| 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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|
