| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Analytic.PointCharts |
| import OAI.AlgebraicGeometry.Seshadri.Jets.EtaleAnalytic |
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|
| namespace OAI |
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| section |
| noncomputable section |
| |
| section |
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| namespace MaximalSeshadri.QuadraticJets |
| noncomputable section |
| open MvPolynomial MaximalSeshadri.AlgebraicJets |
| open scoped Topology |
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|
| 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 |
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| 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] |
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| 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 |
| 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 |
| 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 |
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| end |
| end |
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| end OAI |
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