Documentation

LeanPool.JacobianDiffgeo.Abel.WeakToMero

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:

D2: the dbar-product-rule identity (mathlib-only, pure ℂ → ℂ) #

theorem RS.Abel.wirtingerDbar_exp_neg_mul_eq_zero {u f : } {z : } (hu : DifferentiableAt u z) (hf : DifferentiableAt f z) (h : wirtingerDbar u z = wirtingerDbar f z / f z) (hfz : f z 0) :
wirtingerDbar (fun (w : ) => Complex.exp (-u w) * f w) 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 X] [IsManifold (modelWithCornersSelf ) X] (F : Mero X) (Q : X) (hpole : MeroGermOn.ord F Q = -1) (hreg : ∀ (x : X), x Q0 MeroGermOn.ord F x) :
genus X = 0

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.