| import Mathlib |
| import OAI.Geometry.PrescribedRicci.CalabiTensorAlgebra |
| import OAI.Geometry.PrescribedRicci.ChernPairDifferential |
| import OAI.Geometry.PrescribedRicci.KahlerGradientEnergy |
| import OAI.Geometry.PrescribedRicci.MatrixWirtinger |
|
|
| namespace OAI |
|
|
| /-! Array Wirtinger. -/ |
|
|
| noncomputable section |
| open Matrix Filter Set Topology |
| open scoped ContDiff ComplexOrder Kronecker |
| namespace Anticanonical.SourceSmooth.KaehlerMetric |
| open MongeAmpere |
| variable {d : ℕ} {n : Type*} |
|
|
| lemma holArray_eq_holDerivative {f : Coordinates d → Matrix (Fin d) (Fin d) ℂ} |
| {z : Coordinates d} (hf : DifferentiableAt ℝ f z) (a : Fin d) : |
| holArray f z a = holDerivative f z a := by |
| ext i j |
| exact holDeriv_entry hf a i j |
|
|
| lemma barArray_eq_barDerivative {f : Coordinates d → Matrix (Fin d) (Fin d) ℂ} |
| {z : Coordinates d} (hf : DifferentiableAt ℝ f z) (a : Fin d) : |
| barArray f z a = barDerivative f z a := by |
| ext i j |
| exact barDeriv_entry hf a i j |
|
|
| lemma holArray_const (K : Matrix n n ℂ) (z : Coordinates d) (a : Fin d) : |
| holArray (fun _ => K) z a = 0 := by |
| ext i j |
| simp [holArray,holDeriv] |
|
|
| lemma barArray_const (K : Matrix n n ℂ) (z : Coordinates d) (a : Fin d) : |
| barArray (fun _ => K) z a = 0 := by |
| ext i j |
| simp [barArray,barDeriv] |
|
|
| lemma holArray_transpose (K : Coordinates d → Matrix n n ℂ) (z : Coordinates d) (a : Fin d) : |
| holArray (fun y => (K y).transpose) z a = (holArray K z a).transpose := rfl |
|
|
| lemma barArray_transpose (K : Coordinates d → Matrix n n ℂ) (z : Coordinates d) (a : Fin d) : |
| barArray (fun y => (K y).transpose) z a = (barArray K z a).transpose := rfl |
|
|
| lemma barArray_eq_adjoint_hol {K : Coordinates d → Matrix n n ℂ} {z : Coordinates d} |
| (hK : ∀ᶠ y in nhds z, (K y).IsHermitian) (a : Fin d) : |
| barArray K z a = (holArray K z a)ᴴ := by |
| ext i j |
| change barDeriv (fun y => K y i j) z a = star (holDeriv (fun y => K y j i) z a) |
| have he : (fun y => K y i j) =ᶠ[nhds z] (fun y => star (K y j i)) := by |
| filter_upwards [hK] with y hy |
| exact (Matrix.IsHermitian.ext_iff.mp hy i j).symm |
| rw [barDeriv_congr he,barDeriv_star] |
|
|
| lemma holArray_tensor3 {L U V : Coordinates d → Matrix n n ℂ} {z : Coordinates d} |
| (hL : ∀ i j, DifferentiableAt ℝ (fun y => L y i j) z) |
| (hU : ∀ i j, DifferentiableAt ℝ (fun y => U y i j) z) |
| (hV : ∀ i j, DifferentiableAt ℝ (fun y => V y i j) z) (a : Fin d) : |
| holArray (fun y => tensor3 (L y) (U y) (V y)) z a = |
| tensor3 (holArray L z a) (U z) (V z)+ |
| tensor3 (L z) (holArray U z a) (V z)+tensor3 (L z) (U z) (holArray V z a) := by |
| ext i j |
| change holDeriv (fun y => L y i.1 j.1*(U y i.2.1 j.2.1*V y i.2.2 j.2.2)) z a = _ |
| rw [holDeriv_mul (hL _ _) ((hU _ _).fun_mul (hV _ _)), |
| holDeriv_mul (hU _ _) (hV _ _)] |
| simp only [Matrix.add_apply,tensor3_apply,holArray] |
| ring |
|
|
| lemma barArray_tensor3 {L U V : Coordinates d → Matrix n n ℂ} {z : Coordinates d} |
| (hL : ∀ i j, DifferentiableAt ℝ (fun y => L y i j) z) |
| (hU : ∀ i j, DifferentiableAt ℝ (fun y => U y i j) z) |
| (hV : ∀ i j, DifferentiableAt ℝ (fun y => V y i j) z) (a : Fin d) : |
| barArray (fun y => tensor3 (L y) (U y) (V y)) z a = |
| tensor3 (barArray L z a) (U z) (V z)+ |
| tensor3 (L z) (barArray U z a) (V z)+tensor3 (L z) (U z) (barArray V z a) := by |
| ext i j |
| change barDeriv (fun y => L y i.1 j.1*(U y i.2.1 j.2.1*V y i.2.2 j.2.2)) z a = _ |
| rw [barDeriv_mul (hL _ _) ((hU _ _).fun_mul (hV _ _)), |
| barDeriv_mul (hU _ _) (hV _ _)] |
| simp only [Matrix.add_apply,tensor3_apply,barArray] |
| ring |
|
|
| end Anticanonical.SourceSmooth.KaehlerMetric |
|
|
| end |
|
|
| end OAI |
|
|
