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import OAI.Geometry.PrescribedPotential.KaehlerSecondDerivatives
namespace OAI
/-! Calabi Connection. -/
section
noncomputable section
open Set Filter Topology Matrix
open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
namespace KaehlerCalculus
structure LocalKaehlerField (n : ℕ) where
domain : Set (V n)
isOpen : IsOpen domain
matrix : V n → Matrix (Fin n) (Fin n) ℂ
smooth : ContDiffOn ℝ ∞ matrix domain
positive : ∀ z ∈ domain, (matrix z).PosDef
closed : ∀ y ∈ domain, ∀ u v w : V n,
fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (matrix q) v w) y u +
fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (matrix q) w u) y v +
fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (matrix q) u v) y w = 0
namespace LocalKaehlerField
variable {n : ℕ} (K : LocalKaehlerField n)
def connection (j : Fin n) (z : V n) : Matrix (Fin n) (Fin n) ℂ :=
(K.matrix z)⁻¹ * mderiv (-Complex.I) (e j) K.matrix z
def curvature (i j : Fin n) : V n → Matrix (Fin n) (Fin n) ℂ :=
mderiv Complex.I (e i) (K.connection j)
def logDet : V n → ℝ := fun y => Real.log (K.matrix y).det.re
def ricciHessian (z : V n) : Matrix (Fin n) (Fin n) ℂ :=
PotentialKaehler.potentialMatrix K.logDet z
lemma inverse_smooth : ContDiffOn ℝ ∞ (fun y => (K.matrix y)⁻¹) K.domain :=
MatrixSmoothGeneral.inverse K.smooth (fun y hy => ne_of_gt (K.positive y hy).det_pos)
lemma connection_smooth (j : Fin n) : ContDiffOn ℝ ∞ (K.connection j) K.domain := by
apply K.isOpen.contDiffOn_iff.mpr
intro z hz
exact matrix_smooth_mul (K.inverse_smooth.contDiffAt (K.isOpen.mem_nhds hz))
(mderiv_smooth (K.smooth.contDiffAt (K.isOpen.mem_nhds hz)) _ _)
lemma logDet_smooth : ContDiffOn ℝ ∞ K.logDet K.domain := by
apply K.isOpen.contDiffOn_iff.mpr
intro z hz
exact (Complex.reCLM.contDiff.contDiffAt.comp z
(determinant_smooth.contDiffAt.comp z (K.smooth.contDiffAt (K.isOpen.mem_nhds hz)))).log
(ne_of_gt (Complex.pos_iff.mp (K.positive z hz).det_pos).1)
lemma connection_symmetric {z : V n} (hz : z ∈ K.domain) (i j k : Fin n) :
K.connection i z k j = K.connection j z k i := by
unfold connection
simp only [Matrix.mul_apply]
apply Finset.sum_congr rfl
intro a _
congr 1
exact closed_dz_symmetry K.isOpen K.smooth (fun y hy => (K.positive y hy).isHermitian)
K.closed hz a j i
lemma curvature_formula {z : V n} (hz : z ∈ K.domain) (i j : Fin n) :
K.curvature i j z = (K.matrix z)⁻¹ *
(mderiv Complex.I (e i) (mderiv (-Complex.I) (e j) K.matrix) z -
mderiv Complex.I (e i) K.matrix z * (K.matrix z)⁻¹ *
mderiv (-Complex.I) (e j) K.matrix z) := by
have hs := K.smooth.contDiffAt (K.isOpen.mem_nhds hz)
unfold curvature connection
rw [mderiv_mul (K.inverse_smooth.contDiffAt (K.isOpen.mem_nhds hz))
(mderiv_smooth hs _ _), mderiv_inverse K.isOpen K.smooth
(fun y hy => ne_of_gt (K.positive y hy).det_pos) hz]
noncomm_ring
lemma ricciHessian_trace {z : V n} (hz : z ∈ K.domain) (i j : Fin n) :
K.ricciHessian z i j = (K.curvature i j z).trace := by
rw [ricciHessian,potentialMatrix_eq_dzbar_dz (K.logDet_smooth.contDiffAt (K.isOpen.mem_nhds hz))]
change dzbar (e i) (dz (e j) (fun y => (K.logDet y:ℂ))) z = _
have he : dz (e j) (fun y => (K.logDet y:ℂ)) =ᶠ[𝓝 z]
(fun y => (K.connection j y).trace) := by
filter_upwards [K.isOpen.mem_nhds hz] with y hy
exact wderiv_logdet K.isOpen K.smooth K.positive hy _ _
rw [show dzbar (e i) (dz (e j) (fun y => (K.logDet y:ℂ))) z =
dzbar (e i) (fun y => (K.connection j y).trace) z from wderiv_congr he _ _]
exact mderiv_trace (K.connection_smooth j |>.contDiffAt (K.isOpen.mem_nhds hz)) _ _
lemma curvature_contraction_at_one {z : V n} (hz : z ∈ K.domain)
(hM : K.matrix z = 1) : ∑ k, K.curvature k k z = K.ricciHessian z := by
have hc := ricci_cancellation_at_one K.isOpen K.smooth K.positive K.closed hz hM
have he : (∑ k, K.curvature k k z) =
(∑ k, mderiv Complex.I (e k) (mderiv (-Complex.I) (e k) K.matrix) z) -
∑ k, mderiv Complex.I (e k) K.matrix z*mderiv (-Complex.I) (e k) K.matrix z := by
simp only [K.curvature_formula hz, hM,inv_one,one_mul,mul_one,Finset.sum_sub_distrib]
rw [he,hc,add_sub_cancel_right]
rfl
end LocalKaehlerField
end KaehlerCalculus
end
end
end OAI
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