| import Mathlib |
| import OAI.Geometry.PrescribedPotential.CalabiConnection |
| import OAI.Geometry.PrescribedPotential.CalabiCovariant |
| import OAI.Geometry.PrescribedPotential.CalabiTensorAlgebra |
| import OAI.Geometry.PrescribedPotential.MatrixTensorCalculus |
|
|
| namespace OAI |
|
|
| /-! Calabi Pair Calculus. -/ |
|
|
| section |
|
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| |
| noncomputable section |
| open Set Filter Topology Matrix |
| open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise |
| namespace KaehlerCalculus |
| variable {n : ℕ} |
|
|
| lemma calabi_pair_matrix_trace_smooth {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n} |
| (hM : ContDiffAt ℝ ∞ M z) : ContDiffAt ℝ ∞ (fun y => (M y).trace) z := |
| ContDiffAt.sum (fun i _ => entry_smooth hM i i) |
|
|
| lemma tensorPair_smooth {P M : V n → Matrix (Fin n) (Fin n) ℂ} |
| {T U : Fin n → V n → Matrix (Fin n) (Fin n) ℂ} {z : V n} |
| (hP : ContDiffAt ℝ ∞ P z) (hM : ContDiffAt ℝ ∞ M z) |
| (hT : ∀ i, ContDiffAt ℝ ∞ (T i) z) (hU : ∀ i, ContDiffAt ℝ ∞ (U i) z) : |
| ContDiffAt ℝ ∞ (fun y => tensorPairAt (P y) (M y) (fun i => T i y) (fun i => U i y)) z := by |
| exact ContDiffAt.sum (fun i _ => ContDiffAt.sum (fun j _ => |
| (entry_smooth hP i j).mul (calabi_pair_matrix_trace_smooth (matrix_smooth_mul |
| (matrix_smooth_mul (matrix_smooth_mul (hT i) hP) (matrix_smooth_conjTranspose (hU j))) hM)))) |
|
|
| namespace LocalKaehlerField |
| variable (K : LocalKaehlerField n) |
|
|
| def tensorPair (T U : Fin n → V n → Matrix (Fin n) (Fin n) ℂ) (z : V n) : ℂ := |
| tensorPairAt (K.matrix z)⁻¹ (K.matrix z) (fun i => T i z) (fun i => U i z) |
|
|
| lemma tensorPair_smoothOn {T U : Fin n → V n → Matrix (Fin n) (Fin n) ℂ} |
| (hT : ∀ i, ContDiffOn ℝ ∞ (T i) K.domain) (hU : ∀ i, ContDiffOn ℝ ∞ (U i) K.domain) : |
| ContDiffOn ℝ ∞ (K.tensorPair T U) K.domain := by |
| apply K.isOpen.contDiffOn_iff.mpr |
| intro z hz |
| exact tensorPair_smooth (K.inverse_smooth.contDiffAt (K.isOpen.mem_nhds hz)) |
| (K.smooth.contDiffAt (K.isOpen.mem_nhds hz)) |
| (fun i => (hT i).contDiffAt (K.isOpen.mem_nhds hz)) |
| (fun i => (hU i).contDiffAt (K.isOpen.mem_nhds hz)) |
|
|
| lemma matrix_hol {z : V n} (hz : z ∈ K.domain) (k : Fin n) : |
| mderiv (-Complex.I) (e k) K.matrix z = K.matrix z*K.connection k z := by |
| rw [connection,← mul_assoc,Matrix.mul_nonsing_inv _ |
| (isUnit_iff_ne_zero.mpr (ne_of_gt (K.positive z hz).det_pos)),one_mul] |
|
|
| lemma inverse_hol {z : V n} (hz : z ∈ K.domain) (k : Fin n) : |
| mderiv (-Complex.I) (e k) (fun y => (K.matrix y)⁻¹) z = |
| -K.connection k z*(K.matrix z)⁻¹ := by |
| rw [mderiv_inverse K.isOpen K.smooth (fun y hy => ne_of_gt (K.positive y hy).det_pos) hz] |
| simp only [connection,neg_mul] |
|
|
| lemma tensorPair_hol {T U : Fin n → V n → Matrix (Fin n) (Fin n) ℂ} |
| (hT : ∀ i, ContDiffOn ℝ ∞ (T i) K.domain) (hU : ∀ i, ContDiffOn ℝ ∞ (U i) K.domain) |
| {z : V n} (hz : z ∈ K.domain) (k : Fin n) : |
| dz (e k) (K.tensorPair T U) z = |
| K.tensorPair (K.covHol T k) U z + |
| K.tensorPair T (fun j => mderiv Complex.I (e k) (U j)) z := by |
| have hn := K.isOpen.mem_nhds hz |
| have hP := K.inverse_smooth.contDiffAt hn |
| have hM := K.smooth.contDiffAt hn |
| have ht (i) := (hT i).contDiffAt hn |
| have hu (i) := (hU i).contDiffAt hn |
| have hQ (i j) := matrix_smooth_mul (matrix_smooth_mul |
| (matrix_smooth_mul (ht i) hP) (matrix_smooth_conjTranspose (hu j))) hM |
| have he (i j) : wderiv (-Complex.I) (e k) |
| (fun y => ((K.matrix y)⁻¹ i j) * (T i y*(K.matrix y)⁻¹*(U j y)ᴴ*K.matrix y).trace) z = |
| (-(K.connection k z*(K.matrix z)⁻¹) i j * |
| (T i z*(K.matrix z)⁻¹*(U j z)ᴴ*K.matrix z).trace + |
| (K.matrix z)⁻¹ i j * |
| (mderiv (-Complex.I) (e k) (T i) z*(K.matrix z)⁻¹*(U j z)ᴴ*K.matrix z - |
| T i z*K.connection k z*(K.matrix z)⁻¹*(U j z)ᴴ*K.matrix z + |
| T i z*(K.matrix z)⁻¹*(mderiv Complex.I (e k) (U j) z)ᴴ*K.matrix z + |
| T i z*(K.matrix z)⁻¹*(U j z)ᴴ*K.matrix z*K.connection k z).trace) := by |
| rw [wderiv_mul ((entry_smooth hP i j).differentiableAt (by simp)) |
| ((calabi_pair_matrix_trace_smooth (hQ i j)).differentiableAt (by simp)),mderiv_trace (hQ i j)] |
| change mderiv (-Complex.I) (e k) (fun y => (K.matrix y)⁻¹) z i j * _ + _ = _ |
| rw [K.inverse_hol hz] |
| simp only [neg_mul,Matrix.neg_apply] |
| congr 1 |
| congr 1 |
| congr 1 |
| rw [mderiv_mul (matrix_smooth_mul (matrix_smooth_mul (ht i) hP) |
| (matrix_smooth_conjTranspose (hu j))) hM, |
| mderiv_mul (matrix_smooth_mul (ht i) hP) (matrix_smooth_conjTranspose (hu j)), |
| mderiv_mul (ht i) hP,mderiv_conjTranspose (hu j),K.inverse_hol hz,K.matrix_hol hz] |
| noncomm_ring |
| change wderiv (-Complex.I) (e k) (fun y => ∑ i, ∑ j, |
| ((K.matrix y)⁻¹ i j)*(T i y*(K.matrix y)⁻¹*(U j y)ᴴ*K.matrix y).trace) z = _ |
| rw [wderiv_sum Finset.univ (fun i _ => (ContDiffAt.sum (fun j _ => |
| (entry_smooth hP i j).mul (calabi_pair_matrix_trace_smooth (hQ i j)))).differentiableAt (by simp))] |
| simp only [wderiv_sum Finset.univ (fun j _ => ((entry_smooth hP _ j).mul |
| (calabi_pair_matrix_trace_smooth (hQ _ j))).differentiableAt (by simp)),he] |
| exact covariant_pair_algebra (K.matrix z)⁻¹ (K.matrix z) (K.connection k z) |
| (fun i => T i z) (fun i => U i z) (fun i => mderiv (-Complex.I) (e k) (T i) z) |
| (fun i => mderiv Complex.I (e k) (U i) z) |
|
|
| lemma tensorPair_bar {T U : Fin n → V n → Matrix (Fin n) (Fin n) ℂ} |
| (hT : ∀ i, ContDiffOn ℝ ∞ (T i) K.domain) (hU : ∀ i, ContDiffOn ℝ ∞ (U i) K.domain) |
| {z : V n} (hz : z ∈ K.domain) (k : Fin n) : |
| dzbar (e k) (K.tensorPair T U) z = |
| K.tensorPair (fun i => mderiv Complex.I (e k) (T i)) U z + |
| K.tensorPair T (K.covHol U k) z := by |
| have hc {T U : Fin n → V n → Matrix (Fin n) (Fin n) ℂ} {y : V n} (hy : y ∈ K.domain) : |
| star (K.tensorPair T U y) = K.tensorPair U T y := |
| tensorPairAt_conj (K.positive y hy).isHermitian.inv (K.positive y hy).isHermitian _ _ |
| have he : K.tensorPair T U =ᶠ[𝓝 z] (fun y => star (K.tensorPair U T y)) := by |
| filter_upwards [K.isOpen.mem_nhds hz] with y hy using (hc hy).symm |
| change wderiv Complex.I (e k) (K.tensorPair T U) z = _ |
| rw [wderiv_congr he] |
| change dzbar (e k) (fun y => star (K.tensorPair U T y)) z = _ |
| rw [dzbar_conj (((K.tensorPair_smoothOn hU hT).contDiffAt |
| (K.isOpen.mem_nhds hz)).differentiableAt (by simp)),K.tensorPair_hol hU hT hz] |
| rw [star_add,hc hz,hc hz,add_comm] |
| end LocalKaehlerField |
| end KaehlerCalculus |
|
|
| end |
| end |
|
|
| end OAI |
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