometry.ComplexBundle.Hyperplane
import OAI.Geometry.ComplexBundle.LineStage
namespace OAI
noncomputable section
/-! Hyperplane product sections and regular zeros. -/
section
open Bundle Topology
namespace Kaplansky
/-- Pullback of a genuine continuous section through the actual line-core
pullback construction. -/
theorem continuous_pullbackLineCore_section {B X J : Type*} [TopologicalSpace B] [TopologicalSpace X]
(C : VectorBundleCore ℂ B ℂ J) (g : C(X,B)) (s : ∀ b, C.Fiber b)
(hs : Continuous (fun b => TotalSpace.mk' ℂ b (s b))) :
Continuous (fun x => TotalSpace.mk' ℂ x (show (pullbackLineCore C g).Fiber x from s (g x))) := by
apply continuous_iff_continuousAt.mpr
intro x
rw [FiberBundle.continuousAt_section]
exact ((FiberBundle.continuousAt_section ℂ (g x)).mp (hs.continuousAt (x := g x))).comp g.continuous.continuousAt
open ComplexGrassmann
variable {ι : Type} [Fintype ι] (K : ι → ℕ)
/-- Exactly K_i copies of each actual pulled-back hyperplane line. -/
def hyperplaneProductStage : LineStage K :=
⟨Σ i, Fin (K i),inferInstance,fun s => some s.1⟩
def hyperplaneProductLineSection (s : (hyperplaneProductStage K).Slot) (P : ProjectiveProduct K) :
(productLineCore K (some s.1)).Fiber P :=
hyperplaneFunctionalSection (coordinateFunctional (positiveCoordinate (K s.1) s.2))
(projectiveProjectionMap (K s.1) (P s.1))
theorem continuous_hyperplaneProductLineSection (s : (hyperplaneProductStage K).Slot) :
Continuous (fun P => TotalSpace.mk' ℂ P (hyperplaneProductLineSection K s P)) := by
have h := continuous_pullbackLineCore_section (projectionDualLineCore (1+K s.1))
(projectiveProjectionMap (K s.1))
(hyperplaneFunctionalSection (coordinateFunctional (positiveCoordinate (K s.1) s.2)))
(continuous_hyperplaneFunctionalSection _)
exact continuous_pullbackLineCore_section (projectiveHyperplaneCore (K s.1))
⟨fun P : ProjectiveProduct K => P s.1,continuous_apply s.1⟩ _ h
def hyperplaneProductSection (P : ProjectiveProduct K) : (hyperplaneProductStage K).Fiber P :=
WithLp.toLp 2 (fun s => hyperplaneProductLineSection K s P)
theorem continuous_hyperplaneProductSection :
Continuous (fun P => TotalSpace.mk' (hyperplaneProductStage K).Model P (hyperplaneProductSection K P)) := by
apply (continuous_finiteLineSum_iff (fun s : (hyperplaneProductStage K).Slot =>
productLineCore K (some s.1)) _).mpr
exact ⟨continuous_id,continuous_hyperplaneProductLineSection K⟩
def firstCoordinateProduct : ProjectiveProduct K :=
fun i => projectiveProjectionEquiv (K i) (firstCoordinateProjection (K i))
/-- The canonical section of the actual rank-sum(K_i) hyperplane sum has
one zero on the actual projective-product base. No cohomology identification
or invariance under changes of section is asserted by this zero-set theorem. -/
theorem hyperplaneProductSection_eq_zero_iff (P : ProjectiveProduct K) :
hyperplaneProductSection K P=0 ↔ P=firstCoordinateProduct K := by
constructor
· intro h
funext i
apply (projectiveProjectionEquiv (K i)).symm.injective
change projectiveProjectionMap (K i) (P i)=
(projectiveProjectionEquiv (K i)).symm (projectiveProjectionEquiv (K i) (firstCoordinateProjection (K i)))
rw [Equiv.symm_apply_apply]
apply (hyperplaneCoordinateSections_common_zero _).mp
intro j
exact congrArg (fun v : (hyperplaneProductStage K).Fiber P => (show EuclideanSpace ℂ (Σ i, Fin (K i)) from v) ⟨i,j⟩) h
· rintro rfl
change (show EuclideanSpace ℂ (Σ i, Fin (K i)) from hyperplaneProductSection K (firstCoordinateProduct K))=0
ext s
exact (hyperplaneCoordinateSections_common_zero
(projectiveProjectionMap (K s.1) (firstCoordinateProduct K s.1))).mpr
((projectiveProjectionEquiv (K s.1)).symm_apply_apply _) s.2
theorem hyperplaneProductStage_rank (P : ProjectiveProduct K) :
Module.finrank ℂ ((hyperplaneProductStage K).Fiber P)=∑ i, K i := by
change Module.finrank ℂ (EuclideanSpace ℂ (Σ i, Fin (K i)))=_
simp
end Kaplansky
end
open Bundle Topology
namespace Kaplansky
open ComplexGrassmann
variable {ι : Type} (K : ι → ℕ)
def canonicalProductGraphPoint (A : ProjectiveProductModel K) : ProjectiveProduct K :=
fun i => slopePoint (standardLineGraphFrame (K i)) (A i)
theorem canonicalProductGraphPoint_zero :
canonicalProductGraphPoint K 0=firstCoordinateProduct K := by
funext i
apply (projectiveProjectionEquiv (K i)).symm.injective
change (projectiveProjectionEquiv (K i)).symm (slopePoint (standardLineGraphFrame (K i)) 0)=
(projectiveProjectionEquiv (K i)).symm (projectiveProjectionEquiv (K i) (firstCoordinateProjection (K i)))
rw [projectiveProjectionEquiv_symm_slopePoint,standard_graphProjection_zero,Equiv.symm_apply_apply]
variable [Fintype ι]
/-- The actual global product section expressed componentwise in the
constructed graph evaluation trivializations. -/
def canonicalProductSectionCoordinates (A : ProjectiveProductModel K) : (Σ i, Fin (K i)) → ℂ :=
fun s => (graphHyperplaneTrivialization (standardLineGraphFrame (K s.1))
⟨A s.1,hyperplaneProductLineSection K s (canonicalProductGraphPoint K A)⟩).2
theorem canonicalProductSectionCoordinates_eq (A : ProjectiveProductModel K) :
canonicalProductSectionCoordinates K A = fun s => A s.1 (fun _ => 1) s.2 := by
funext s
have he : (show ℂ from hyperplaneProductLineSection K s (canonicalProductGraphPoint K A))=
hyperplaneFunctionalSection (graphVerticalFunctional (standardLineGraphFrame (K s.1)) s.2)
(graphLineProjection (standardLineGraphFrame (K s.1)) (A s.1)) := by
change (show ℂ from hyperplaneFunctionalSection (coordinateFunctional (positiveCoordinate (K s.1) s.2))
((projectiveProjectionEquiv (K s.1)).symm (slopePoint (standardLineGraphFrame (K s.1)) (A s.1))))=_
rw [projectiveProjectionEquiv_symm_slopePoint,standard_verticalFunctional]
change (graphHyperplaneTrivialization (standardLineGraphFrame (K s.1))
⟨A s.1,show ℂ from hyperplaneProductLineSection K s (canonicalProductGraphPoint K A)⟩).2=_
rw [he]
exact congrFun (graphHyperplaneSectionCoordinates_eq (standardLineGraphFrame (K s.1)) (A s.1)) s.2
def productSlopeEvaluationEquiv : ProjectiveProductModel K ≃L[ℂ] ((Σ i, Fin (K i)) → ℂ) :=
((LinearEquiv.piCongrRight (fun i => (slopeEvaluationOneEquiv (K i)).toLinearEquiv)).trans
(LinearEquiv.piCurry ℂ (fun i (_ : Fin (K i)) => ℂ)).symm).toContinuousLinearEquiv
/-- The unique zero of the actual product section has an invertible complex
linear derivative in the genuine graph bundle coordinates. The Euler-index
invariance and Chern-class identification are not assumed or proved here. -/
theorem canonicalProductSection_regular_zero :
HasFDerivAt (canonicalProductSectionCoordinates K)
(productSlopeEvaluationEquiv K).toContinuousLinearMap 0 ∧
Function.Bijective (productSlopeEvaluationEquiv K) := by
have he : canonicalProductSectionCoordinates K=productSlopeEvaluationEquiv K :=
funext (canonicalProductSectionCoordinates_eq K)
rw [he]
exact ⟨(productSlopeEvaluationEquiv K).toContinuousLinearMap.hasFDerivAt,
(productSlopeEvaluationEquiv K).bijective⟩
theorem canonicalProductSection_regular_zero_real :
HasFDerivAt (canonicalProductSectionCoordinates K)
((productSlopeEvaluationEquiv K).toContinuousLinearMap.restrictScalars ℝ) 0 :=
(canonicalProductSection_regular_zero K).1.restrictScalars ℝ
end Kaplansky
end
end OAI
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