| import Mathlib |
| import OAI.AlgebraicGeometry.SectionFields.Recovery |
|
|
| namespace OAI |
|
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| /-! Complete linear systems, rational maps and geometric effective-base definitions. -/ |
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| noncomputable section |
| open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace Order Polynomial |
| open scoped TensorProduct WithZero |
| universe u |
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| namespace RelativeDenominators.EffectiveBase |
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| |
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| structure ComplexVariety where |
| scheme : Scheme |
| structural : scheme ⟶ Spec (CommRingCat.of ℂ) |
| integral : IsIntegral scheme |
| finiteType : LocallyOfFiniteType structural |
| quasiCompact : QuasiCompact structural |
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| attribute [instance] ComplexVariety.integral ComplexVariety.finiteType |
| ComplexVariety.quasiCompact |
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| namespace ComplexVariety |
| |
| instance (X : ComplexVariety) : IsNoetherian X.scheme where |
| toIsLocallyNoetherian := LocallyOfFiniteType.isLocallyNoetherian X.structural |
| toCompactSpace := QuasiCompact.compactSpace_of_compactSpace X.structural |
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| |
| def IsNormal (X : ComplexVariety) : Prop := |
| ∀ x : X.scheme, IsIntegrallyClosed (X.scheme.presheaf.stalk x) |
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| |
| def scalarMap (X : ComplexVariety) (x : X.scheme) : |
| ℂ →+* X.scheme.presheaf.stalk x := |
| ((Scheme.ΓSpecIso (CommRingCat.of ℂ)).inv ≫ X.structural.appTop ≫ |
| X.scheme.presheaf.germ ⊤ x (by trivial)).hom |
| |
| instance stalkComplexAlgebra (X : ComplexVariety) (x : X.scheme) : |
| Algebra ℂ (X.scheme.presheaf.stalk x) := (X.scalarMap x).toAlgebra |
| |
| instance functionFieldComplexAlgebra (X : ComplexVariety) : |
| Algebra ℂ X.scheme.functionField := |
| stalkComplexAlgebra X (genericPoint X.scheme) |
| |
| instance stalkFunctionFieldTower (X : ComplexVariety) (x : X.scheme) : |
| IsScalarTower ℂ (X.scheme.presheaf.stalk x) X.scheme.functionField := by |
| apply IsScalarTower.of_algebraMap_eq' |
| change (X.scalarMap (genericPoint X.scheme)) = |
| (algebraMap (X.scheme.presheaf.stalk x) X.scheme.functionField).comp (X.scalarMap x) |
| change ((Scheme.ΓSpecIso (CommRingCat.of ℂ)).inv ≫ X.structural.appTop ≫ |
| X.scheme.presheaf.germ ⊤ (genericPoint X.scheme) (by trivial)).hom = |
| ((Scheme.ΓSpecIso (CommRingCat.of ℂ)).inv ≫ X.structural.appTop ≫ |
| X.scheme.presheaf.germ ⊤ x (by trivial) ≫ |
| X.scheme.presheaf.stalkSpecializes |
| ((genericPoint_spec X.scheme).specializes (Set.mem_univ x))).hom |
| congr 1 |
| erw [TopCat.Presheaf.germ_stalkSpecializes] |
| |
| end ComplexVariety |
| |
| attribute [local instance] MvPolynomial.gradedAlgebra |
| |
| |
| def projectiveSpace (n : ℕ) : Scheme := |
| Proj (MvPolynomial.homogeneousSubmodule (Fin (n + 1)) ℂ) |
| |
| |
| def constantToDegreeZero (n : ℕ) : |
| ℂ →+* MvPolynomial.homogeneousSubmodule (Fin (n + 1)) ℂ 0 where |
| toFun z := ⟨MvPolynomial.C z, MvPolynomial.isHomogeneous_C _ z⟩ |
| map_one' := Subtype.ext (map_one MvPolynomial.C) |
| map_zero' := Subtype.ext (map_zero MvPolynomial.C) |
| map_mul' x y := Subtype.ext (map_mul MvPolynomial.C x y) |
| map_add' x y := Subtype.ext (map_add MvPolynomial.C x y) |
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| def projectiveSpaceStructural (n : ℕ) : |
| projectiveSpace n ⟶ Spec (CommRingCat.of ℂ) := |
| Proj.toSpecZero (MvPolynomial.homogeneousSubmodule (Fin (n + 1)) ℂ) ≫ |
| Spec.map (CommRingCat.ofHom (constantToDegreeZero n)) |
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| |
| def ComplexVariety.IsProjective (X : ComplexVariety) : Prop := |
| ∃ n : ℕ, ∃ e : X.scheme ⟶ projectiveSpace n, |
| IsClosedImmersion e ∧ e ≫ projectiveSpaceStructural n = X.structural |
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| |
| def IsProjectiveMorphism (X Z : ComplexVariety) (f : X.scheme ⟶ Z.scheme) : Prop := |
| ∃ n : ℕ, ∃ e : X.scheme ⟶ pullback Z.structural (projectiveSpaceStructural n), |
| IsClosedImmersion e ∧ e ≫ pullback.fst _ _ = f |
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| |
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| structure Contraction (X Z : ComplexVariety) where |
| morphism : X.scheme ⟶ Z.scheme |
| over_base : morphism ≫ Z.structural = X.structural |
| projective : IsProjectiveMorphism X Z morphism |
| surjective : Surjective morphism |
| sheaf_iso : IsIso morphism.c |
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| attribute [instance] Contraction.surjective Contraction.sheaf_iso |
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| |
| def rationalize (X : Scheme) (K : WeilDivisor X) : RationalWeilDivisor X := |
| K.mapRange (Int.castRingHom ℚ) (by simp) |
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| |
| def exteriorPullback {R S : Type*} [CommRing R] [CommRing S] [Algebra R S] |
| {V W : Type*} [AddCommGroup V] [AddCommGroup W] [Module R V] [Module S W] |
| [Module R W] [IsScalarTower R S W] (d : ℕ) (f : V →ₗ[R] W) : |
| (⋀[R]^d V) →ₗ[R] (⋀[S]^d W) := by |
| let a : W [⋀^Fin d]→ₗ[R] (⋀[S]^d W) := |
| { toMultilinearMap := (exteriorPower.ιMulti S d).toMultilinearMap.restrictScalars R |
| map_eq_zero_of_eq' := fun v i j hv hij => |
| (exteriorPower.ιMulti S d).map_eq_zero_of_eq v hv hij } |
| exact exteriorPower.alternatingMapLinearEquiv (a.compLinearMap f) |
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|
| abbrev RationalTopForms (X : ComplexVariety) (d : ℕ) := |
| ⋀[X.scheme.functionField]^d Ω[X.scheme.functionField⁄ℂ] |
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| |
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| def RegularAt (X : ComplexVariety) (d : ℕ) (x : X.scheme) |
| (ω : RationalTopForms X d) : Prop := |
| ∃ α : ⋀[X.scheme.presheaf.stalk x]^d Ω[X.scheme.presheaf.stalk x⁄ℂ], |
| exteriorPullback (S := X.scheme.functionField) d |
| (KaehlerDifferential.map ℂ ℂ (X.scheme.presheaf.stalk x) X.scheme.functionField) α = ω |
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| def IsDivisorOfForm (X : ComplexVariety) (d : ℕ) (ω : RationalTopForms X d) |
| (K : WeilDivisor X.scheme) : Prop := |
| ω ≠ 0 ∧ ∀ x : PrimeDivisor X.scheme, ∀ v : X.scheme.functionFieldˣ, |
| RegularAt X d x.1 ((v : X.scheme.functionField) • ω) ↔ |
| 0 ≤ X.scheme.ord (v : X.scheme.functionField) x.1 + K x |
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| |
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|
| structure BirationalModel (X : ComplexVariety) where |
| variety : ComplexVariety |
| normal : variety.IsNormal |
| morphism : variety.scheme ⟶ X.scheme |
| proper : IsProper morphism |
| over_base : morphism ≫ X.structural = variety.structural |
| dominant : IsDominant morphism |
| fieldEquiv : X.scheme.functionField ≃ₐ[ℂ] variety.scheme.functionField |
| induced : fieldEquiv.toRingEquiv.toRingHom = |
| @functionFieldPullback _ _ variety.integral X.integral morphism dominant |
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| attribute [instance] BirationalModel.dominant |
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| |
| def fieldPullback {K L : Type*} [Field K] [Field L] [Algebra ℂ K] [Algebra ℂ L] |
| (e : K ≃ₐ[ℂ] L) (d : ℕ) : (⋀[K]^d Ω[K⁄ℂ]) → (⋀[L]^d Ω[L⁄ℂ]) := by |
| letI : Algebra K L := e.toRingEquiv.toRingHom.toAlgebra |
| letI : IsScalarTower ℂ K L := IsScalarTower.of_algebraMap_eq' (by |
| ext z |
| exact (e.commutes z).symm) |
| exact fun ω => exteriorPullback (S := L) d (KaehlerDifferential.map ℂ ℂ K L) ω |
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| def IsLogCanonical (X : ComplexVariety) (hNX : X.IsNormal) (d : ℕ) |
| (ω : RationalTopForms X d) (K : WeilDivisor X.scheme) |
| (B : RationalWeilDivisor X.scheme) |
| (hAdj : IsQCartier X.scheme (rationalize X.scheme K + B)) : Prop := |
| IsDivisorOfForm X d ω K ∧ (∀ E, 0 ≤ B E) ∧ |
| ∀ Y : BirationalModel X, ∃ KY : WeilDivisor Y.variety.scheme, |
| IsDivisorOfForm Y.variety d (fieldPullback Y.fieldEquiv d ω) KY ∧ |
| ∀ E : PrimeDivisor Y.variety.scheme, |
| (-1 : ℚ) ≤ (KY E : ℚ) - |
| pullbackQCartier hNX Y.morphism (rationalize X.scheme K + B) hAdj E |
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| |
| def CoefficientsIn {X : Scheme} (B : RationalWeilDivisor X) (Φ : Finset ℚ) : Prop := |
| ∀ E : PrimeDivisor X, B E ≠ 0 → B E ∈ Φ |
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| def sectionSpace (X : ComplexVariety) (A : RationalWeilDivisor X.scheme) : |
| Submodule ℂ X.scheme.functionField := |
| Submodule.span ℂ ((fun u : X.scheme.functionFieldˣ => (u : X.scheme.functionField)) '' |
| divisorialSections X.scheme A) |
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| def IsBig (X : ComplexVariety) (A : RationalWeilDivisor X.scheme) : Prop := |
| ∃ e : ℕ, topologicalKrullDim X.scheme = e ∧ |
| ∃ c : ℝ, 0 < c ∧ ∀ N : ℕ, ∃ n : ℕ, N < n ∧ |
| c * (n : ℝ) ^ e ≤ (Module.finrank ℂ (sectionSpace X ((n : ℚ) • A)) : ℝ) |
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| |
| def IsSectionBasis (X : ComplexVariety) (A : RationalWeilDivisor X.scheme) |
| (n : ℕ) (b : Fin (n + 1) → X.scheme.functionFieldˣ) : Prop := |
| (∀ i, b i ∈ divisorialSections X.scheme A) ∧ |
| LinearIndependent ℂ (fun i => (b i : X.scheme.functionField)) ∧ |
| Submodule.span ℂ (Set.range (fun i => (b i : X.scheme.functionField))) = sectionSpace X A |
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| def coordinatesEvaluation (X : ComplexVariety) (n : ℕ) |
| (b : Fin (n + 1) → X.scheme.functionFieldˣ) : |
| MvPolynomial (Fin (n + 1)) ℂ →+* Γ(Spec X.scheme.functionField, ⊤) := |
| (Scheme.ΓSpecIso X.scheme.functionField).inv.hom.comp |
| (MvPolynomial.eval₂Hom (algebraMap ℂ X.scheme.functionField) |
| (fun i => (b i : X.scheme.functionField))) |
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| |
| theorem coordinates_irrelevant_eq_top (X : ComplexVariety) (n : ℕ) |
| (b : Fin (n + 1) → X.scheme.functionFieldˣ) : |
| (HomogeneousIdeal.irrelevant (MvPolynomial.homogeneousSubmodule (Fin (n + 1)) ℂ)).toIdeal.map |
| (coordinatesEvaluation X n b) = ⊤ := by |
| let i : Fin (n + 1) := ⟨0, Nat.succ_pos n⟩ |
| apply Ideal.eq_top_of_isUnit_mem (x := coordinatesEvaluation X n b (MvPolynomial.X i)) |
| · apply Ideal.mem_map_of_mem |
| exact HomogeneousIdeal.mem_irrelevant_of_mem _ (by decide : 0 < (1 : ℕ)) |
| (MvPolynomial.isHomogeneous_X ℂ i) |
| · change IsUnit ((Scheme.ΓSpecIso X.scheme.functionField).inv.hom |
| ((MvPolynomial.eval₂Hom (algebraMap ℂ X.scheme.functionField) |
| (fun j => (b j : X.scheme.functionField))) (MvPolynomial.X i))) |
| simp only [MvPolynomial.eval₂Hom_X'] |
| exact (b i).isUnit.map _ |
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| |
| def projectiveGenericPoint (X : ComplexVariety) (n : ℕ) |
| (b : Fin (n + 1) → X.scheme.functionFieldˣ) : |
| Spec X.scheme.functionField ⟶ projectiveSpace n := |
| Proj.fromOfGlobalSections (MvPolynomial.homogeneousSubmodule (Fin (n + 1)) ℂ) |
| (coordinatesEvaluation X n b) (coordinates_irrelevant_eq_top X n b) |
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| def IsCompleteSystemMap (X : ComplexVariety) (A : RationalWeilDivisor X.scheme) |
| (Y : ComplexVariety) (φ : X.scheme ⤏ Y.scheme) : Prop := |
| φ.compHom Y.structural = X.structural.toRationalMap ∧ |
| IsDominant φ.fromFunctionField ∧ |
| ∃ n : ℕ, ∃ b : Fin (n + 1) → X.scheme.functionFieldˣ, |
| IsSectionBasis X A n b ∧ |
| ∃ e : Y.scheme ⟶ projectiveSpace n, |
| IsClosedImmersion e ∧ e ≫ projectiveSpaceStructural n = Y.structural ∧ |
| φ.fromFunctionField ≫ e = projectiveGenericPoint X n b |
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| |
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| def BirationallyEquivalentMaps (X Y Z : ComplexVariety) |
| (φ : X.scheme ⤏ Y.scheme) (ψ : X.scheme ⤏ Z.scheme) : Prop := |
| ∃ U : Y.scheme.Opens, ∃ V : Z.scheme.Opens, |
| Dense (U : Set Y.scheme) ∧ Dense (V : Set Z.scheme) ∧ |
| ∃ e : U.toScheme ≅ V.toScheme, |
| e.hom ≫ V.ι ≫ Z.structural = U.ι ≫ Y.structural ∧ |
| ∃ t : Spec X.scheme.functionField ⟶ U.toScheme, |
| t ≫ U.ι = φ.fromFunctionField ∧ |
| t ≫ e.hom ≫ V.ι = ψ.fromFunctionField |
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| def IsIitakaFibration (X : ComplexVariety) (A : RationalWeilDivisor X.scheme) |
| (I : ComplexVariety) (ι : X.scheme ⤏ I.scheme) : Prop := |
| I.IsNormal ∧ I.IsProjective ∧ |
| ι.compHom I.structural = X.structural.toRationalMap ∧ |
| IsDominant ι.fromFunctionField ∧ |
| ∃ r : ℕ, 0 < r ∧ ∀ k : ℕ, 0 < k → |
| ∃ Y : ComplexVariety, ∃ φ : X.scheme ⤏ Y.scheme, |
| IsCompleteSystemMap X (((r * k : ℕ) : ℚ) • A) Y φ ∧ |
| BirationallyEquivalentMaps X Y I φ ι |
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| def RecoversBaseAndIitaka (X Z : ComplexVariety) (f : Contraction X Z) |
| (A : RationalWeilDivisor X.scheme) (l : ℕ) : Prop := |
| (divisorialSections X.scheme ((l : ℚ) • A)).Nonempty ∧ |
| sectionField X.scheme ((l : ℚ) • A) = (functionFieldPullback f.morphism).fieldRange ∧ |
| ∃ Y : ComplexVariety, ∃ φ : X.scheme ⤏ Y.scheme, |
| IsCompleteSystemMap X ((l : ℚ) • A) Y φ ∧ |
| BirationallyEquivalentMaps X Y Z φ f.morphism.toRationalMap ∧ |
| ∃ I : ComplexVariety, ∃ ι : X.scheme ⤏ I.scheme, |
| IsIitakaFibration X A I ι ∧ BirationallyEquivalentMaps X Y I φ ι |
|
|
| end RelativeDenominators.EffectiveBase |
| end |
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| end OAI |
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|
