| import Mathlib |
| import OAI.Geometry.PrescribedPotential.CalabiCovariant |
| import OAI.Geometry.PrescribedPotential.CalabiPairCalculus |
|
|
| namespace OAI |
|
|
| /-! Calabi Bochner. -/ |
|
|
| section |
|
|
| |
|
|
| noncomputable section |
| open Set Filter Topology Matrix |
| open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise |
| namespace KaehlerCalculus |
| variable {n : ℕ} |
|
|
| lemma tensorPairAt_sum_left {α : Type*} (s : Finset α) |
| (P M : Matrix (Fin n) (Fin n) ℂ) (T : α → ConnectionTensor n) (U : ConnectionTensor n) : |
| tensorPairAt P M (fun i => ∑ a ∈ s, T a i) U = ∑ a ∈ s, tensorPairAt P M (T a) U := by |
| simp only [tensorPairAt,Matrix.sum_mul,Matrix.trace_sum,Finset.mul_sum] |
| calc |
| _ = ∑ i, ∑ a ∈ s, ∑ j, P i j*(T a i*P*(U j)ᴴ*M).trace := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [Finset.sum_comm] |
| _ = _ := Finset.sum_comm |
|
|
| lemma tensorPairAt_sum_right {α : Type*} (s : Finset α) |
| (P M : Matrix (Fin n) (Fin n) ℂ) (T : ConnectionTensor n) (U : α → ConnectionTensor n) : |
| tensorPairAt P M T (fun i => ∑ a ∈ s, U a i) = ∑ a ∈ s, tensorPairAt P M T (U a) := by |
| simp only [tensorPairAt,Matrix.conjTranspose_sum,Matrix.sum_mul, |
| Matrix.trace_sum,Finset.mul_sum] |
| calc |
| _ = ∑ i, ∑ a ∈ s, ∑ j, P i j*(T i*P*(U a j)ᴴ*M).trace := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [Finset.sum_comm] |
| _ = _ := Finset.sum_comm |
|
|
| namespace LocalKaehlerField |
| variable (K : LocalKaehlerField n) |
|
|
| lemma covHol_bar_commute {T : Fin n → V n → Matrix (Fin n) (Fin n) ℂ} |
| (hT : ∀ i, ContDiffOn ℝ ∞ (T i) K.domain) {z : V n} (hz : z ∈ K.domain) (k i : Fin n) : |
| K.covHol (fun j => mderiv Complex.I (e k) (T j)) k i z = |
| mderiv Complex.I (e k) (K.covHol T k i) z - K.curvature k k z*T i z + |
| T i z*K.curvature k k z + ∑ q, K.curvature k k z q i • T q z := by |
| have hn := K.isOpen.mem_nhds hz |
| have ht (j) := (hT j).contDiffAt hn |
| have hg := K.connection_smooth k |>.contDiffAt hn |
| have hs := ContDiffAt.sum (fun q (_ : q ∈ (Finset.univ : Finset (Fin n))) => |
| matrix_smooth_smul (entry_smooth hg q i) (ht q)) |
| unfold covHol |
| rw [mderiv_sub (((mderiv_smooth (ht i) _ _).add (matrix_smooth_mul hg (ht i))).sub |
| (matrix_smooth_mul (ht i) hg)) hs, |
| mderiv_sub ((mderiv_smooth (ht i) _ _).add (matrix_smooth_mul hg (ht i))) |
| (matrix_smooth_mul (ht i) hg), |
| mderiv_add (mderiv_smooth (ht i) _ _) (matrix_smooth_mul hg (ht i)), |
| mderiv_mul hg (ht i),mderiv_mul (ht i) hg, |
| mderiv_sum Finset.univ (fun q _ => matrix_smooth_smul (entry_smooth hg q i) (ht q))] |
| simp only [mderiv_smul (entry_smooth hg _ i) (ht _),Finset.sum_add_distrib, |
| mderiv_comm (ht i) Complex.I (-Complex.I) (e k) (e k)] |
| change _ = _ - (K.curvature k k z*T i z) + _ + _ |
| unfold curvature |
| have he : (∑ q, wderiv Complex.I (e k) (fun y => K.connection k y q i) z • T q z) = |
| ∑ q, mderiv Complex.I (e k) (K.connection k) z q i • T q z := rfl |
| rw [he] |
| abel |
|
|
| lemma calabi_curvature_divergence_at_one {z : V n} (hz : z ∈ K.domain) |
| (hM : K.matrix z = 1) (i : Fin n) : |
| (∑ k, K.covHol (K.curvature k) k i z) = |
| mderiv (-Complex.I) (e i) K.ricciHessian z - K.ricciHessian z*K.connection i z := by |
| have he (k : Fin n) : K.covHol (K.curvature k) k i z = |
| mderiv Complex.I (e k) (K.covHol K.connection k i) z - K.curvature k k z*K.connection i z + |
| K.connection i z*K.curvature k k z + ∑ q, K.curvature k k z q i • K.connection q z := |
| K.covHol_bar_commute (T := K.connection) K.connection_smooth hz k i |
| simp_rw [he] |
| simp only [Finset.sum_add_distrib,Finset.sum_sub_distrib,← Matrix.sum_mul,← Matrix.mul_sum] |
| rw [K.calabi_connection_laplacian_at_one hz hM i,K.curvature_contraction_at_one hz hM] |
| have hs : (∑ k, ∑ q, K.curvature k k z q i • K.connection q z) = |
| ∑ q, K.ricciHessian z q i • K.connection q z := by |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro q _ |
| rw [← Finset.sum_smul] |
| congr 1 |
| simpa only [Matrix.sum_apply] using congrArg |
| (fun A : Matrix (Fin n) (Fin n) ℂ => A q i) (K.curvature_contraction_at_one hz hM) |
| rw [hs] |
| abel |
|
|
| def calabiNorm (z : V n) : ℝ := (K.tensorPair K.connection K.connection z).re |
|
|
| lemma calabiNorm_smooth : ContDiffOn ℝ ∞ K.calabiNorm K.domain := |
| Complex.reCLM.contDiff.comp_contDiffOn (K.tensorPair_smoothOn K.connection_smooth K.connection_smooth) |
|
|
| lemma calabiNorm_complex {z : V n} (hz : z ∈ K.domain) : |
| (K.calabiNorm z : ℂ) = K.tensorPair K.connection K.connection z := by |
| apply Complex.ext |
| · rfl |
| · exact (tensorPairAt_self_real (K.positive z hz).isHermitian.inv |
| (K.positive z hz).isHermitian _).symm |
|
|
| lemma calabiNorm_second {z : V n} (hz : z ∈ K.domain) (k : Fin n) : |
| dzbar (e k) (dz (e k) (fun y => (K.calabiNorm y : ℂ))) z = |
| K.tensorPair (fun i => mderiv Complex.I (e k) (K.covHol K.connection k i)) K.connection z + |
| K.tensorPair (K.covHol K.connection k) (K.covHol K.connection k) z + |
| K.tensorPair (K.curvature k) (K.curvature k) z + |
| K.tensorPair K.connection (K.covHol (K.curvature k) k) z := by |
| have hn := K.isOpen.mem_nhds hz |
| have hc : (fun y => (K.calabiNorm y : ℂ)) =ᶠ[𝓝 z] K.tensorPair K.connection K.connection := by |
| filter_upwards [hn] with y hy using K.calabiNorm_complex hy |
| have hd : dz (e k) (fun y => (K.calabiNorm y : ℂ)) =ᶠ[𝓝 z] |
| (fun y => K.tensorPair (K.covHol K.connection k) K.connection y + |
| K.tensorPair K.connection (K.curvature k) y) := by |
| filter_upwards [hn] with y hy |
| have he : (fun y => (K.calabiNorm y : ℂ)) =ᶠ[𝓝 y] K.tensorPair K.connection K.connection := by |
| filter_upwards [K.isOpen.mem_nhds hy] with x hx using K.calabiNorm_complex hx |
| change wderiv (-Complex.I) (e k) _ y = _ |
| rw [wderiv_congr he] |
| exact K.tensorPair_hol K.connection_smooth K.connection_smooth hy k |
| change wderiv Complex.I (e k) _ z = _ |
| rw [wderiv_congr hd,wderiv_add |
| ((K.tensorPair_smoothOn (T := K.covHol K.connection k) (U := K.connection) |
| (K.covHol_smooth (T := K.connection) K.connection_smooth k) K.connection_smooth).contDiffAt hn |
| |>.differentiableAt (by simp)) |
| ((K.tensorPair_smoothOn (T := K.connection) (U := K.curvature k) K.connection_smooth (K.curvature_smooth k)).contDiffAt hn |
| |>.differentiableAt (by simp))] |
| change dzbar (e k) _ z + dzbar (e k) _ z = _ |
| rw [K.tensorPair_bar (T := K.covHol K.connection k) (U := K.connection) |
| (K.covHol_smooth (T := K.connection) K.connection_smooth k) K.connection_smooth hz, |
| K.tensorPair_bar (T := K.connection) (U := K.curvature k) K.connection_smooth (K.curvature_smooth k) hz] |
| exact (add_assoc _ _ _).symm |
|
|
| lemma calabi_bochner_at_one {z : V n} (hz : z ∈ K.domain) (hM : K.matrix z = 1) : |
| (PotentialKaehler.potentialMatrix K.calabiNorm z).trace = |
| (∑ k, tensorPairAt 1 1 (fun i => K.covHol K.connection k i z) |
| (fun i => K.covHol K.connection k i z)) + |
| (∑ k, tensorPairAt 1 1 (fun i => K.curvature k i z) (fun i => K.curvature k i z)) + |
| tensorPairAt 1 1 (fun i => mderiv (-Complex.I) (e i) K.ricciHessian z - |
| K.connection i z*K.ricciHessian z - ∑ q, K.ricciHessian z q i • K.connection q z) |
| (fun i => K.connection i z) + |
| tensorPairAt 1 1 (fun i => K.connection i z) |
| (fun i => mderiv (-Complex.I) (e i) K.ricciHessian z - K.ricciHessian z*K.connection i z) := by |
| change (∑ k, PotentialKaehler.potentialMatrix K.calabiNorm z k k) = _ |
| simp_rw [potentialMatrix_eq_dzbar_dz (K.calabiNorm_smooth.contDiffAt (K.isOpen.mem_nhds hz))] |
| change (∑ k, dzbar (e k) (dz (e k) (fun y => (K.calabiNorm y : ℂ))) z) = _ |
| simp_rw [K.calabiNorm_second hz] |
| simp only [tensorPair,hM,inv_one,Finset.sum_add_distrib] |
| rw [← tensorPairAt_sum_left,← tensorPairAt_sum_right] |
| simp_rw [K.calabi_connection_laplacian_at_one hz hM,K.calabi_curvature_divergence_at_one hz hM] |
| abel |
|
|
| end LocalKaehlerField |
| end KaehlerCalculus |
|
|
| end |
| end |
|
|
| end OAI |
|
|
