openAI-math-dataset / lean /OAI /AlgebraicGeometry /Seshadri /Intersection /GeneralSurfaceEuler.lean
| import Mathlib | |
| import OAI.AlgebraicGeometry.Seshadri.Sheaves.TensorPowerDistribution | |
| import OAI.AlgebraicGeometry.Seshadri.Intersection.MixedIntersection | |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.AmpleFromCover | |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.GeneralQuadraticNumerics | |
| namespace OAI | |
| section | |
| namespace MaximalSeshadri.Geometry | |
| noncomputable section | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace | |
| open MaximalSeshadri.Frames MaximalSeshadri.Projective | |
| theorem Surface.arbitrary_power_euler_quadratic (S : Surface) | |
| (L : LineBundle S.scheme) (hL : L.IsAmple) (M : LineBundle S.scheme) (n : ℕ) : | |
| 2*eulerCharacteristic S.structureMap 2 (M.pow n).sheaf = | |
| (n : ℤ)*((n : ℤ)-1)*selfIntersection S M + | |
| 2*(n : ℤ)*(eulerCharacteristic S.structureMap 2 M.sheaf - | |
| eulerCharacteristic S.structureMap 2 (O S.scheme)) + | |
| 2*eulerCharacteristic S.structureMap 2 (O S.scheme) := by | |
| obtain ⟨d,_,hB⟩ := L.exists_ample_twist M hL | |
| let B := (L.pow d).tensor M | |
| let f (n : ℕ) := eulerCharacteristic S.structureMap 2 (M.pow n).sheaf | |
| let g (n : ℕ) := eulerCharacteristic S.structureMap 2 (B.pow n).sheaf | |
| let h (n : ℕ) := eulerCharacteristic S.structureMap 2 (L.pow n).sheaf | |
| have hQ (A : LineBundle S.scheme) (hA : A.IsAmple) : | |
| EulerNumerics.Quadratic (fun n => eulerCharacteristic S.structureMap 2 (A.pow n).sheaf) := by | |
| intro j | |
| dsimp only | |
| rw [S.euler_power_second_difference A,S.euler_power_one A] | |
| exact S.power_euler_quadratic A hA j | |
| have he (j : ℕ) : f j = g j - h (d*j) + | |
| eulerCharacteristic S.structureMap 2 (O S.scheme) - | |
| (d : ℤ)*(j : ℤ)^2*mixedEuler S L M := by | |
| have H := S.twist_euler L hL (M.pow j) (d*j) | |
| have E := eulerCharacteristic_iso S.structureMap (lineTwistPower L M d j) 2 | |
| change g j = eulerCharacteristic S.structureMap 2 ((L.pow (d*j)).tensor (M.pow j)).sheaf at E | |
| rw [← E,mixedEuler_pow_right S L hL] at H | |
| dsimp only [f,h] | |
| push_cast at H | |
| linear_combination -H | |
| have H := EulerNumerics.quadratic_of_twisted f g h d | |
| (eulerCharacteristic S.structureMap 2 (O S.scheme)) (mixedEuler S L M) | |
| (hQ B hB) (hQ L hL) he n | |
| dsimp only [f] at H | |
| rw [S.euler_power_second_difference M,S.euler_power_one M] at H | |
| exact H | |
| end | |
| end MaximalSeshadri.Geometry | |
| end | |
| end OAI | |