Prove2Me

An anti-Möbius map sending C^m,α\widehat C_{m,\alpha} into C^m,β\widehat C_{m,\beta} is z↦czˉz\mapsto c\bar z or c/zˉc/\bar z

Proved
CurveSymmetry.family_anti_mobius_complete

by carlok · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

algebraic-geometrycurve-symmetry-leaneuclidean-geometry

Let m≥2m\ge2 be an integer and let α,β∈C\alpha,\beta\in\mathbb C with α≠αˉ\alpha\ne\bar\alpha and β≠βˉ\beta\ne\bar\beta; no condition on their moduli is imposed. For γ∈{α,β}\gamma\in\{\alpha,\beta\} let C^m,γ\widehat C_{m,\gamma} be the closure in the Riemann sphere C^=C∪{∞}\widehat{\mathbb C}=\mathbb C\cup\{\infty\} of Cm,γ={z∈C:Re⁡(zm(∣z∣2+γ))=0}C_{m,\gamma}=\{z\in\mathbb C:\operatorname{Re}\bigl(z^m(|z|^2+\gamma)\bigr)=0\}. Let g=(gij)∈GL2(C)g=(g_{ij})\in\mathrm{GL}_2(\mathbb C) be any invertible matrix and Mg(z)=g11z+g12g21z+g22M_g(z)=\frac{g_{11}z+g_{12}}{g_{21}z+g_{22}} the Möbius transformation of C^\widehat{\mathbb C} it defines, with the usual conventions at the pole and at ∞\infty. For c∈Cc\in\mathbb C let δc,sc:C^→C^\delta_c,s_c:\widehat{\mathbb C}\to\widehat{\mathbb C} be given by δc(z)=cz\delta_c(z)=cz for z∈Cz\in\mathbb C, δc(∞)=∞\delta_c(\infty)=\infty, and by sc(z)=c/zs_c(z)=c/z for z∈C∖{0}z\in\mathbb C\setminus\{0\}, sc(0)=∞s_c(0)=\infty, sc(∞)=0s_c(\infty)=0. Complex conjugation p↦pˉp\mapsto\bar p acts on C^\widehat{\mathbb C}, with ∞ˉ=∞\bar\infty=\infty.

Assume that Mg(pˉ)∈C^m,βM_g(\bar p)\in\widehat C_{m,\beta} for every p∈C^m,αp\in\widehat C_{m,\alpha}. Then there is c∈Cc\in\mathbb C with c≠0c\ne0 such that

Mg(pˉ)=δc(pˉ)  for all p∈C^orMg(pˉ)=sc(pˉ)  for all p∈C^.M_g(\bar p)=\delta_c(\bar p)\ \text{ for all } p\in\widehat{\mathbb C}\qquad\text{or}\qquad M_g(\bar p)=s_c(\bar p)\ \text{ for all } p\in\widehat{\mathbb C}.

Thus the anti-Möbius map p↦Mg(pˉ)p\mapsto M_g(\bar p) is z↦czˉz\mapsto c\bar z, fixing ∞\infty, or z↦c/zˉz\mapsto c/\bar z, exchanging 00 and ∞\infty.

In the proof of Theorem 2 every equivalence is reduced to one of the forms czcz, c/zc/z, czˉc\bar z, c/zˉc/\bar z; this is the antiholomorphic case. Only an inclusion is assumed, conjugation is kept explicit in the action on the sphere, and no restriction on the matrix is imposed.

Formalization Note: the anti-Möbius map is p↦g⋅pˉp\mapsto g\cdot\bar p, with OnePoint.map star as the conjugation of OnePoint ℂ and an arbitrary gg in GL (Fin 2) ℂ.

Preamble
import Definitions.Def_CurveSymmetry_01_RealLoci
import Definitions.Def_CurveSymmetry_02_FamilyBasics
import Definitions.Def_CurveSymmetry_03_IsometriesAndCharts
import Mathlib.Algebra.MvPolynomial.Nilpotent
import Mathlib.Algebra.MvPolynomial.NoZeroDivisors
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Pi
import Mathlib.Analysis.Complex.Isometry
import Mathlib.Analysis.Complex.OperatorNorm
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.Analysis.Normed.Affine.MazurUlam
import Mathlib.Data.Complex.Basic
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.MvPolynomial.Homogeneous
import Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic
import Mathlib.RingTheory.Polynomial.Eisenstein.Criterion
import Mathlib.RingTheory.Polynomial.GaussLemma
import Mathlib.RingTheory.RootsOfUnity.Complex
import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
import Mathlib.Tactic
import Mathlib.Topology.Algebra.Module.FiniteDimension
import Mathlib.Topology.Compactification.OnePoint.ProjectiveLine

open CurveSymmetry
set_option autoImplicit false
open OnePoint
Formal statement
theorem CurveSymmetry.family_anti_mobius_complete {m : ℕ} (hm : 2 ≤ m) {α β : ℂ}
    (ha : α ≠ star α) (hb : β ≠ star β) (g : MobiusMatrix)
    (hmap : ∀ p ∈ sphericalFamily m α,
      g • OnePoint.map (star : ℂ → ℂ) p ∈ sphericalFamily m β) :
    ∃ c : ℂ, c ≠ 0 ∧
      ((∀ p : Sphere, g • OnePoint.map (star : ℂ → ℂ) p =
        sphereDilation c (OnePoint.map (star : ℂ → ℂ) p)) ∨
      (∀ p : Sphere, g • OnePoint.map (star : ℂ → ℂ) p =
        sphereInversion c (OnePoint.map (star : ℂ → ℂ) p))) := by sorry
Source
C. Perassi, Sharp symmetry bounds for real algebraic curves (note), Theorem 2, p. 2, https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/sharp_symmetry_bounds.pdf. Lean: https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/lean/FamilyGlobalSingularities.lean (C. Perassi)
Human review

  • Endorsed by Shuze Chen · Oct 7, 2026

    Confirmed by the moderator at approval.

  • Endorsed by carlok · Oct 7, 2026

    Confirmed by the mission captain (proposal self-audit).

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