abel-theorem: the CR-converse promotion (docs/design/abel-theorem.md §4.2 D2, §2.1 step 7) #
Unit: abel-theorem. Namespace RS.Abel. Two deliverables:
wirtingerDbar_exp_neg_mul_eq_zero— thedbarF = 0computation underlying the meromorphic promotion: ifu's Wirtingerdbar-derivative matchesf's own logarithmicdbar-derivative (dbaru = dbarf / f) at a point wheref ≠ 0, thenF := exp(-u) * fhasdbarF = 0there — a direct Leibniz/chain-rule computation (wirtingerDbar_mul+ the holomorphic-outer chain rule forComplex.exp, via mathlib'sHasDerivAt.comp_hasFDerivAt).genus_eq_zero_of_exists_simple_pole_zero— Forster's necessity shortcut (§2.2): a meromorphic function with exactly one simple pole and one simple zero (elsewhere holomorphic and non-vanishing) forcesgenus X = 0, composing two ALREADY-BUILT, fully-proved facts (no admitted goals) —RS.homeoSphere_of_exists_simple_pole(proper-map-degree) andRS.SphereTopology.genus_eq_zero_of_homeo_sphere(sphere-topology).form-trace-toweris not imported anywhere in this unit, matching the design's finding (§2.2).
D2: the dbar-product-rule identity (mathlib-only, pure ℂ → ℂ) #
theorem
RS.Abel.wirtingerDbar_exp_neg_mul_eq_zero
{u f : ℂ → ℂ}
{z : ℂ}
(hu : DifferentiableAt ℝ u z)
(hf : DifferentiableAt ℝ f z)
(h : wirtingerDbar u z = wirtingerDbar f z / f z)
(hfz : f z ≠ 0)
:
D2, the dbarF = 0 computation (§2.1 step 7): the exponential-corrected function
F := fun w => exp(-(u w)) * f w is genuinely dbar-flat wherever u's dbar-derivative matches
f's own logarithmic one.
§2.2: Forster's necessity shortcut (via ALREADY-BUILT proper-map-degree/sphere-topology) #
theorem
RS.Abel.genus_eq_zero_of_exists_simple_pole
{X : Type u_1}
[TopologicalSpace X]
[T2Space X]
[CompactSpace X]
[ConnectedSpace X]
[ChartedSpace ℂ X]
[IsManifold (modelWithCornersSelf ℂ ℂ) ⊤ X]
(F : Mero X)
(Q : X)
(hpole : MeroGermOn.ord F Q = -1)
(hreg : ∀ (x : X), x ≠ Q → 0 ≤ MeroGermOn.ord F x)
:
The necessity shortcut this unit actually needs (§2.2): NOT Forster's general
Trace(ω)-on-ℙ¹ necessity construction — just the "one simple pole" specialization, already
fully discharged by two other units. form-trace-tower is not imported anywhere here.