openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Ampleness /CurveNormalizationPolarization.lean
| import OAI.NumberTheory.PiExponent.Ampleness.FinitePullbackAmple | |
| import OAI.NumberTheory.PiExponent.Ampleness.ProjectiveO1Ample | |
| import OAI.NumberTheory.PiExponent.Cohomology.CurveNormalizationCohomology | |
| import OAI.NumberTheory.PiExponent.Geometry.CurveDegreeAdditivity | |
| import OAI.NumberTheory.PiExponent.Geometry.CurveNormalizationNonaffine | |
| import OAI.NumberTheory.PiExponent.Geometry.ProjectiveFinitePencil | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponent.CurveNormalizationModel | |
| open AlgebraicGeometry CategoryTheory | |
| open PiExponentSeshadri.Geometry | |
| variable {E : Type} [Field E] [Algebra ℂ E] | |
| variable (f : E) (hf : Transcendental ℂ f) | |
| variable [FiniteDimensional (IntermediateField.adjoin ℂ {f}) E] | |
| def parameterCurvePolarization : LineBundle (parameterCurve f hf) := | |
| (ProjectiveO1.lineBundle (R := ℂ) (σ := Bool)).pullback (parameterCurveProjection f hf) | |
| theorem parameterCurvePolarization_ample : (parameterCurvePolarization f hf).IsAmple := by | |
| let : CompactSpace (ProjectiveO1.projectiveSpace ℂ Bool) := | |
| QuasiCompact.compactSpace_of_compactSpace (polynomialProjectiveProjection ℂ Bool) | |
| exact LineBundle.IsAmple.pullback_finite _ ProjectiveO1.lineBundle_ample | |
| (parameterCurveProjection f hf) | |
| theorem parameterCurve_finiteLineCohomology : | |
| CurveDegree.FiniteLineCohomology (parameterCurveStructureMap f hf) := by | |
| intro L n hn | |
| have h0 := ProjectivePencil.H0_finite_complex (parameterCurveProjection f hf) | |
| (parameterCurve_not_isAffine f hf) L | |
| have h1 := ProjectivePencil.H1_finite_complex (parameterCurveProjection f hf) L | |
| rcases Nat.le_one_iff_eq_zero_or_eq_one.mp hn with rfl | rfl | |
| · exact h0 | |
| · exact h1 | |
| theorem parameterCurve_lineCohomologyTwoZero : | |
| CurveDegree.LineCohomologyTwoZero (parameterCurve f hf) := by | |
| intro L | |
| apply subsingleton_of_forall_eq 0 | |
| intro x | |
| exact parameterCurve_lineBundle_ext_zero f hf L 2 (by omega) x | |
| theorem parameterCurve_tensor_euler_add (L M : LineBundle (parameterCurve f hf)) : | |
| eulerCharacteristic (parameterCurveStructureMap f hf) 1 (L.tensor M).sheaf - | |
| eulerCharacteristic (parameterCurveStructureMap f hf) 1 M.sheaf = | |
| eulerCharacteristic (parameterCurveStructureMap f hf) 1 L.sheaf - | |
| eulerCharacteristic (parameterCurveStructureMap f hf) 1 | |
| (structureSheaf (parameterCurve f hf)) := | |
| CurveDegree.tensor_euler_add _ (parameterCurve_dimension_le_one f hf) | |
| (parameterCurvePolarization f hf) (parameterCurvePolarization_ample f hf) | |
| (parameterCurve_finiteLineCohomology f hf) (parameterCurve_lineCohomologyTwoZero f hf) L M | |
| theorem parameterCurve_power_euler (L : LineBundle (parameterCurve f hf)) (n : ℕ) : | |
| eulerCharacteristic (parameterCurveStructureMap f hf) 1 (L.pow n).sheaf - | |
| eulerCharacteristic (parameterCurveStructureMap f hf) 1 | |
| (structureSheaf (parameterCurve f hf)) = | |
| (n : ℤ) * (eulerCharacteristic (parameterCurveStructureMap f hf) 1 L.sheaf - | |
| eulerCharacteristic (parameterCurveStructureMap f hf) 1 | |
| (structureSheaf (parameterCurve f hf))) := | |
| CurveDegree.power_euler _ (parameterCurve_dimension_le_one f hf) | |
| (parameterCurvePolarization f hf) (parameterCurvePolarization_ample f hf) | |
| (parameterCurve_finiteLineCohomology f hf) (parameterCurve_lineCohomologyTwoZero f hf) L n | |
| end PiExponent.CurveNormalizationModel | |
| end | |
| end OAI | |