Dual boundary complexes of Betti moduli spaces over the two-sphere with one irregular singularity

TL;DR

This paper verifies the weak geometric P=W conjecture for certain wild character varieties over the two-sphere with one irregular singularity, using microlocal geometry and braid theory.

Contribution

It provides the first verification of the conjecture for wild character varieties associated with Stokes Legendrian links from positive braids.

Findings

Homotopy equivalence of boundary complex to a sphere of dimension d-1

Verification of the weak geometric P=W conjecture in new wild setting

Application of microlocal geometric methods to character varieties

Abstract

The weak geometric P=W conjecture of L. Katzarkov, A. Noll, P. Pandit, and C. Simpson states that, a smooth Betti moduli space of complex dimension $d$ over a punctured Riemann surface has the dual boundary complex homotopy equivalent to a sphere of dimension $d-1$. Via a microlocal geometric perspective, we verify this conjecture for a class of rank $n$ wild character varieties over the two-sphere with one puncture, associated with any Stokes Legendrian link defined by an $n$-strand positive braid.