| import OAI.NumberTheory.PiExponent.Ampleness.ClosedAmpleRestriction |
| import OAI.NumberTheory.PiExponent.Approximation.NumericalMarginRestriction |
| import OAI.NumberTheory.PiExponent.Geometry.CartierDegreeLength |
| import OAI.NumberTheory.PiExponent.Geometry.CartierPowerFrames |
| import OAI.NumberTheory.PiExponent.Geometry.CurveEffectiveTwist |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.SectionZeroIdeal |
|
|
| namespace OAI |
|
|
| namespace PiExponent.CurveDegree |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace |
| open PiExponentSeshadri.Geometry PiExponentSeshadri.Frames |
| open PiExponent.SectionZeroIdeal PiExponent.CartierPowerFrames |
| variable {X : Scheme.{0}} |
|
|
| def FiniteLineCohomology (p : X ⟶ Spec (CommRingCat.of ℂ)) : Prop := |
| ∀ L : LineBundle X, ∀ n ≤ 1, |
| let := Module.compHom (cohomology L.sheaf n) (baseScalars p) |
| FiniteDimensional ℂ (cohomology L.sheaf n) |
|
|
| def LineCohomologyTwoZero (X : Scheme.{0}) : Prop := |
| ∀ L : LineBundle X, Subsingleton (cohomology L.sheaf 2) |
|
|
| theorem section_tensor_euler_add [IsIntegral X] |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p] |
| (hd : topologicalKrullDim X ≤ 1) |
| (hfinite : FiniteLineCohomology p) (hH2 : LineCohomologyTwoZero X) |
| (L M : LineBundle X) (s : GlobalSections X L.sheaf) (hs : s ≠ 0) : |
| eulerCharacteristic p 1 (L.tensor M).sheaf - eulerCharacteristic p 1 M.sheaf = |
| eulerCharacteristic p 1 L.sheaf - eulerCharacteristic p 1 (structureSheaf X) := by |
| let hsMono : Mono s := L.mono_section s hs |
| let I := zeroIdeal L s |
| have hI : I ≠ ⊥ := zeroIdeal_ne_bot L s hs |
| let φ := sectionMultiplyLeft L.sheaf M.sheaf s |
| let : Mono (moduleTensorMap s (𝟙 M.sheaf)) := moduleTensorMap_mono s M |
| let hφMono : Mono φ := mono_comp _ _ |
| have heqTensor : ∀ x : X, ∃ U : X.affineOpens, x ∈ U.1 ∧ |
| ∃ f : M.sheaf.restrict U.1.ι ≅ O U.1.toScheme, |
| ∃ e : (L.tensor M).sheaf.restrict U.1.ι ≅ O U.1.toScheme, |
| I.ideal U = Ideal.span {U.1.topIso.hom |
| (endValue (f.inv ≫ (Scheme.Modules.restrictFunctor U.1.ι).map φ ≫ e.hom))} := by |
| intro x |
| obtain ⟨U, hx, ⟨e⟩, ⟨f⟩⟩ := common_affine_frames L M x |
| refine ⟨U, hx, f, tensorFrame L.sheaf M.sheaf U.1 e f, ?_⟩ |
| exact (zeroIdeal_on_frame L s U e).trans |
| (tensor_multiply_ideal L.sheaf M.sheaf s U e f).symm |
| have heqSection : ∀ x : X, ∃ U : X.affineOpens, x ∈ U.1 ∧ |
| ∃ f : (L.pow 0).sheaf.restrict U.1.ι ≅ O U.1.toScheme, |
| ∃ e : L.sheaf.restrict U.1.ι ≅ O U.1.toScheme, |
| I.ideal U = Ideal.span {U.1.topIso.hom |
| (endValue (f.inv ≫ (Scheme.Modules.restrictFunctor U.1.ι).map s ≫ e.hom))} := by |
| intro x |
| obtain ⟨U, hx, ⟨e⟩, ⟨f⟩⟩ := common_affine_frames L M x |
| refine ⟨U, hx, Scheme.Modules.restrictUnitIso U.1.ι, e, ?_⟩ |
| simpa only [coefficient, restrictSection, Category.assoc] using! zeroIdeal_on_frame L s U e |
| let : Subsingleton (cohomology M.sheaf 2) := hH2 M |
| let : Subsingleton (cohomology (L.pow 0).sheaf 2) := hH2 (L.pow 0) |
| have hTensor := @CartierDegreeLength.cartier_euler_difference_eq_sum_local_lengths |
| X inferInstance p inferInstance hd M (L.tensor M) φ hφMono I hI heqTensor (hfinite M) (hfinite (L.tensor M)) |
| have hSection := @CartierDegreeLength.cartier_euler_difference_eq_sum_local_lengths |
| X inferInstance p inferInstance hd (L.pow 0) L s hsMono I hI heqSection (hfinite (L.pow 0)) (hfinite L) |
| exact hTensor.trans hSection.symm |
|
|
| theorem tensor_euler_add [IsIntegral X] |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p] |
| (hd : topologicalKrullDim X ≤ 1) (H : LineBundle X) (hH : H.IsAmple) |
| (hfinite : FiniteLineCohomology p) (hH2 : LineCohomologyTwoZero X) |
| (L M : LineBundle X) : |
| eulerCharacteristic p 1 (L.tensor M).sheaf - eulerCharacteristic p 1 M.sheaf = |
| eulerCharacteristic p 1 L.sheaf - eulerCharacteristic p 1 (structureSheaf X) := by |
| let : IsLocallyNoetherian X := LocallyOfFiniteType.isLocallyNoetherian p |
| let : CompactSpace X := QuasiCompact.compactSpace_of_compactSpace p |
| let : IsNoetherian X := ⟨⟩ |
| obtain ⟨A, ⟨t, ht⟩, ⟨u, hu⟩⟩ := exists_effective_twist H hH L |
| have h₀ := section_tensor_euler_add p hd hfinite hH2 A L t ht |
| have h₁ := section_tensor_euler_add p hd hfinite hH2 (A.tensor L) M u hu |
| have h₂ := section_tensor_euler_add p hd hfinite hH2 A (L.tensor M) t ht |
| have he := eulerCharacteristic_iso p (lineTensorAssoc A L M) 1 |
| change eulerCharacteristic p 1 ((A.tensor L).tensor M).sheaf = |
| eulerCharacteristic p 1 (A.tensor (L.tensor M)).sheaf at he |
| omega |
|
|
| theorem power_euler [IsIntegral X] |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p] |
| (hd : topologicalKrullDim X ≤ 1) (H : LineBundle X) (hH : H.IsAmple) |
| (hfinite : FiniteLineCohomology p) (hH2 : LineCohomologyTwoZero X) |
| (L : LineBundle X) (n : ℕ) : |
| eulerCharacteristic p 1 (L.pow n).sheaf - eulerCharacteristic p 1 (structureSheaf X) = |
| (n : ℤ) * (eulerCharacteristic p 1 L.sheaf - eulerCharacteristic p 1 (structureSheaf X)) := by |
| induction n with |
| | zero => simp [LineBundle.pow, modulePow, structureSheaf] |
| | succ n ih => |
| have he := tensor_euler_add p hd H hH hfinite hH2 L (L.pow n) |
| change eulerCharacteristic p 1 (L.pow (n+1)).sheaf - |
| eulerCharacteristic p 1 (L.pow n).sheaf = _ at he |
| push_cast |
| nlinarith |
|
|
| theorem inverse_euler [IsIntegral X] |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p] |
| (hd : topologicalKrullDim X ≤ 1) (H : LineBundle X) (hH : H.IsAmple) |
| (hfinite : FiniteLineCohomology p) (hH2 : LineCohomologyTwoZero X) |
| (L : LineBundle X) : |
| eulerCharacteristic p 1 L.inverse.sheaf - eulerCharacteristic p 1 (structureSheaf X) = |
| -(eulerCharacteristic p 1 L.sheaf - eulerCharacteristic p 1 (structureSheaf X)) := by |
| have h := tensor_euler_add p hd H hH hfinite hH2 L L.inverse |
| have he := eulerCharacteristic_iso p (lineTensorInverseIso L) 1 |
| change eulerCharacteristic p 1 (L.tensor L.inverse).sheaf = |
| eulerCharacteristic p 1 (structureSheaf X) at he |
| omega |
|
|
| end |
| end PiExponent.CurveDegree |
|
|
| namespace PiExponent.NumericalAmpleness |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open PiExponentSeshadri.Geometry |
| variable {X : Scheme.{0}} |
|
|
| theorem curveDegree_tensor (p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p] |
| (H : LineBundle X) (hH : H.IsAmple) (L M : LineBundle X) (C : IntegralCurve X) |
| (hfinite : CurveDegree.FiniteLineCohomology (C.embedding ≫ p)) |
| (hH2 : CurveDegree.LineCohomologyTwoZero C.scheme) : |
| curveDegree p (L.tensor M) C = curveDegree p L C + curveDegree p M C := by |
| have he := CurveDegree.tensor_euler_add (C.embedding ≫ p) C.dimension.le |
| (H.pullback C.embedding) (LineBundle.IsAmple.pullback_closedImmersion H hH C.embedding) |
| hfinite hH2 (L.pullback C.embedding) (M.pullback C.embedding) |
| have hi := eulerCharacteristic_iso (C.embedding ≫ p) |
| (PiExponentSeshadri.PullbackTensor.iso C.embedding L M) 1 |
| unfold curveDegree |
| rw [hi] |
| change _ - _ = (_ - _) + (_ - _) |
| dsimp only [LineBundle.pullback, LineBundle.tensor] at he ⊢ |
| omega |
|
|
| theorem curveDegree_pow (p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p] |
| (H : LineBundle X) (hH : H.IsAmple) (L : LineBundle X) (C : IntegralCurve X) |
| (hfinite : CurveDegree.FiniteLineCohomology (C.embedding ≫ p)) |
| (hH2 : CurveDegree.LineCohomologyTwoZero C.scheme) (n : ℕ) : |
| curveDegree p (L.pow n) C = (n : ℤ) * curveDegree p L C := by |
| have hi := eulerCharacteristic_iso (C.embedding ≫ p) |
| (PiExponentSeshadri.PullbackTensor.powIso C.embedding L n) 1 |
| unfold curveDegree |
| rw [hi] |
| exact CurveDegree.power_euler (C.embedding ≫ p) C.dimension.le |
| (H.pullback C.embedding) (LineBundle.IsAmple.pullback_closedImmersion H hH C.embedding) |
| hfinite hH2 (L.pullback C.embedding) n |
|
|
| theorem curveDegree_iso (p : X ⟶ Spec (CommRingCat.of ℂ)) |
| (L M : LineBundle X) (C : IntegralCurve X) (e : L.sheaf ≅ M.sheaf) : |
| curveDegree p L C = curveDegree p M C := by |
| unfold curveDegree |
| rw [eulerCharacteristic_iso (C.embedding ≫ p) |
| ((Scheme.Modules.pullback C.embedding).mapIso e) 1] |
|
|
| theorem curveDegree_inverse (p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p] |
| (H : LineBundle X) (hH : H.IsAmple) (L : LineBundle X) (C : IntegralCurve X) |
| (hfinite : CurveDegree.FiniteLineCohomology (C.embedding ≫ p)) |
| (hH2 : CurveDegree.LineCohomologyTwoZero C.scheme) : |
| curveDegree p L.inverse C = -curveDegree p L C := by |
| have h := curveDegree_tensor p H hH L L.inverse C hfinite hH2 |
| have hi := eulerCharacteristic_iso (C.embedding ≫ p) |
| (((Scheme.Modules.pullback C.embedding).mapIso (lineTensorInverseIso L)) ≪≫ |
| PiExponentSeshadri.Geometry.pullbackUnitIso C.embedding) 1 |
| change eulerCharacteristic (C.embedding ≫ p) 1 |
| ((Scheme.Modules.pullback C.embedding).obj (L.tensor L.inverse).sheaf) = |
| eulerCharacteristic (C.embedding ≫ p) 1 (structureSheaf C.scheme) at hi |
| have hz : curveDegree p (L.tensor L.inverse) C = 0 := by |
| unfold curveDegree |
| rw [hi] |
| exact sub_self _ |
| omega |
|
|
| theorem curveDegree_eq_rat_of_power_iso |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p] |
| (H : LineBundle X) (hH : H.IsAmple) (L A B : LineBundle X) (C : IntegralCurve X) |
| (hfinite : CurveDegree.FiniteLineCohomology (C.embedding ≫ p)) |
| (hH2 : CurveDegree.LineCohomologyTwoZero C.scheme) |
| (n a b : ℕ) (hn : 0 < n) |
| (e : (L.pow n).sheaf ≅ ((A.pow a).tensor (B.inverse.pow b)).sheaf) : |
| (curveDegree p L C : ℚ) = |
| ((a : ℚ) / n) * (curveDegree p A C : ℚ) - |
| ((b : ℚ) / n) * (curveDegree p B C : ℚ) := by |
| have hd := curveDegree_iso p (L.pow n) ((A.pow a).tensor (B.inverse.pow b)) C e |
| rw [curveDegree_pow p H hH L C hfinite hH2 n, |
| curveDegree_tensor p H hH (A.pow a) (B.inverse.pow b) C hfinite hH2, |
| curveDegree_pow p H hH A C hfinite hH2 a, |
| curveDegree_pow p H hH B.inverse C hfinite hH2 b, |
| curveDegree_inverse p H hH B C hfinite hH2] at hd |
| have hq := congrArg (fun z : ℤ => (z : ℚ)) hd |
| push_cast at hq |
| have hnq : (n : ℚ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hn) |
| calc |
| (curveDegree p L C : ℚ) = |
| ((a : ℚ) * (curveDegree p A C : ℚ) - (b : ℚ) * (curveDegree p B C : ℚ)) / n := |
| (eq_div_iff hnq).mpr (by nlinarith [hq]) |
| _ = _ := by ring |
|
|
| end |
| end PiExponent.NumericalAmpleness |
|
|
| end OAI |
|
|
