| import Mathlib |
| import OAI.Geometry.PrescribedPotential.InverseTraceFunction |
| import OAI.Geometry.PrescribedPotential.LogHessianCalculus |
| import OAI.Geometry.PrescribedPotential.MatrixTraceInequalities |
|
|
| namespace OAI |
|
|
| /-! Chern Lu Local Inequality. -/ |
|
|
| section |
|
|
| |
|
|
| noncomputable section |
| open Set Filter Topology Matrix |
| open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise |
| namespace KaehlerCalculus |
| variable {n : ℕ} |
|
|
| def curvatureDiagonal (G : V n → Matrix (Fin n) (Fin n) ℂ) (z v : V n) := |
| mderiv Complex.I v (mderiv (-Complex.I) v G) z - |
| (mderiv (-Complex.I) v G z)ᴴ*(G z)⁻¹*mderiv (-Complex.I) v G z |
|
|
| lemma trace_gradient_sum_bound (G : Matrix (Fin n) (Fin n) ℂ) (hG : G.PosDef) |
| (X Y W : Fin n → Matrix (Fin n) (Fin n) ℂ) : |
| ∑ k, ‖(Y k-X k*G).trace‖^2 ≤ G.trace.re * |
| ((∑ k, (X k*(X k)ᴴ*G-X k*(Y k)ᴴ-(X k)ᴴ*Y k+W k).trace.re) - |
| ∑ k, (W k-(Y k)ᴴ*G⁻¹*Y k).trace.re) := by |
| have h := Finset.sum_le_sum (s := Finset.univ) |
| (fun k _ => trace_gradient_square_bound hG (X k) (Y k)) |
| rw [← Finset.mul_sum] at h |
| have he : ((∑ k, (X k*(X k)ᴴ*G-X k*(Y k)ᴴ-(X k)ᴴ*Y k+W k).trace.re) - |
| ∑ k, (W k-(Y k)ᴴ*G⁻¹*Y k).trace.re) = |
| ∑ k, ((X k*(X k)ᴴ*G).trace.re-(X k*(Y k)ᴴ).trace.re- |
| ((X k)ᴴ*Y k).trace.re+((Y k)ᴴ*G⁻¹*Y k).trace.re) := by |
| rw [← Finset.sum_sub_distrib] |
| apply Finset.sum_congr rfl |
| intro k _ |
| simp only [Matrix.trace_add,Matrix.trace_sub,Complex.add_re,Complex.sub_re] |
| ring |
| rw [he] |
| exact h |
|
|
| lemma log_hessian_trace {U : Set (V n)} (hU : IsOpen U) |
| {f : V n → ℝ} (hf : ContDiffOn ℝ ∞ f U) (hp : ∀ y ∈ U, f y ≠ 0) |
| {z : V n} (hz : z ∈ U) : |
| (PotentialKaehler.potentialMatrix (fun y => Real.log (f y)) z).trace.re = |
| (f z)⁻¹*(PotentialKaehler.potentialMatrix f z).trace.re - |
| (f z)^(-2 : ℤ)*(∑ k, ‖dz (e k) (fun y => (f y : ℂ)) z‖^2) := by |
| simp only [Matrix.trace,Matrix.diag_apply,Complex.re_sum] |
| simp_rw [log_hessian_diagonal hU hf hp hz] |
| rw [Finset.sum_sub_distrib,← Finset.mul_sum,← Finset.mul_sum] |
|
|
| lemma scalar_log_cancellation {s d l r q c a : ℝ} (hs : 0 < s) |
| (hl : l = s⁻¹*d - s^(-2 : ℤ)*a) (hd : d = -r+q) |
| (ha : a ≤ s*(q-c)) : -r+c ≤ s*l := by |
| have he : s^2*l = s*d-a := by |
| rw [hl] |
| simp only [_root_.zpow_neg,zpow_ofNat] |
| field_simp |
| have hh : s*(-r+c) ≤ s^2*l := by rw [he,hd]; nlinarith |
| have hf : s^2*l = s*(s*l) := by ring |
| rw [hf] at hh |
| exact (mul_le_mul_iff_right₀ hs).mp hh |
|
|
| lemma inverseTrace_log_inequality_at_one [Nonempty (Fin n)] |
| {U : Set (V n)} (hU : IsOpen U) |
| {M G : V n → Matrix (Fin n) (Fin n) ℂ} (hM : ContDiffOn ℝ ∞ M U) |
| (hp : ∀ y ∈ U, (M y).PosDef) |
| (hclosed : ∀ y ∈ U, ∀ u v w : V n, |
| fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) v w) y u + |
| fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) w u) y v + |
| fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) u v) y w = 0) |
| (hG : ContDiffOn ℝ ∞ G U) (hGp : ∀ y ∈ U, (G y).PosDef) |
| {z : V n} (hz : z ∈ U) (hMz : M z = 1) : |
| -(PotentialKaehler.potentialMatrix (fun y => Real.log (M y).det.re) z*G z).trace.re + |
| ∑ k, (curvatureDiagonal G z (e k)).trace.re ≤ |
| inverseTrace M G z * |
| (PotentialKaehler.potentialMatrix (fun y => Real.log (inverseTrace M G y)) z).trace.re := by |
| let X := fun k => mderiv (-Complex.I) (e k) M z |
| let Y := fun k => mderiv (-Complex.I) (e k) G z |
| let W := fun k => mderiv Complex.I (e k) (mderiv (-Complex.I) (e k) G) z |
| have hGh : ∀ y ∈ U, (G y).IsHermitian := fun y hy => (hGp y hy).isHermitian |
| have hpos : ∀ y ∈ U, 0 < inverseTrace M G y := fun y hy => inverseTrace_pos (hp y hy) (hGp y hy) |
| have hl := log_hessian_trace hU (inverseTrace_smooth hU hM hp hG) |
| (fun y hy => ne_of_gt (hpos y hy)) hz |
| have hd := inverseTrace_second_at_one hU hM hp hclosed hG hGh hz hMz |
| have hg := trace_gradient_sum_bound (G z) (hGp z hz) X Y W |
| have hs : inverseTrace M G z = (G z).trace.re := by simp [inverseTrace,hMz] |
| have he : (∑ k, ‖dz (e k) (fun y => (inverseTrace M G y : ℂ)) z‖^2) = |
| ∑ k, ‖(Y k-X k*G z).trace‖^2 := by |
| apply Finset.sum_congr rfl |
| intro k _ |
| rw [inverseTrace_dz_at_one hU hM hp hG hGh hz hMz] |
| rw [he] at hl |
| rw [← hs] at hg |
| exact scalar_log_cancellation (hpos z hz) hl hd hg |
| end KaehlerCalculus |
|
|
| end |
| end |
|
|
| end OAI |
|
|
