| import Mathlib |
| import OAI.Geometry.PrescribedRicci.ChernLuAlgebra |
| import OAI.Geometry.PrescribedRicci.MatrixWirtingerExtra |
| import OAI.Geometry.PrescribedRicci.MatrixWirtingerSecond |
|
|
| namespace OAI |
|
|
| /-! Chern Connection. -/ |
|
|
| noncomputable section |
| open Matrix Filter Set Topology |
| open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise |
| namespace Anticanonical.SourceSmooth.KaehlerMetric |
| variable {d : ℕ} |
| local notation "Mat" => Matrix (Fin d) (Fin d) ℂ |
|
|
| lemma holDerivative_congr {f g : Coordinates d → Mat} {z : Coordinates d} |
| (he : f =ᶠ[nhds z] g) (a : Fin d) : holDerivative f z a = holDerivative g z a := by |
| unfold holDerivative |
| rw [he.fderiv_eq] |
|
|
| lemma holDerivative_sub {f g : Coordinates d → Mat} {z : Coordinates d} |
| (hf : DifferentiableAt ℝ f z) (hg : DifferentiableAt ℝ g z) (a : Fin d) : |
| holDerivative (fun y => f y-g y) z a = holDerivative f z a-holDerivative g z a := by |
| unfold holDerivative |
| have he : fderiv ℝ (fun y => f y-g y) z = fderiv ℝ f z-fderiv ℝ g z := by |
| convert! (hf.hasFDerivAt.sub hg.hasFDerivAt).fderiv using 1 |
| rw [he] |
| simp only [_root_.sub_apply,smul_sub] |
| abel |
|
|
| lemma hol_hol_comm {f : Coordinates d → Mat} {z : Coordinates d} |
| (hf : ContDiffAt ℝ ∞ f z) (a b : Fin d) : |
| holDerivative (fun y => holDerivative f y a) z b = |
| holDerivative (fun y => holDerivative f y b) z a := by |
| change (2:ℂ)⁻¹ • (fderiv ℝ (fun y => holDerivative f y a) z (coordinateVector b) - |
| Complex.I • fderiv ℝ (fun y => holDerivative f y a) z (Complex.I • coordinateVector b)) = |
| (2:ℂ)⁻¹ • (fderiv ℝ (fun y => holDerivative f y b) z (coordinateVector a) - |
| Complex.I • fderiv ℝ (fun y => holDerivative f y b) z (Complex.I • coordinateVector a)) |
| rw [fderiv_holDerivative hf,fderiv_holDerivative hf, |
| fderiv_holDerivative hf,fderiv_holDerivative hf] |
| have hs (u v : Coordinates d) : fderiv ℝ (fderiv ℝ f) z u v = fderiv ℝ (fderiv ℝ f) z v u := by |
| exact hf.isSymmSndFDerivAt (by |
| simp only [minSmoothness_of_isRCLikeNormedField] |
| change ((2:ℕ∞):WithTop ℕ∞) ≤ ((⊤:ℕ∞):WithTop ℕ∞) |
| exact WithTop.coe_le_coe.mpr le_top) u v |
| rw [hs (coordinateVector a) (coordinateVector b), |
| hs (coordinateVector a) (Complex.I • coordinateVector b), |
| hs (Complex.I • coordinateVector a) (coordinateVector b), |
| hs (Complex.I • coordinateVector a) (Complex.I • coordinateVector b)] |
| ext i j |
| simp only [Matrix.smul_apply,Matrix.sub_apply,smul_eq_mul] |
| ring |
|
|
| variable {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X} |
|
|
| def chernConnection (g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d) (a : Fin d) : Mat := |
| (g.matrix q z)⁻¹*holDerivative (g.matrix q) z a |
|
|
| def chernCurvature (g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d) |
| (b a : Fin d) : Mat := (g.matrix q z)⁻¹*g.curvatureMatrix q z b a |
|
|
| lemma chernConnection_smooth (g : KaehlerMetric A) (q : Fin A.count) |
| {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) : |
| ContDiffAt ℝ ∞ (fun y => g.chernConnection q y a) z := by |
| have hg := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz) |
| exact matrix_mul_contDiffAt |
| (((MongeAmpere.contDiffAt_inv _ (g.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hg) |
| (contDiffAt_holDerivative hg a) |
| |
| lemma chernConnection_torsion (g : KaehlerMetric A) (q : Fin A.count) |
| {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a k j : Fin d) : |
| g.chernConnection q z a k j = g.chernConnection q z j k a := by |
| simp only [chernConnection,Matrix.mul_apply] |
| apply Finset.sum_congr rfl |
| intro u _ |
| rw [g.holDerivative_symm q hz a u j] |
| |
| lemma chernConnection_bar (g : KaehlerMetric A) (q : Fin A.count) |
| {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a b : Fin d) : |
| barDerivative (fun y => g.chernConnection q y a) z b = -g.chernCurvature q z b a := by |
| have hg := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz) |
| have hi : ContDiffAt ℝ ∞ (fun y => (g.matrix q y)⁻¹) z := |
| ((MongeAmpere.contDiffAt_inv _ (g.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hg |
| unfold chernConnection |
| rw [barDerivative_mul (hi.differentiableAt (by simp)) |
| ((contDiffAt_holDerivative hg a).differentiableAt (by simp)), |
| barDerivative_inv (hg.differentiableAt (by simp)) (isUnit_iff_ne_zero.mpr (g.positive q z hz).det_pos.ne')] |
| simp only [chernCurvature,curvatureMatrix,mul_add,neg_mul,mul_neg,mul_assoc] |
| abel |
| |
| lemma chernConnection_hol (g : KaehlerMetric A) (q : Fin A.count) |
| {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a u : Fin d) : |
| holDerivative (fun y => g.chernConnection q y a) z u = |
| -(g.chernConnection q z u*g.chernConnection q z a)+ |
| (g.matrix q z)⁻¹*holDerivative (fun y => holDerivative (g.matrix q) y a) z u := by |
| have hg := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz) |
| have hi : ContDiffAt ℝ ∞ (fun y => (g.matrix q y)⁻¹) z := |
| ((MongeAmpere.contDiffAt_inv _ (g.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hg |
| unfold chernConnection |
| rw [holDerivative_mul (hi.differentiableAt (by simp)) |
| ((contDiffAt_holDerivative hg a).differentiableAt (by simp)), |
| holDerivative_inv (hg.differentiableAt (by simp)) (isUnit_iff_ne_zero.mpr (g.positive q z hz).det_pos.ne')] |
| simp only [neg_mul,mul_assoc] |
| |
| lemma chernConnection_maurerCartan (g : KaehlerMetric A) (q : Fin A.count) |
| {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a u : Fin d) : |
| holDerivative (fun y => g.chernConnection q y a) z u - |
| holDerivative (fun y => g.chernConnection q y u) z a = |
| g.chernConnection q z a*g.chernConnection q z u- |
| g.chernConnection q z u*g.chernConnection q z a := by |
| rw [g.chernConnection_hol q hz,g.chernConnection_hol q hz, |
| hol_hol_comm ((g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)) a u] |
| abel |
| |
| end Anticanonical.SourceSmooth.KaehlerMetric |
| |
| end |
| |
| end OAI |
| |
