File size: 4,476 Bytes
2fca478 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 | 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
|