| import OAI.Geometry.Kahler.BaseChartLow |
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| namespace OAI |
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| open Complex |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder |
| open scoped ContDiff ComplexOrder |
| open scoped ContDiff ENNReal |
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ENNReal Pointwise |
| open Set Filter Topology |
| open scoped ContDiff |
| noncomputable section |
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| open Set Filter Topology |
| open scoped ContDiff |
| namespace PinchedHartogs.BaseConstruction |
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|
| def chartScale (k : ℕ) (δ : ℝ) : ℝ := |
| if (k:ℝ)*δ ≤ 1 then Real.sqrt ((k:ℝ)*δ) else Real.exp (-(k:ℝ)*δ/16) |
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| lemma logarithmicTest_chart_jets (a : ℝ) {ε : ℝ} (he : ε ≠ 0) {n : ℕ} (hn : 1 ≤ n) : |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ {D : ℝ}, 1 ≤ D → ∀ {k : ℕ}, 1 ≤ k → |
| ∀ {P : Finset Sphere}, ProjectivelySeparated (D/Real.sqrt k) P → |
| ∀ {r : ℝ}, 0 ≤ r → r < 1 → ∀ U : Base ≃ₗᵢ[ℂ] Base, |
| ‖iteratedFDeriv ℝ n (logarithmicTest a ε P k ∘ centeredChart r U) 0‖ ≤ C*chartScale k (1-r) := by |
| obtain ⟨B,hB,hBlow⟩ := peak_chart_low |
| let B' := B+2*gaussianConstant |
| have hB' : 0 ≤ B' := by dsimp [B']; linarith [gaussianConstant_pos] |
| obtain ⟨C,hC,hCb⟩ := regularizedLog_positive_jet_bound a he (by norm_num : (0:ℝ) < 1/32) |
| gaussianConstant_pos.le hB' hn |
| refine ⟨C,hC,?_⟩ |
| intro D hD k hk P hP r hr hr1 U |
| let f := peakPolynomial P k ∘ centeredChart r U |
| have hf : AnalyticOnNhd ℂ f (Metric.ball (0:Base) (1/32)) := by |
| intro z hz |
| have hzb : z ∈ ball := by |
| have hz' : ‖z‖ < 1/32 := by simpa using hz |
| change ‖z‖ < 1 |
| linarith |
| exact (peakPolynomial_analytic P k _ (mem_univ _)).comp |
| (centeredChart_analyticAt U (centeredChart_den_ne_zero hr hr1 hzb)) |
| have hfc : ‖f 0‖ ≤ gaussianConstant := by |
| apply (peak_global_bound hD hk hP _).trans |
| apply mul_le_of_le_one_right gaussianConstant_pos.le |
| exact pow_le_one₀ (norm_nonneg _) (centeredChart_zero_mem hr hr1 U).le |
| have hkδ : 0 < (k:ℝ)*(1-r) := mul_pos (by exact_mod_cast (show 0 < k by omega)) (by linarith) |
| by_cases hlow : (k:ℝ)*(1-r) ≤ 1 |
| · rw [chartScale,ite_eq_left hlow] |
| apply hCb f 0 (Real.sqrt ((k:ℝ)*(1-r))) hf hfc (Real.sqrt_pos.mpr hkδ) (Real.sqrt_le_one.mpr hlow) |
| intro z hz |
| apply (hBlow hD hk hP hr hr1 hlow U z (by simpa using hz)).trans |
| exact mul_le_mul_of_nonneg_right (by dsimp [B']; linarith [gaussianConstant_pos]) (Real.sqrt_nonneg _) |
| · rw [chartScale,ite_eq_right hlow] |
| apply hCb f 0 (Real.exp (-(k:ℝ)*(1-r)/16)) hf hfc (Real.exp_pos _) |
| (Real.exp_le_one_iff.mpr (by nlinarith)) |
| intro z hz |
| have hb := peak_chart_high hD hk hP hr hr1 U (by simpa using hz : ‖z‖ < (1/32:ℝ)) |
| have hb0 := peak_chart_high hD hk hP hr hr1 U (z := 0) (by simp) |
| apply (norm_sub_le _ _).trans |
| dsimp [f,Function.comp_def] |
| dsimp [B'] |
| nlinarith [Real.exp_pos (-(k:ℝ)*(1-r)/16)] |
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| end PinchedHartogs.BaseConstruction |
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| end |
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| end OAI |
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