| import Mathlib |
| import OAI.AlgebraicGeometry.LogKodaira.Descent |
| import OAI.AlgebraicGeometry.LogKodaira.Blowup |
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| namespace OAI |
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| noncomputable section |
| open CategoryTheory AlgebraicGeometry |
| open scoped TensorProduct |
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| universe u v |
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| namespace ReverseLogKodaira |
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| theorem normal_root_zero_test |
| {R : Type u} {K : Type v} |
| [CommRing R] [IsDomain R] [Field K] [Algebra R K] |
| [IsFractionRing R K] [IsIntegrallyClosed R] |
| {m e : ℕ} (hm : 0 < m) (he : m ∣ e) |
| {t s : R} (ht : t ≠ 0) (hs : t ^ e ∣ s) |
| {r : K} (hr : r ^ m = algebraMap R K s) : |
| ∃ c : R, algebraMap R K (t ^ (e / m) * c) = r := by |
| obtain ⟨c, hc⟩ := hs |
| have htK : algebraMap R K t ≠ 0 := |
| (map_ne_zero_iff (algebraMap R K) (IsFractionRing.injective R K)).mpr ht |
| have heq : (e / m) * m = e := Nat.div_mul_cancel he |
| have hpow : (r / algebraMap R K (t ^ (e / m))) ^ m = algebraMap R K c := by |
| rw [div_pow, hr, hc, map_mul, map_pow, map_pow, ← pow_mul, heq] |
| exact mul_div_cancel_left₀ _ (pow_ne_zero _ htK) |
| obtain ⟨d, hd⟩ := IsIntegrallyClosed.exists_algebraMap_eq_of_isIntegral_pow |
| (R := R) hm (hpow ▸ isIntegral_algebraMap) |
| refine ⟨d, ?_⟩ |
| rw [map_mul, hd] |
| exact mul_div_cancel₀ _ (by simpa only [map_pow] using pow_ne_zero (e / m) htK) |
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| end ReverseLogKodaira |
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| namespace ReverseLogKodaira.PluriformDescent |
| open SmoothProjectiveVariety DifferentialBaseChange CanonicalSpecialization |
| open FiberChartSpecialization |
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| theorem original_fiber_zero_ramified_root_factor |
| {X Y : SmoothProjectiveVariety} {E : X.ReducedSNCBoundary} {D : Y.ReducedSNCBoundary} |
| (f : StratumSmoothFibration X Y E D) (m : ℕ) (s : X.RationalPluriform m) |
| (y : Y.ComplexPoint) (hy : y.point ∈ D.complement) |
| (hzero : FiberVanishesAt f m s y hy) |
| (F : FiberModel f.toBoundaryFibration y) |
| (U : Y.scheme.affineOpens) (hU : U.1 ≤ D.complement) (hyU : y.point ∈ U.1) |
| (b : Module.Basis (Fin Y.dimension) Γ(Y.scheme, U.1) |
| (KaehlerDifferential ℂ Γ(Y.scheme, U.1))) |
| (P : NumeratorChart f.toBoundaryFibration y F U (fiber_dimension_eq f y hy F) m s) |
| {B K : Type*} [CommRing B] [IsDomain B] [IsIntegrallyClosed B] |
| [Field K] [Algebra B K] [IsFractionRing B K] |
| (z : RingHom.ker (complexPointEvaluation y U.1 hyU)) |
| (g : Γ(X.scheme, P.V.1) →+* B) |
| (h : Blowup.Chart (RingHom.ker (complexPointEvaluation y U.1 hyU)) z →+* B) |
| (square : g.comp (f.hom.appLE U.1 P.V.1 P.overOpen).hom = |
| h.comp (Blowup.chartMap (RingHom.ker (complexPointEvaluation y U.1 hyU)) z)) |
| {e : ℕ} (hm : 0 < m) (he : m ∣ e) (τ : B) (hτ : τ ≠ 0) |
| (ramification : h (Blowup.chartMap |
| (RingHom.ker (complexPointEvaluation y U.1 hyU)) z z) = τ ^ e) : |
| letI : Algebra Γ(Y.scheme, U.1) Γ(X.scheme, P.V.1) := |
| (f.hom.appLE U.1 P.V.1 P.overOpen).hom.toAlgebra |
| letI : IsScalarTower ℂ Γ(Y.scheme, U.1) Γ(X.scheme, P.V.1) := |
| IsScalarTower.of_algebraMap_eq fun a => |
| (appLE_constants f.hom f.over_base U.1 P.V.1 P.overOpen a).symm |
| letI := P.formallySmooth |
| ∀ r : K, |
| r ^ m = algebraMap B K (g (pluricanonicalCoordinate |
| ℂ Γ(Y.scheme, U.1) Γ(X.scheme, P.V.1) |
| Y.dimension F.variety.dimension m b P.relativeBasis P.numerator)) → |
| ∃ c : B, algebraMap B K (τ ^ (e / m) * c) = r := by |
| let : Algebra Γ(Y.scheme, U.1) Γ(X.scheme, P.V.1) := |
| (f.hom.appLE U.1 P.V.1 P.overOpen).hom.toAlgebra |
| let : IsScalarTower ℂ Γ(Y.scheme, U.1) Γ(X.scheme, P.V.1) := |
| IsScalarTower.of_algebraMap_eq fun a => |
| (appLE_constants f.hom f.over_base U.1 P.V.1 P.overOpen a).symm |
| let := P.formallySmooth |
| intro r hr |
| have hmem := original_coefficient_in_point_ideal_of_fiber_zero |
| f m s y hy hzero F U hU hyU b P |
| have hdiv := Blowup.parameter_dvd_image |
| (RingHom.ker (complexPointEvaluation y U.1 hyU)) z |
| (f.hom.appLE U.1 P.V.1 P.overOpen).hom g h square hmem |
| rw [ramification] at hdiv |
| exact normal_root_zero_test hm he hτ hdiv hr |
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| end ReverseLogKodaira.PluriformDescent |
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| end |
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| end OAI |
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