Asymptotic geometry of non-abelian Hodge theory and Riemann--Hilbert correspondence, rank three $\widetilde{E}_6$ case

TL;DR

This paper proves the Geometric P=W conjecture for rank 3 on the three-punctured sphere, describing the topology at infinity of the character variety using asymptotic analysis of harmonic bundles and Stokes phenomena.

Contribution

It establishes the Geometric P=W conjecture in rank 3 for the three-punctured sphere and analyzes the asymptotic geometry and topology of the associated character variety.

Findings

Proof of the Geometric P=W conjecture in rank 3

Description of the topology at infinity of the character variety

Analysis of the Stokes phenomenon and asymptotic behavior

Abstract

We prove the Geometric P=W conjecture in rank 3 on the three-punctured sphere. We describe the topology at infinity of the related character variety. We use asymptotic abelianization of harmonic bundles away from the ramification divisor and an equivariant approach near the branch points to find the WKB (also known as Liouville--Green or phase-integral) expansion of the involved maps. We analyze the Stokes phenomenon governing their behavior.