| import Mathlib | |
| import OAI.Geometry.PrescribedRicci.CurvatureRicci | |
| import OAI.Geometry.PrescribedRicci.KahlerGradientEnergy | |
| import OAI.Geometry.PrescribedRicci.KaehlerTrace | |
| import OAI.Geometry.PrescribedRicci.MatrixWirtingerExtra | |
| namespace OAI | |
| /-! Chern Lu Connection. -/ | |
| section | |
| noncomputable section | |
| open Matrix Filter Set Topology | |
| open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise | |
| namespace MongeAmpere | |
| variable {n : Type*} [Fintype n] [DecidableEq n] | |
| lemma trace_lu_W_identity (B G Pg Qg Ph Qh Rg Rh : Matrix n n ℂ) | |
| (hG : IsUnit G.det) : | |
| (-(B*Qh*B)*(Pg-G*B*Ph)+ | |
| B*(Rg-(Qg*B*Ph-G*B*Qh*B*Ph+G*B*Rh))).trace = | |
| (-(B*(-Rg+Qg*G⁻¹*Pg))).trace + | |
| (B*(Qg-Qh*B*G)*G⁻¹*(Pg-G*B*Ph)).trace + | |
| (B*(-Rh+Qh*B*Ph)*B*G).trace := by | |
| have he : B*(Qg-Qh*B*G)*G⁻¹*(Pg-G*B*Ph) = | |
| B*Qg*G⁻¹*Pg-B*Qg*B*Ph-B*Qh*B*Pg+B*Qh*B*G*B*Ph := by | |
| simp only [mul_sub,sub_mul] | |
| simp only [mul_assoc,Matrix.mul_nonsing_inv_cancel_left G _ hG, | |
| Matrix.nonsing_inv_mul_cancel_left G _ hG] | |
| noncomm_ring | |
| rw [he] | |
| simp only [mul_add,add_mul,mul_sub,mul_neg,neg_mul,trace_add,trace_sub,trace_neg] | |
| have h1 : (B*G*B*Qh*B*Ph).trace = (B*Qh*B*Ph*B*G).trace := by | |
| simpa only [mul_assoc] using Matrix.trace_mul_comm (B*G) (B*Qh*B*Ph) | |
| have h2 : (B*G*B*Rh).trace = (B*Rh*B*G).trace := by | |
| simpa only [mul_assoc] using Matrix.trace_mul_comm (B*G) (B*Rh) | |
| simp only [mul_assoc] at h1 h2 ⊢ | |
| linear_combination h1-h2 | |
| end MongeAmpere | |
| namespace Anticanonical.SourceSmooth.KaehlerMetric | |
| variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X} | |
| local notation "Mat" => Matrix (Fin d) (Fin d) ℂ | |
| def connectionDifference (h g : KaehlerMetric A) (q : Fin A.count) | |
| (z : Coordinates d) (a : Fin d) : Mat := | |
| holDerivative (g.matrix q) z a - g.matrix q z*(h.matrix q z)⁻¹*holDerivative (h.matrix q) z a | |
| lemma connectionDifference_smooth (h g : KaehlerMetric A) (q : Fin A.count) | |
| {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) : | |
| ContDiffAt ℝ ∞ (fun y => h.connectionDifference g q y a) z := by | |
| have hg := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz) | |
| have hh := (h.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz) | |
| have hi : ContDiffAt ℝ ∞ (fun y => (h.matrix q y)⁻¹) z := | |
| ((MongeAmpere.contDiffAt_inv _ (h.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hh | |
| exact (contDiffAt_holDerivative hg a).sub | |
| (matrix_mul_contDiffAt (matrix_mul_contDiffAt hg hi) (contDiffAt_holDerivative hh a)) | |
| lemma connectionDifference_adjoint (h g : KaehlerMetric A) (q : Fin A.count) | |
| {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) : | |
| (h.connectionDifference g q z a)ᴴ = | |
| barDerivative (g.matrix q) z a - barDerivative (h.matrix q) z a*(h.matrix q z)⁻¹*g.matrix q z := by | |
| unfold connectionDifference | |
| rw [conjTranspose_sub,conjTranspose_mul,conjTranspose_mul, | |
| g.holDerivative_adjoint q hz,h.holDerivative_adjoint q hz, | |
| (h.positive q z hz).inv.isHermitian.eq,(g.positive q z hz).isHermitian.eq] | |
| simp only [mul_assoc] | |
| lemma holRealDeriv_traceMetric (h g : KaehlerMetric A) (q : Fin A.count) | |
| {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) : | |
| holRealDeriv ((h.traceMetric g).localExpression q) z a = | |
| ((h.matrix q z)⁻¹*h.connectionDifference g q z a).trace := by | |
| have hs : ContDiffAt ℝ ∞ ((h.traceMetric g).localExpression q) z := | |
| ((h.traceMetric g).smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz) | |
| have hg := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz) | |
| have hh := (h.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz) | |
| have hi : ContDiffAt ℝ ∞ (fun y => (h.matrix q y)⁻¹) z := | |
| ((MongeAmpere.contDiffAt_inv _ (h.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hh | |
| have he : (fun y => (((h.traceMetric g).localExpression q y):ℂ)) =ᶠ[nhds z] | |
| (fun y => ((h.matrix q y)⁻¹*g.matrix q y).trace) := by | |
| filter_upwards [(A.chart q).open_target.mem_nhds hz] with y hy | |
| rw [h.traceMetric_local g q hy] | |
| exact MongeAmpere.trace_mul_real (h.positive q y hy).inv.isHermitian (g.positive q y hy).isHermitian | |
| rw [← holDeriv_ofReal (hs.differentiableAt (by simp)),holDeriv_congr he, | |
| holDeriv_trace (matrix_mul_differentiableAt (hi.differentiableAt (by simp)) (hg.differentiableAt (by simp))), | |
| holDerivative_mul (hi.differentiableAt (by simp)) (hg.differentiableAt (by simp)), | |
| holDerivative_inv (hh.differentiableAt (by simp)) (isUnit_iff_ne_zero.mpr (h.positive q z hz).det_pos.ne')] | |
| simp only [connectionDifference,neg_mul,mul_sub,trace_add,trace_sub,trace_neg] | |
| have hc : ((h.matrix q z)⁻¹*holDerivative (h.matrix q) z a*(h.matrix q z)⁻¹*g.matrix q z).trace = | |
| ((h.matrix q z)⁻¹*g.matrix q z*(h.matrix q z)⁻¹*holDerivative (h.matrix q) z a).trace := by | |
| simpa only [mul_assoc] using Matrix.trace_mul_comm | |
| ((h.matrix q z)⁻¹*holDerivative (h.matrix q) z a) ((h.matrix q z)⁻¹*g.matrix q z) | |
| simp only [mul_assoc] at hc ⊢ | |
| linear_combination -hc | |
| lemma barDerivative_connectionDifference (h g : KaehlerMetric A) (q : Fin A.count) | |
| {z : Coordinates d} (hz : z ∈ (A.chart q).target) (b a : Fin d) : | |
| barDerivative (fun y => h.connectionDifference g q y a) z b = | |
| barDerivative (fun y => holDerivative (g.matrix q) y a) z b - | |
| (barDerivative (g.matrix q) z b*(h.matrix q z)⁻¹*holDerivative (h.matrix q) z a - | |
| g.matrix q z*(h.matrix q z)⁻¹*barDerivative (h.matrix q) z b*(h.matrix q z)⁻¹*holDerivative (h.matrix q) z a + | |
| g.matrix q z*(h.matrix q z)⁻¹*barDerivative (fun y => holDerivative (h.matrix q) y a) z b) := by | |
| have hg := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz) | |
| have hh := (h.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz) | |
| have hi : ContDiffAt ℝ ∞ (fun y => (h.matrix q y)⁻¹) z := | |
| ((MongeAmpere.contDiffAt_inv _ (h.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hh | |
| have pd := (contDiffAt_holDerivative hh a).differentiableAt (by simp) | |
| have gd := hg.differentiableAt (by simp) | |
| have id := hi.differentiableAt (by simp) | |
| unfold connectionDifference | |
| rw [barDerivative_sub ((contDiffAt_holDerivative hg a).differentiableAt (by simp)) | |
| (matrix_mul_differentiableAt (matrix_mul_differentiableAt gd id) pd), | |
| barDerivative_mul (matrix_mul_differentiableAt gd id) pd, | |
| barDerivative_mul gd id, | |
| barDerivative_inv (hh.differentiableAt (by simp)) (isUnit_iff_ne_zero.mpr (h.positive q z hz).det_pos.ne')] | |
| simp only [mul_neg,neg_mul,add_mul,mul_assoc,sub_eq_add_neg] | |
| lemma hessian_traceMetric (h g : KaehlerMetric A) (q : Fin A.count) | |
| {z : Coordinates d} (hz : z ∈ (A.chart q).target) (b a : Fin d) : | |
| (h.traceMetric g).hessian q z b a = | |
| (-((h.matrix q z)⁻¹*g.curvatureMatrix q z b a)).trace + | |
| ((h.matrix q z)⁻¹*(h.connectionDifference g q z b)ᴴ*(g.matrix q z)⁻¹*h.connectionDifference g q z a).trace + | |
| ((h.matrix q z)⁻¹*h.curvatureMatrix q z b a*(h.matrix q z)⁻¹*g.matrix q z).trace := by | |
| have hh := (h.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz) | |
| have hi : ContDiffAt ℝ ∞ (fun y => (h.matrix q y)⁻¹) z := | |
| ((MongeAmpere.contDiffAt_inv _ (h.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hh | |
| have hw := (h.connectionDifference_smooth g q hz a).differentiableAt (by simp) | |
| have he : (fun y => holRealDeriv ((h.traceMetric g).localExpression q) y a) =ᶠ[nhds z] | |
| (fun y => ((h.matrix q y)⁻¹*h.connectionDifference g q y a).trace) := by | |
| filter_upwards [(A.chart q).open_target.mem_nhds hz] with y hy | |
| exact h.holRealDeriv_traceMetric g q hy a | |
| have hs : ContDiffAt ℝ ∞ ((h.traceMetric g).localExpression q) z := | |
| ((h.traceMetric g).smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz) | |
| change PotentialKaehler.potentialMatrix ((h.traceMetric g).localExpression q) z b a = _ | |
| rw [← barDeriv_holRealDeriv hs, | |
| barDeriv_congr he,barDeriv_trace (matrix_mul_differentiableAt (hi.differentiableAt (by simp)) hw), | |
| barDerivative_mul (hi.differentiableAt (by simp)) hw, | |
| barDerivative_inv (hh.differentiableAt (by simp)) (isUnit_iff_ne_zero.mpr (h.positive q z hz).det_pos.ne'), | |
| h.barDerivative_connectionDifference g q hz,h.connectionDifference_adjoint g q hz] | |
| exact MongeAmpere.trace_lu_W_identity _ _ _ _ _ _ _ _ | |
| (isUnit_iff_ne_zero.mpr (g.positive q z hz).det_pos.ne') | |
| end Anticanonical.SourceSmooth.KaehlerMetric | |
| end | |
| end | |
| end OAI | |