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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 | import OAI.Analysis.Kaplansky.PolynomialTest
import OAI.Geometry.ComplexBundle.Complement
namespace OAI
noncomputable section
/-! Regrouping retained frames and their complement bundles. -/
section
open Bundle
open scoped BigOperators
namespace Kaplansky
open ComplexGrassmann
attribute [local instance 2000] CStarAlgebra.spectralOrder CStarAlgebra.spectralOrderedRing
namespace ContinuousOrthonormalStageFrame
variable {ι : Type*} [Fintype ι] {K : ι → ℕ} {a b : LineStage K} {d : ℕ}
def congr (f : ContinuousOrthonormalStageFrame a d)
(e : UnitaryBundleEquiv a.Model b.Model a.Fiber b.Fiber) :
ContinuousOrthonormalStageFrame b d where
vector y i := e.fiber y (f.vector y i)
continuous_vector i := e.continuous_forward.comp (f.continuous_vector i)
orthonormal y := ((e.fiber y).toLinearIsometry.orthonormal_comp_iff).mpr (f.orthonormal y)
end ContinuousOrthonormalStageFrame
def retainedGroupedStage (m R : ℕ) : LineStage (testProjectiveDimension m R) :=
⟨RetainedGroupedSlot m R,inferInstance,fun s => s.1⟩
/-- The polynomial image frame is transported to the actual line-bundle
presentation used in the Chern computation. No characteristic-class
nonvanishing or obstruction is a premise. -/
theorem properlyInfinite_regrouped_frame {m : ℕ} (hm : 0<m)
(hpi : ProperlyInfinite (CStarMatrix (Fin m) (Fin m) ActualAlgebra)) :
∃ R : ℕ, 0<R ∧ Nonempty (ContinuousOrthonormalStageFrame (retainedGroupedStage m R)
(2*m*testRank m R 0)) := by
obtain ⟨R,hR,⟨f⟩⟩ := properlyInfinite_retained_frame hm hpi
exact ⟨R,hR,⟨f.congr (retainedRegroupingUnitary m R)⟩⟩
/-- A hypothetical proper-infiniteness witness yields a genuine continuous
frame whose orthogonal complement has rank below the Chern degree on every
fiber. Identifying that Chern class and proving it nonzero remain separate. -/
theorem properlyInfinite_retained_complement {m : ℕ} (hm : 0<m)
(hpi : ProperlyInfinite (CStarMatrix (Fin m) (Fin m) ActualAlgebra)) :
∃ (R : ℕ) (_ : 0<R) (f : ContinuousOrthonormalStageFrame (retainedLineStage m R)
(2*m*testRank m R 0)),
∀ y : SmoothTestBase m R,
Module.finrank ℂ (Submodule.span ℂ (Set.range (f.vector y)))ᗮ <
∑ i : Fin R, testProjectiveDimension m R i := by
obtain ⟨R,hR,⟨f⟩⟩ := properlyInfinite_retained_frame hm hpi
refine ⟨R,hR,f,fun y => ?_⟩
exact retained_complement_rank_lt hm y _ (f.image_finrank y).ge
end Kaplansky
end
open Bundle Topology
open scoped BigOperators
namespace Kaplansky
open ComplexGrassmann
attribute [local instance 2000] CStarAlgebra.spectralOrder CStarAlgebra.spectralOrderedRing
theorem retainedGrouped_complementBundle_rank_lt {m R : ℕ} (hm : 0<m)
(f : ContinuousOrthonormalStageFrame (retainedGroupedStage m R) (2*m*testRank m R 0))
(y : SmoothTestBase m R) :
Module.finrank ℂ (f.ComplementFiber y) < ∑ i : Fin R, testProjectiveDimension m R i := by
have hr := f.complement_finrank y
have hc : Fintype.card (retainedGroupedStage m R).Slot = retainedTrivialMultiplicity m R +
∑ i : Fin R, testProjectiveDimension m R i := by
have hd := retainedBundle_rank_decomposition m R y
change Module.finrank ℂ (EuclideanSpace ℂ (RetainedSlot m R)) = _ at hd
rw [finrank_euclideanSpace, Fintype.card_congr (retainedGroupingEquiv m R)] at hd
exact hd
rw [hc] at hr
have ht := retainedTrivialMultiplicity_bound (R := R) hm
change Module.finrank ℂ (Submodule.span ℂ (Set.range (f.vector y)))ᗮ < _
omega
/-- From a proper-infiniteness witness in the actual matrix algebra, construct
the ordinary complement vector bundle and the actual global bundle splitting.
No characteristic-class hypothesis, bundle realization or norm bound is input. -/
theorem properlyInfinite_retained_bundle_split {m : ℕ} (hm : 0<m)
(hpi : ProperlyInfinite (CStarMatrix (Fin m) (Fin m) ActualAlgebra)) :
∃ (R : ℕ) (_ : 0<R)
(f : ContinuousOrthonormalStageFrame (retainedGroupedStage m R) (2*m*testRank m R 0))
(H : f.ComplementSplitTotal ≃ₜ
TotalSpace (retainedGroupedStage m R).Model (retainedGroupedStage m R).Fiber),
(∀ z, (H z).proj=z.1.proj) ∧
(∀ y : SmoothTestBase m R, Module.finrank ℂ (f.ComplementFiber y) <
∑ i : Fin R, testProjectiveDimension m R i) := by
obtain ⟨R,hR,⟨f⟩⟩ := properlyInfinite_regrouped_frame hm hpi
exact ⟨R,hR,f,f.complementSplitHomeomorph,f.complementSplitHomeomorph_base,
retainedGrouped_complementBundle_rank_lt hm f⟩
end Kaplansky
end
end OAI
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