Character varieties, hexagonal 3-webs and Hess connection

TL;DR

This paper explores affine structures on SL 2(C)-character varieties of the four-punctured sphere using web and symplectic geometry, offering a new proof of Painlevé VI equations' irreducibility.

Contribution

It introduces a novel geometric approach to prove the non-integrability of Painlevé VI equations, expanding understanding of character varieties and their geometric structures.

Findings

Established affine structures on character varieties.

Provided a new proof of Painlevé VI irreducibility.

Linked web geometry with symplectic structures.

Abstract

We investigate the existence of some affine structure on SL 2 (C)-character varieties on the four-punctured sphere through web and symplectic geometry. This work provides a new proof of non integrability (i.e. irreducibility) of Painlev{\'e} VI equations, previously proved by S. Cantat and the third author.