| import Mathlib | |
| import OAI.Geometry.PrescribedPotential.NormalizedPoisson | |
| import OAI.Geometry.PrescribedPotential.VolumePath | |
| namespace OAI | |
| /-! Dimension One Potential. -/ | |
| section | |
| noncomputable section | |
| open Set Filter Topology Matrix | |
| open scoped ContDiff Classical ComplexOrder | |
| namespace MongeAmpere | |
| lemma singleton_det_add (H K : Matrix (Fin 1) (Fin 1) ℂ) (hH : H.PosDef) : | |
| (H + K).det.re = H.det.re * (1 + (H⁻¹ * K).trace.re) := by | |
| have hr : ((H 0 0).re : ℂ) = H 0 0 := | |
| Complex.conj_eq_iff_re.mp (hH.isHermitian.apply 0 0) | |
| have hn : (H 0 0).re ≠ 0 := ne_of_gt (Complex.pos_iff.mp hH.diag_pos).1 | |
| simp only [Matrix.det_fin_one, Matrix.add_apply, Complex.add_re, | |
| Matrix.trace_fin_one, Matrix.mul_apply, Fin.sum_univ_one, | |
| Matrix.inv_def, Matrix.adjugate_fin_one, Matrix.det_fin_one] | |
| rw [← hr] | |
| simp [Complex.mul_re, Complex.inv_re, Complex.inv_im, Complex.normSq, Ring.inverse_eq_inv] | |
| field_simp | |
| lemma singleton_posDef_of_det (H : Matrix (Fin 1) (Fin 1) ℂ) | |
| (hH : H.IsHermitian) (hp : 0 < H.det.re) : H.PosDef := by | |
| have hd : H = Matrix.diagonal (fun _ : Fin 1 => H 0 0) := by | |
| ext i j | |
| fin_cases i | |
| fin_cases j | |
| simp | |
| rw [hd, Matrix.posDef_diagonal_iff] | |
| intro i | |
| apply Complex.pos_iff.mpr | |
| refine ⟨?_, ?_⟩ | |
| · simpa only [Matrix.det_fin_one] using hp | |
| · exact (Complex.conj_eq_iff_im.mp (hH.apply 0 0)).symm | |
| end MongeAmpere | |
| namespace Anticanonical.SourceSmooth | |
| variable {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X] | |
| [ConnectedSpace X] {A : ComplexAtlas 1 X} | |
| theorem dimensionOne_positiveDensity (g : KaehlerMetric A) (q : SmoothRealFunction A) | |
| (hq : ∀ x, 0 < q.value x) : | |
| ∃ (φ : SmoothRealFunction A) (c : ℝ), 0 < c ∧ g.PositivePotential φ ∧ | |
| ∀ i z, z ∈ (A.chart i).target → | |
| (g.matrix i z + φ.hessian i z).det.re = | |
| c * q.localExpression i z * g.volumeCoefficient i z := by | |
| let x₀ : X := Classical.arbitrary X | |
| obtain ⟨u,a,hn,he⟩ := normalizedPoisson 1 X A g x₀ q | |
| have ha : 0 < a := by | |
| obtain ⟨x,_,hx⟩ := isCompact_univ.exists_isMaxOn Set.univ_nonempty u.continuous.continuousOn | |
| obtain ⟨i,hi⟩ := A.covers x | |
| have hm := g.linearized_nonpos_at_globalMax u (fun y => hx (mem_univ y)) i hi | |
| have heq := he i (A.chart i x) ((A.chart i).mapsTo hi) | |
| have hqe : q.localExpression i (A.chart i x) = q.value x := by | |
| simp only [SmoothRealFunction.localExpression, Function.comp_apply, (A.chart i).left_inv hi] | |
| rw [hqe] at heq | |
| linarith [hq x] | |
| let φ := u.realSMul a⁻¹ | |
| have heq (i : Fin A.count) (z : Coordinates 1) (hz : z ∈ (A.chart i).target) : | |
| (g.matrix i z + φ.hessian i z).det.re = | |
| a⁻¹ * q.localExpression i z * g.volumeCoefficient i z := by | |
| rw [MongeAmpere.singleton_det_add _ _ (g.positive i z hz)] | |
| change g.volumeCoefficient i z * (1 + g.linearizedMongeAmpere φ i z) = _ | |
| rw [show φ = u.realSMul a⁻¹ from rfl, g.linearizedMongeAmpere_realSMul] | |
| have hh := he i z hz | |
| have hae : a ≠ 0 := ne_of_gt ha | |
| field_simp | |
| rw [add_comm a _,hh] | |
| refine ⟨φ,a⁻¹,inv_pos.mpr ha,?_,heq⟩ | |
| intro i z hz | |
| apply MongeAmpere.singleton_posDef_of_det | |
| · exact (g.positive i z hz).isHermitian.add (φ.hessian_hermitian i hz) | |
| · rw [heq i z hz] | |
| exact mul_pos (mul_pos (inv_pos.mpr ha) (hq ((A.chart i).symm z))) | |
| (g.volumeCoefficient_pos i hz) | |
| theorem prescribedPotential_dimensionOne (X : Type) [TopologicalSpace X] [T2Space X] | |
| [CompactSpace X] [ConnectedSpace X] (A : ComplexAtlas 1 X) | |
| (P : A.ProjectiveEmbedding) (h : SemipositiveAnticanonicalMetric A) : | |
| HasPrescribedVolumePotential P h := by | |
| let g := projectiveBackground P | |
| let F := prescribedForcing g h | |
| let q : SmoothRealFunction A := { | |
| value := fun x => Real.exp (F.value x) | |
| smooth := fun i => Real.contDiff_exp.contDiffOn.comp (F.smooth i) (fun _ _ => mem_univ _) } | |
| obtain ⟨φ,c,hc,hp,he⟩ := dimensionOne_positiveDensity g q (fun x => Real.exp_pos _) | |
| refine ⟨φ,c,hc,hp,fun i z hz => ?_⟩ | |
| rw [he i z hz] | |
| change c * Real.exp (F.localExpression i z) * g.volumeCoefficient i z = _ | |
| rw [prescribedForcing_local g h i hz, Real.exp_sub, | |
| Real.exp_log (g.volumeCoefficient_pos i hz)] | |
| field_simp [ne_of_gt (g.volumeCoefficient_pos i hz)] | |
| end Anticanonical.SourceSmooth | |
| end | |
| end | |
| end OAI | |