| import OAI.Geometry.Kahler.BasePeakBounds |
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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 Set Filter Topology |
| open scoped ContDiff |
| open scoped ContDiff ENNReal |
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ENNReal Pointwise |
| noncomputable section |
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|
| open Set Filter Topology |
| open scoped ContDiff |
| namespace PinchedHartogs.BaseConstruction |
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| lemma chart_small_ball {r : ℝ} (hr : 0 ≤ r) (hr1 : r < 1) |
| {z : Base} (hz : ‖z‖ < (1/32:ℝ)) : |
| 1/2 ≤ ‖1+(r:ℂ)*z 0‖ ∧ ‖1+(r:ℂ)*z 0‖ ≤ 2 := by |
| have hz0 := PiLp.norm_apply_le (p := (2:ENNReal)) z 0 |
| have hm : ‖(r:ℂ)*z 0‖ ≤ 1/32 := by |
| rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hr] |
| have := norm_nonneg (z 0) |
| nlinarith |
| have hupper := norm_add_le (1:ℂ) ((r:ℂ)*z 0) |
| have hlower := norm_add_le (1+(r:ℂ)*z 0) (-((r:ℂ)*z 0)) |
| simp only [norm_one] at hupper |
| simp only [add_neg_cancel_right,norm_one,norm_neg] at hlower |
| constructor <;> linarith |
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| lemma chart_norm_deficit {r : ℝ} (hr : 0 ≤ r) (hr1 : r < 1) |
| (U : Base ≃ₗᵢ[ℂ] Base) {z : Base} (hz : ‖z‖ < (1/32:ℝ)) : |
| ‖centeredChart r U z‖ ≤ 1-(1-r)/16 := by |
| have hzb : z ∈ ball := by change ‖z‖ < 1; linarith |
| have hd := centeredChart_den_ne_zero hr hr1 hzb |
| have he := centeredChart_norm_identity hr hr1 U z hd |
| have hnorm : ‖centeredChart r U z‖ < 1 := centeredChart_mapsTo hr hr1 U hzb |
| have hden := chart_small_ball hr hr1 hz |
| have hsqden : ‖1+(r:ℂ)*z 0‖^2 ≤ 4 := by nlinarith [norm_nonneg (1+(r:ℂ)*z 0)] |
| have hnum : (1-r)/2 ≤ (1-r^2)*(1-‖z‖^2) := by |
| have hn : ‖z‖^2 ≤ 1/2 := by nlinarith [norm_nonneg z] |
| have hrr : 1-r ≤ 1-r^2 := by nlinarith |
| nlinarith |
| have hdiv : (1-r)/8 ≤ (1-r^2)*(1-‖z‖^2)/‖1+(r:ℂ)*z 0‖^2 := by |
| apply (le_div_iff₀ (sq_pos_of_pos (by linarith : 0 < ‖1+(r:ℂ)*z 0‖))).2 |
| nlinarith |
| rw [← he] at hdiv |
| nlinarith [norm_nonneg (centeredChart r U z)] |
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| lemma peak_chart_high {D : ℝ} (hD : 1 ≤ D) {k : ℕ} (hk : 1 ≤ k) |
| {P : Finset Sphere} (hP : ProjectivelySeparated (D/Real.sqrt k) P) |
| {r : ℝ} (hr : 0 ≤ r) (hr1 : r < 1) (U : Base ≃ₗᵢ[ℂ] Base) |
| {z : Base} (hz : ‖z‖ < (1/32:ℝ)) : |
| ‖peakPolynomial P k (centeredChart r U z)‖ ≤ gaussianConstant*Real.exp (-(k:ℝ)*(1-r)/16) := by |
| have hb := chart_norm_deficit hr hr1 U hz |
| have he : ‖centeredChart r U z‖ ≤ Real.exp (-(1-r)/16) := |
| hb.trans (by simpa only [neg_div] using Real.one_sub_le_exp_neg ((1-r)/16)) |
| have hp := pow_le_pow_left₀ (norm_nonneg (centeredChart r U z)) he k |
| rw [← Real.exp_nat_mul] at hp |
| have hex : (k:ℝ)*(-(1-r)/16)=-(k:ℝ)*(1-r)/16 := by ring |
| rw [hex] at hp |
| exact (peak_global_bound hD hk hP _).trans (mul_le_mul_of_nonneg_left hp gaussianConstant_pos.le) |
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| end PinchedHartogs.BaseConstruction |
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| end |
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| end OAI |
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