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2fca478 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | import Mathlib
import OAI.Geometry.PrescribedPotential.CalabiBochner
import OAI.Geometry.PrescribedPotential.CalabiFrameAlgebra
import OAI.Geometry.PrescribedPotential.HolomorphicDirection
import OAI.Geometry.PrescribedPotential.MatrixFrameNormalization
import OAI.Geometry.PrescribedPotential.ScalarFrameCalculus
namespace OAI
/-! Calabi Frame Calculus. -/
section
noncomputable section
open Set Filter Topology Matrix
open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
namespace KaehlerCalculus.LocalKaehlerField
variable {n : ℕ} (K : LocalKaehlerField n)
def pull (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B)) : LocalKaehlerField n where
domain := B ⁻¹' K.domain
isOpen := K.isOpen.preimage B.continuous
matrix := pullMetric B K.matrix
smooth := pullMetric_smooth K.isOpen B K.smooth
positive := pullMetric_positive B hB K.positive
closed := pullMetric_closed K.isOpen B K.smooth K.closed
lemma pull_connection (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B))
{w : V n} (hw : B w ∈ K.domain) (i : Fin n) :
(K.pull B hB).connection i w = pullConnectionTensor (frameMatrix B) (fun p => K.connection p (B w)) i := by
let C := frameMatrix B
have hc : C*C⁻¹ = 1 := Matrix.mul_nonsing_inv _ (Matrix.isUnit_iff_isUnit_det _ |>.mp hB)
have hd : (Cᴴ)⁻¹*Cᴴ = 1 := by
rw [← Matrix.conjTranspose_nonsing_inv,← Matrix.conjTranspose_mul,hc,Matrix.conjTranspose_one]
change (pullMetric B K.matrix w)⁻¹*mderiv (-Complex.I) (e i) (pullMetric B K.matrix) w = _
rw [mderiv_pullMetric B (K.smooth.contDiffAt (K.isOpen.mem_nhds hw))]
unfold pullMetric
change (Cᴴ*K.matrix (B w)*C)⁻¹*(Cᴴ*mderiv (-Complex.I) (B (e i)) K.matrix (B w)*C) = _
simp only [Matrix.mul_inv_rev,mul_assoc]
rw [← mul_assoc (Cᴴ)⁻¹ Cᴴ,hd,one_mul,mderiv_hol_coordinates]
have he (p) : B (e i) p = C p i := rfl
simp only [he,pullConnectionTensor,connection,Matrix.mul_sum,Matrix.mul_smul,Matrix.sum_mul,Matrix.smul_mul,mul_assoc,C]
lemma pull_calabiNorm (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B))
{w : V n} (hw : B w ∈ K.domain) :
(K.pull B hB).calabiNorm w = K.calabiNorm (B w) := by
unfold calabiNorm tensorPair
simp_rw [K.pull_connection B hB hw]
change (tensorPairAt (pullMetric B K.matrix w)⁻¹ (pullMetric B K.matrix w) _ _).re = _
unfold pullMetric
simp only [Matrix.mul_inv_rev]
exact congrArg Complex.re (by
simpa only [mul_assoc] using tensorPairAt_changeBasis (frameMatrix B) (K.matrix (B w))⁻¹
(K.matrix (B w)) hB (fun p => K.connection p (B w)) (fun p => K.connection p (B w)))
lemma pull_ricciHessian (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B))
{w : V n} (hw : B w ∈ K.domain) :
(K.pull B hB).ricciHessian w = pullMetric B K.ricciHessian w :=
potentialMatrix_logdet_pullMetric K.isOpen B hB K.smooth K.positive hw
lemma pull_ricciHessian_hol (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B))
{w : V n} (hw : B w ∈ K.domain) (i : Fin n) :
mderiv (-Complex.I) (e i) (K.pull B hB).ricciHessian w =
(frameMatrix B)ᴴ * mderiv (-Complex.I) (B (e i)) K.ricciHessian (B w) * frameMatrix B := by
have he : (K.pull B hB).ricciHessian =ᶠ[𝓝 w] pullMetric B K.ricciHessian := by
filter_upwards [B.continuous.continuousAt.preimage_mem_nhds (K.isOpen.mem_nhds hw)] with y hy
exact K.pull_ricciHessian B hB hy
rw [mderiv_congr he]
exact mderiv_pullMetric B (K.ricciHessian_smooth.contDiffAt (K.isOpen.mem_nhds hw)) _ _
end KaehlerCalculus.LocalKaehlerField
end
end
end OAI
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