| import OAI.Geometry.Kahler.BaseHessianJet |
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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 |
|
|
| open Set Filter Topology |
| open scoped ContDiff |
| namespace PinchedHartogs.BaseConstruction |
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| lemma testSeries_chart_summable_jets (a : ℝ) {ε D : ℝ} (he : ε ≠ 0) (hD : 1 ≤ D) |
| {Q : ℕ} (hQ : 2 ≤ Q) (P : ℕ → Finset Sphere) |
| (hP : ∀ j, ProjectivelySeparated (D/Real.sqrt (Q^(j+1):ℕ)) (P (Q^(j+1)))) |
| {r : ℝ} (hr : 0 ≤ r) (hr1 : r < 1) (U : Base ≃ₗᵢ[ℂ] Base) (n : ℕ) : |
| Summable (fun j => iteratedFDeriv ℝ n (logarithmicTest a ε (P (Q^(j+1))) (Q^(j+1)) ∘ centeredChart r U) 0) := by |
| obtain ⟨C,hC,hb⟩ := logarithmicTest_chart_local_jets a he hD hr hr1 n |
| apply Summable.of_norm_bounded |
| ((summable_lacunary (by linarith : 0 ≤ 1-(1-r)/16) (by linarith : 1-(1-r)/16 < 1) hQ).mul_left C) |
| intro j |
| exact hb _ (Nat.one_le_pow _ _ (by omega)) _ (hP j) U 0 (Metric.mem_ball_self (by norm_num)) |
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| lemma testSeries_chart_uniform_jets (a : ℝ) {ε D : ℝ} (he : ε ≠ 0) (hD : 1 ≤ D) |
| {Q : ℕ} (hQ : 2 ≤ Q) (P : ℕ → Finset Sphere) |
| (hP : ∀ j, ProjectivelySeparated (D/Real.sqrt (Q^(j+1):ℕ)) (P (Q^(j+1)))) |
| {n : ℕ} (hn : 1 ≤ n) : |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ r : ℝ, 0 ≤ r → r < 1 → ∀ U : Base ≃ₗᵢ[ℂ] Base, |
| ‖iteratedFDeriv ℝ n ((fun y => ∑' j, logarithmicTest a ε (P (Q^(j+1))) (Q^(j+1)) y) ∘ centeredChart r U) 0‖ ≤ C := by |
| obtain ⟨C,hC,hCb⟩ := logarithmicTest_chart_jets a he hn |
| obtain ⟨B,hB,hBb⟩ := chartScale_sum_uniform hQ |
| refine ⟨C*B,mul_nonneg hC hB,?_⟩ |
| intro r hr hr1 U |
| let f := fun j => iteratedFDeriv ℝ n (logarithmicTest a ε (P (Q^(j+1))) (Q^(j+1)) ∘ centeredChart r U) 0 |
| have hs : Summable (fun j => C*chartScale (Q^(j+1)) (1-r)) := |
| (chartScale_summable hQ (by linarith : 0 < 1-r)).mul_left C |
| have hb (j : ℕ) : ‖f j‖ ≤ C*chartScale (Q^(j+1)) (1-r) := |
| hCb hD (Nat.one_le_pow _ _ (by omega)) (hP j) hr hr1 U |
| have hnS := Summable.of_nonneg_of_le (fun j => norm_nonneg (f j)) hb hs |
| rw [testSeries_chart_iteratedFDeriv a he hD hQ P hP hr hr1 U n] |
| exact (norm_tsum_le_tsum_norm hnS).trans ((hnS.tsum_le_tsum hb hs).trans (by |
| rw [tsum_mul_left] |
| exact mul_le_mul_of_nonneg_left (hBb (1-r) (by linarith)) hC)) |
|
|
| lemma testSeries_chart_psh (a : ℝ) {ε D : ℝ} (he : ε ≠ 0) (hD : 1 ≤ D) |
| {Q : ℕ} (hQ : 2 ≤ Q) (P : ℕ → Finset Sphere) |
| (hP : ∀ j, ProjectivelySeparated (D/Real.sqrt (Q^(j+1):ℕ)) (P (Q^(j+1)))) |
| {r : ℝ} (hr : 0 ≤ r) (hr1 : r < 1) (U : Base ≃ₗᵢ[ℂ] Base) (v : Base) : |
| 0 ≤ (∑ i, ∑ j, baseHessian |
| ((fun y => ∑' l, logarithmicTest a ε (P (Q^(l+1))) (Q^(l+1)) y) ∘ centeredChart r U) 0 i j * v i * star (v j)).re := by |
| have hf := chartWeight_contDiffAt (testSeries_contDiffOn a he hD hQ P hP) hr hr1 U |
| rw [baseHessian_value_eq_jet hf, testSeries_chart_iteratedFDeriv a he hD hQ P hP hr hr1 U 2, |
| (realHessianValueJet v).map_tsum (testSeries_chart_summable_jets a he hD hQ P hP hr hr1 U 2)] |
| apply tsum_nonneg |
| intro l |
| rw [← baseHessian_value_eq_jet (logarithmicTest_chart_contDiffAt a he _ _ hr hr1 U (by change ‖(0:Base)‖ < 1; simp))] |
| apply regularizedLog_holomorphic_psh a he |
| exact (peakPolynomial_analytic _ _ _ (mem_univ _)).comp |
| (centeredChart_analyticAt U (z := 0) (by simp)) |
|
|
| end PinchedHartogs.BaseConstruction |
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| end |
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| end OAI |
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