openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /ConstantPullbackDegree.lean
| import OAI.NumberTheory.PiExponent.Approximation.LinePullback | |
| import OAI.NumberTheory.PiExponent.Approximation.NumericalMarginRestriction | |
| import OAI.NumberTheory.PiExponent.Approximation.SectionPullback | |
| namespace OAI | |
| namespace PiExponent.ConstantPullbackDegree | |
| noncomputable section | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace | |
| open PiExponentSeshadri.Geometry PiExponentSeshadri.Frames | |
| variable {X Y : Scheme.{0}} | |
| def pullbackFrame_of_range_subset (K : LineBundle Y) (g : X ⟶ Y) | |
| (U : Y.Opens) (h : Set.range g ⊆ (U : Set Y)) | |
| (e : K.sheaf.restrict U.ι ≅ structureSheaf U.toScheme) : | |
| (K.pullback g).sheaf ≅ structureSheaf X := by | |
| have hr : Set.range g ⊆ Set.range U.ι := by | |
| rw [Scheme.Opens.range_ι] | |
| exact h | |
| let l := IsOpenImmersion.lift U.ι g hr | |
| have hl : l ≫ U.ι = g := IsOpenImmersion.lift_fac U.ι g hr | |
| exact (Scheme.Modules.pullbackCongr hl.symm).app K.sheaf ≪≫ | |
| ((Scheme.Modules.pullbackComp l U.ι).app K.sheaf).symm ≪≫ | |
| (Scheme.Modules.pullback l).mapIso ((Scheme.Modules.restrictFunctorIsoPullback U.ι).app | |
| K.sheaf).symm ≪≫ (Scheme.Modules.pullback l).mapIso e ≪≫ pullbackUnitIso l | |
| def constantPullbackIso (K : LineBundle Y) (g : X ⟶ Y) (y : Y) | |
| (hg : Set.range g ⊆ ({y} : Set Y)) : | |
| (K.pullback g).sheaf ≅ structureSheaf X := by | |
| apply Classical.choice | |
| obtain ⟨U, hy, ⟨e⟩⟩ := K.locallyRankOne y | |
| refine ⟨pullbackFrame_of_range_subset K g U ?_ e⟩ | |
| intro z hz | |
| have hz' : z = y := hg hz | |
| exact hz'.symm ▸ hy | |
| theorem euler_difference_eq_zero (p : X ⟶ Spec (CommRingCat.of ℂ)) | |
| (K : LineBundle Y) (g : X ⟶ Y) (y : Y) | |
| (hg : Set.range g ⊆ ({y} : Set Y)) (d : ℕ) : | |
| eulerCharacteristic p d (K.pullback g).sheaf - | |
| eulerCharacteristic p d (structureSheaf X) = 0 := by | |
| rw [eulerCharacteristic_iso p (constantPullbackIso K g y hg) d] | |
| exact sub_self _ | |
| theorem euler_difference_eq_zero_of_constant (p : X ⟶ Spec (CommRingCat.of ℂ)) | |
| (K : LineBundle Y) (g : X ⟶ Y) (y : Y) | |
| (hg : ∀ x, g x = y) (d : ℕ) : | |
| eulerCharacteristic p d (K.pullback g).sheaf - | |
| eulerCharacteristic p d (structureSheaf X) = 0 := | |
| euler_difference_eq_zero p K g y (by rintro z ⟨x,rfl⟩; exact hg x) d | |
| theorem curveDegree_eq_zero_of_contracted | |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) (K : LineBundle Y) | |
| (g : X ⟶ Y) (C : NumericalAmpleness.IntegralCurve X) (y : Y) | |
| (hg : Set.range (C.embedding ≫ g) ⊆ ({y} : Set Y)) : | |
| NumericalAmpleness.curveDegree p (K.pullback g) C = 0 := by | |
| have he := eulerCharacteristic_iso (C.embedding ≫ p) | |
| ((Scheme.Modules.pullbackComp C.embedding g).app K.sheaf) 1 | |
| simp only [CategoryTheory.Functor.comp_obj] at he | |
| unfold NumericalAmpleness.curveDegree | |
| change eulerCharacteristic (C.embedding ≫ p) 1 | |
| ((Scheme.Modules.pullback C.embedding).obj ((Scheme.Modules.pullback g).obj K.sheaf)) - | |
| eulerCharacteristic (C.embedding ≫ p) 1 (structureSheaf C.scheme) = 0 | |
| rw [he] | |
| exact euler_difference_eq_zero (C.embedding ≫ p) K (C.embedding ≫ g) y hg 1 | |
| end | |
| end PiExponent.ConstantPullbackDegree | |
| end OAI | |