| import OAI.Geometry.Kahler.HartogsTransferTools |
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| namespace OAI |
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| open scoped ContDiff |
| open Complex |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise |
| open Set Filter Topology |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder |
| noncomputable section |
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| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ComplexOrder |
| namespace PinchedHartogs |
| namespace BaseConstruction |
|
|
| abbrev Sphere := Metric.sphere (0 : Base) 1 |
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| def bracket (z p : Base) : ℂ := inner ℂ p z |
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| def peakPolynomial (P : Finset Sphere) (k : ℕ) (z : Base) : ℂ := |
| ∑ p ∈ P, bracket z p ^ k |
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| def regularizedLog (a ε : ℝ) (v : ℂ) : ℝ := |
| Real.log ((‖1-(a:ℂ)*v‖^2+ε^2)/(1+ε^2)) |
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| def logarithmicTest (a ε : ℝ) (P : Finset Sphere) (k : ℕ) (z : Base) : ℝ := |
| regularizedLog a ε (peakPolynomial P k z) |
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| def basePotential (P : ℕ → Finset Sphere) (a ε L : ℝ) (Q : ℕ) (z : Base) : ℝ := |
| psi z + L * ∑' j : ℕ, logarithmicTest a ε (P (Q^(j+1))) (Q^(j+1)) z |
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| lemma bracket_analytic (p : Base) : AnalyticOnNhd ℂ (fun z => bracket z p) univ := |
| fun z _ => (innerSL ℂ p).analyticAt z |
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| lemma peakPolynomial_analytic (P : Finset Sphere) (k : ℕ) : |
| AnalyticOnNhd ℂ (peakPolynomial P k) univ := by |
| intro z hz |
| apply Finset.analyticAt_fun_sum |
| intro p hp |
| exact (bracket_analytic p z (mem_univ _)).pow k |
|
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| lemma regularizedLog_eq (a ε : ℝ) (v : ℂ) (hε : ε ≠ 0) : |
| regularizedLog a ε v = Real.log (‖1-(a:ℂ)*v‖^2+ε^2)-Real.log (1+ε^2) := by |
| apply Real.log_div <;> positivity |
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| @[simp] lemma regularizedLog_zero (a ε : ℝ) : regularizedLog a ε 0 = 0 := by |
| simp [regularizedLog] |
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| lemma regularizedLog_contDiff (a : ℝ) {ε : ℝ} (hε : ε ≠ 0) : |
| ContDiff ℝ ∞ (regularizedLog a ε) := by |
| have hn : ContDiff ℝ ∞ (fun v : ℂ => ‖1-(a:ℂ)*v‖^2+ε^2) := |
| ((contDiff_norm_sq ℂ).comp (contDiff_const.sub (contDiff_const.mul contDiff_id))).add contDiff_const |
| have hh : ContDiff ℝ ∞ (fun v : ℂ => Real.log (‖1-(a:ℂ)*v‖^2+ε^2)-Real.log (1+ε^2)) := |
| (hn.log (fun v => by positivity)).sub contDiff_const |
| have he : regularizedLog a ε = (fun v : ℂ => Real.log (‖1-(a:ℂ)*v‖^2+ε^2)-Real.log (1+ε^2)) := |
| funext (fun v => regularizedLog_eq a ε v hε) |
| rw [he] |
| exact hh |
|
|
| lemma logarithmicTest_contDiff (a : ℝ) {ε : ℝ} (hε : ε ≠ 0) (P : Finset Sphere) (k : ℕ) : |
| ContDiff ℝ ∞ (logarithmicTest a ε P k) := by |
| apply contDiff_iff_contDiffAt.mpr |
| intro z |
| exact (regularizedLog_contDiff a hε).contDiffAt.comp z |
| (((peakPolynomial_analytic P k) z (mem_univ _)).contDiffAt.restrict_scalars ℝ) |
|
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| lemma peakPolynomial_zero (P : Finset Sphere) {k : ℕ} (hk : k ≠ 0) : |
| peakPolynomial P k 0 = 0 := by simp [peakPolynomial,bracket,hk] |
|
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| lemma logarithmicTest_zero (a ε : ℝ) (P : Finset Sphere) {k : ℕ} (hk : k ≠ 0) : |
| logarithmicTest a ε P k 0 = 0 := by simp [logarithmicTest,peakPolynomial_zero P hk] |
|
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| lemma peakPolynomial_homogeneous (P : Finset Sphere) (k : ℕ) (c : ℂ) (z : Base) : |
| peakPolynomial P k (c • z) = c^k * peakPolynomial P k z := by |
| simp only [peakPolynomial,bracket, inner_smul_right,mul_pow,Finset.mul_sum] |
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| lemma regularizedLog_lipschitz_bound (a : ℝ) {ε : ℝ} (hε : ε ≠ 0) {A : ℝ} (hA : 0 ≤ A) : |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ v : ℂ, ‖v‖ ≤ A → |regularizedLog a ε v| ≤ C*‖v‖ := by |
| have hs := regularizedLog_contDiff a hε |
| have hc : Continuous (fderiv ℝ (regularizedLog a ε)) := hs.continuous_fderiv (by norm_num) |
| obtain ⟨C,hC⟩ := (isCompact_closedBall (0:ℂ) A).bddAbove_image hc.norm.continuousOn |
| refine ⟨max C 0, le_max_right _ _, ?_⟩ |
| intro v hv |
| have hbound : ∀ x ∈ Metric.closedBall (0:ℂ) A, ‖fderiv ℝ (regularizedLog a ε) x‖ ≤ max C 0 := by |
| intro x hx |
| exact (hC (mem_image_of_mem _ hx)).trans (le_max_left _ _) |
| have hm := (convex_closedBall (0:ℂ) A).norm_image_sub_le_of_norm_fderiv_le |
| (fun x _ => hs.differentiable (by norm_num) x) hbound |
| (show (0:ℂ) ∈ Metric.closedBall 0 A by simpa using hA) |
| (show v ∈ Metric.closedBall 0 A by simpa using hv) |
| simpa [Real.norm_eq_abs] using hm |
|
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| private def regularizedFiberLog (ε : ℝ) (q : Ambient) : ℝ := |
| Real.log (‖q.2‖^2+ε^2)-Real.log (1+ε^2) |
|
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| private lemma regularizedFiberLog_contDiff {ε : ℝ} (hε : ε ≠ 0) : |
| ContDiff ℝ ∞ (regularizedFiberLog ε) := by |
| have he := sq_pos_of_ne_zero hε |
| have hn : ContDiff ℝ ∞ (fun q : Ambient => ‖q.2‖^2+ε^2) := |
| ((contDiff_norm_sq ℂ).comp contDiff_snd).add contDiff_const |
| exact (hn.log (fun q => ne_of_gt (add_pos_of_nonneg_of_pos (sq_nonneg _) he))).sub contDiff_const |
|
|
| private lemma regularizedFiberLog_hessian {ε : ℝ} (hε : ε ≠ 0) (q : Ambient) (i j : Fin 3) : |
| complexHessian (regularizedFiberLog ε) q i j = |
| ((ε^2/(‖q.2‖^2+ε^2)^2 : ℝ):ℂ) * (if i=2 then 1 else 0)*(if j=2 then 1 else 0) := by |
| have hden : ‖q.2‖^2+ε^2 ≠ 0 := by positivity |
| have hρ : ∀ᶠ y in 𝓝 (‖q.2‖^2), |
| HasDerivAt (fun s => Real.log (s+ε^2)-Real.log (1+ε^2)) ((y+ε^2)⁻¹) y := by |
| filter_upwards [(continuousAt_id.add continuousAt_const).eventually_ne hden] with y hy |
| simpa [one_div] using (((hasDerivAt_id y).add_const (ε^2)).log hy).sub_const (Real.log (1+ε^2)) |
| have hρ' : HasDerivAt (fun s : ℝ => (s+ε^2)⁻¹) (-( (‖q.2‖^2+ε^2)^2)⁻¹) (‖q.2‖^2) := by |
| convert ((hasDerivAt_id (‖q.2‖^2)).add_const (ε^2)).inv hden using 1 <;> try rfl |
| simp [one_div, neg_div] |
| have hh := complexHessian_real_comp |
| (((contDiff_norm_sq ℂ).comp contDiff_snd).contDiffAt : ContDiffAt ℝ 2 (fun p : Ambient => ‖p.2‖^2) q) |
| hρ hρ' i j |
| dsimp only [Function.comp_def] at hh |
| rw [complexHessian_fiber_norm_sq,dz_fiber_norm_sq,dbar_fiber_norm_sq] at hh |
| change complexHessian (regularizedFiberLog ε) q i j = _ at hh |
| rw [hh] |
| have hn : star q.2 * q.2 = (‖q.2‖^2 : ℝ) := by |
| rw [Complex.star_def, ← Complex.normSq_eq_conj_mul_self, Complex.normSq_eq_norm_sq] |
| push_cast |
| by_cases hi : i=2 <;> by_cases hj : j=2 <;> simp [hi,hj] |
| rw [mul_assoc, ← Complex.star_def, hn] |
| have hd : (↑‖q.2‖^2 + (ε:ℂ)^2) ≠ 0 := by exact_mod_cast hden |
| field_simp |
| ; push_cast; ring |
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| lemma regularizedLog_holomorphic_psh (a : ℝ) {ε : ℝ} (he : ε ≠ 0) |
| {f : Base → ℂ} {z : Base} (hf : AnalyticAt ℂ f z) (v : Base) : |
| 0 ≤ (∑ i, ∑ j, baseHessian (regularizedLog a ε ∘ f) z i j * v i * star (v j)).re := by |
| let F : Ambient → Ambient := fun q => (q.1,1-(a:ℂ)*f q.1) |
| have hF : AnalyticAt ℂ F (z,0) := |
| ((ContinuousLinearMap.fst ℂ Base ℂ).analyticAt _).prod |
| (analyticAt_const.sub (analyticAt_const.mul (hf.comp ((ContinuousLinearMap.fst ℂ Base ℂ).analyticAt _)))) |
| have hfc : ContDiffAt ℝ 2 (regularizedLog a ε ∘ f) z := |
| ((regularizedLog_contDiff a he).contDiffAt.comp z (hf.contDiffAt.restrict_scalars ℝ)).of_le (WithTop.coe_le_coe.mpr le_top) |
| have heq : regularizedFiberLog ε ∘ F = fun q : Ambient => (regularizedLog a ε ∘ f) q.1 := by |
| funext q |
| exact (regularizedLog_eq a ε (f q.1) he).symm |
| have hh := congrArg (fun A => (hermitianValue A (v,0) (v,0)).re) |
| (complexHessian_pullback ((regularizedFiberLog_contDiff he).contDiffAt.of_le (WithTop.coe_le_coe.mpr le_top)) hF) |
| rw [heq,hermitianValue_lift_base hfc,hermitianValue_pullback] at hh |
| rw [hh] |
| let u := fderiv ℂ F (z,0) (v,0) |
| change 0 ≤ (hermitianValue (complexHessian (regularizedFiberLog ε) (F (z,0))) u u).re |
| simp only [hermitianValue,regularizedFiberLog_hessian he,Fin.sum_univ_three] |
| simp only [Fin.reduceEq,ite_false,ite_true,mul_zero,zero_mul,zero_add,add_zero] |
| have hu : u.2*star u.2 = (‖u.2‖^2:ℝ) := by |
| rw [Complex.star_def,Complex.mul_conj,Complex.normSq_eq_norm_sq] |
| change 0 ≤ (((ε^2/(‖(F (z,0)).2‖^2+ε^2)^2:ℝ):ℂ)*1*1*u.2*star u.2).re |
| rw [mul_one,mul_one,mul_assoc,hu] |
| simp only [← Complex.ofReal_mul,Complex.ofReal_re] |
| positivity |
|
|
| lemma logarithmicTest_psh (a : ℝ) {ε : ℝ} (he : ε ≠ 0) (P : Finset Sphere) |
| (k : ℕ) (z v : Base) : |
| 0 ≤ (∑ i, ∑ j, baseHessian (logarithmicTest a ε P k) z i j * v i * star (v j)).re := |
| regularizedLog_holomorphic_psh a he (peakPolynomial_analytic P k z (mem_univ _)) v |
|
|
| end BaseConstruction |
| end PinchedHartogs |
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| end |
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| end OAI |
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