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import OAI.AlgebraicGeometry.Seshadri.Analytic.PointCharts
import OAI.AlgebraicGeometry.Seshadri.Jets.EtaleAnalytic
namespace OAI
section
noncomputable section
section
namespace MaximalSeshadri.QuadraticJets
noncomputable section
open MvPolynomial MaximalSeshadri.AlgebraicJets
open scoped Topology
lemma polynomial_binary_standardSmooth :
Algebra.IsStandardSmoothOfRelativeDimension 2 ℂ (MvPolynomial (Fin 2) ℂ) := by
let b := KaehlerDifferential.mvPolynomialBasis ℂ (Fin 2)
have : Algebra.IsStandardSmooth ℂ (MvPolynomial (Fin 2) ℂ) :=
Algebra.IsStandardSmooth.of_basis_kaehlerDifferential b (by
rintro _ ⟨i,rfl⟩
exact ⟨X i, (KaehlerDifferential.mvPolynomialBasis_apply ℂ (Fin 2) i).symm⟩)
rw [Algebra.IsStandardSmoothOfRelativeDimension.iff_of_isStandardSmooth]
rw [rank_eq_card_basis b]
simp
variable {A : Type} [CommRing A] [Algebra ℂ A]
[Algebra (MvPolynomial (Fin 2) ℂ) A] [IsScalarTower ℂ (MvPolynomial (Fin 2) ℂ) A]
[Algebra.Etale (MvPolynomial (Fin 2) ℂ) A]
theorem exists_etale_analytic_chart
(ρ : A →ₐ[ℂ] ℂ)
(hρ : ∀ i : Fin 2, ρ (algebraMap (MvPolynomial (Fin 2) ℂ) A (X i)) = 0) :
∃ q : (ℂ × ℂ) → (A →ₐ[ℂ] ℂ), q 0 = ρ ∧
(∀ s : A, AnalyticAt ℂ (fun z => q z s) 0) ∧
∀ᶠ z in 𝓝 0, ∀ i : Fin 2,
q z (algebraMap (MvPolynomial (Fin 2) ℂ) A (X i)) =
if i = 0 then z.2 else z.1 := by
classical
have : Algebra.IsStandardSmoothOfRelativeDimension 2 ℂ (MvPolynomial (Fin 2) ℂ) :=
polynomial_binary_standardSmooth
have : Algebra.IsStandardSmoothOfRelativeDimension 0 (MvPolynomial (Fin 2) ℂ) A :=
Algebra.Etale.iff_isStandardSmoothOfRelativeDimension_zero.mp inferInstance
have : Algebra.IsStandardSmoothOfRelativeDimension 2 ℂ A :=
Algebra.IsStandardSmoothOfRelativeDimension.trans 2 0 ℂ (MvPolynomial (Fin 2) ℂ) A
obtain ⟨ι,σ,hσ,hι,P,hP⟩ :=
Algebra.IsStandardSmoothOfRelativeDimension.out (n := 2) (R := ℂ) (S := A)
let : Fintype ι := Fintype.ofFinite ι
let : Fintype σ := Fintype.ofFinite σ
have hcard : Fintype.card {i : ι // i ∉ Set.range P.map} = 2 := by
rw [← hP]
change Fintype.card ((Set.range P.map)ᶜ : Set ι) = P.dimension
simp only [Algebra.Presentation.dimension, Nat.card_eq_fintype_card, Fintype.card_compl_set, Set.card_range_of_injective P.map_inj]
let e : {i : ι // i ∉ Set.range P.map} ≃ Fin 2 :=
Fintype.equivFinOfCardEq hcard
obtain ⟨q,hq₀,hq,hcoord⟩ := exists_binary_point_chart P e ρ
have hu : ∀ᶠ z in 𝓝 0, q z (P.val (e.symm 1)) = ρ (P.val (e.symm 1)) + z.1 := by
filter_upwards [hcoord] with z hz
simpa using hz 1
have hw : ∀ᶠ z in 𝓝 0, q z (P.val (e.symm 0)) = ρ (P.val (e.symm 0)) + z.2 := by
filter_upwards [hcoord] with z hz
simpa using hz 0
obtain ⟨ψ,hψ₀,hψ,hψcoord,-,-⟩ :=
etale_coordinates_analytic_inverse ρ hρ q hq₀ hq _ _ hu hw
refine ⟨fun z => q (ψ z), by simp [hψ₀,hq₀], ?_, hψcoord⟩
intro s
exact (hq s).comp_of_eq hψ hψ₀
end
end MaximalSeshadri.QuadraticJets
end
end
end
end OAI
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