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Towards a nonlinear Schwarz's list

Philip Boalch

TL;DR

The work investigates algebraic solutions of the Painlevé VI equation by extending Schwarz's finite monodromy framework to a nonlinear setting. It develops multiple construction pathways—nonrigid Schwarz-type lists (A) and (B), isomonodromic deformations, and nonabelian Gauss–Manin connections—augmented by middle convolution, pullbacks, and folding transformations. Key contributions include explicit icosahedral, Klein, and Valentiner solutions, elliptic and genus-one examples, and a broad mechanism to generate new solutions from finite and complex reflection groups, up to degree 72. This broadens the nonlinear Schwarz’s list and highlights deep connections between differential equations, algebraic geometry (moduli of cubic surfaces, Belyi maps, dessins), and representation theory, with potential implications for the study of monodromy, moduli, and special function theory.

Abstract

This is basically the text of a survey talk (entitled 'Painleve, Klein and the icosahedron') given at Hitchin's 60th birthday conference. It discusses the search for and construction of algebraic solutions of the sixth Painleve differential equation, which may be viewed as a nonlinear analogue of the Gauss hypergeometric equation. Both algebraic and transcendental methods are used and the story involves affine Weyl groups, braid groups and cubic surfaces. Some emphasis is given to the interpretation of the sixth Painleve equation as the explicit form of the simplest nonabelian Gauss-Manin connection, i.e. as a nonlinear differential equation 'coming from geometry', much as Picard-Fuchs equations arise in the case of cohomology with abelian coefficients.

Towards a nonlinear Schwarz's list

TL;DR

The work investigates algebraic solutions of the Painlevé VI equation by extending Schwarz's finite monodromy framework to a nonlinear setting. It develops multiple construction pathways—nonrigid Schwarz-type lists (A) and (B), isomonodromic deformations, and nonabelian Gauss–Manin connections—augmented by middle convolution, pullbacks, and folding transformations. Key contributions include explicit icosahedral, Klein, and Valentiner solutions, elliptic and genus-one examples, and a broad mechanism to generate new solutions from finite and complex reflection groups, up to degree 72. This broadens the nonlinear Schwarz’s list and highlights deep connections between differential equations, algebraic geometry (moduli of cubic surfaces, Belyi maps, dessins), and representation theory, with potential implications for the study of monodromy, moduli, and special function theory.

Abstract

This is basically the text of a survey talk (entitled 'Painleve, Klein and the icosahedron') given at Hitchin's 60th birthday conference. It discusses the search for and construction of algebraic solutions of the sixth Painleve differential equation, which may be viewed as a nonlinear analogue of the Gauss hypergeometric equation. Both algebraic and transcendental methods are used and the story involves affine Weyl groups, braid groups and cubic surfaces. Some emphasis is given to the interpretation of the sixth Painleve equation as the explicit form of the simplest nonabelian Gauss-Manin connection, i.e. as a nonlinear differential equation 'coming from geometry', much as Picard-Fuchs equations arise in the case of cohomology with abelian coefficients.

Paper Structure

This paper contains 19 sections, 3 theorems, 42 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Naive generalisations
  3. Nonlinear analogue---the Painlevé VI equation
  4. What is Painlevé VI?
  5. Conceptual approach to $\text{\bf P}_{\text{\bf VI}}$
  6. Explicit nonlinear equations
  7. Monodromy of Painlevé VI
  8. Algebraic solutions from finite subgroups of $\mathbf{{\mathop{\rm SL}}_2(C)}$
  9. What exactly is an algebraic solution?
  10. (A) $\mapsto$ (C)
  11. Beyond Platonic $\text{\bf P}_{\text{\bf VI}}$ solutions
  12. Construction
  13. Relating connections (A) and (B)
  14. New solutions
  15. Pullbacks
  16. ...and 4 more sections

Key Result

Theorem 1

The isomonodromic deformations of type (B) connections (on rank three vector bundles) are also controlled by the Painlevé VI equation, and all $\text{\rm P}_{\text{\rm VI}}$ equations arise in this way.

Figures (2)

  • Figure 1: Degeneration to two hypergeometric systems
  • Figure 2: $237$ degree $10$ rational map $f$

Theorems & Definitions (3)

  • Theorem 1: pecrk2p
  • Theorem 2: srops
  • Theorem 3: icosa