P=W conjectures for character varieties with symplectic resolution

TL;DR

This paper proves the P=W and PI=WI conjectures for certain character varieties with symplectic resolutions, focusing on genus 1 and 2 cases, and develops new topological tools for Higgs bundle moduli spaces.

Contribution

It establishes the P=W conjecture for symplectic resolutions of character varieties and introduces new constructions of moduli space compactifications.

Findings

Proved P=W conjecture for genus 1 and 2 character varieties with symplectic resolution.

Constructed a relative compactification of the Hodge moduli space for reductive groups.

Showed projectivity of the compactified de Rham moduli space.

Abstract

We establish P=W and PI=WI conjectures for character varieties with structural group $\mathrm{GL}_n$ and $\mathrm{SL}_n$ which admit a symplectic resolution, i.e. for genus 1 and arbitrary rank, and genus 2 and rank 2. We formulate the P=W conjecture for resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-\'{e}tale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, and the projectivity of the compactification of the de Rham moduli space. In particular, we study in detail a Dolbeault moduli space which is specialization of the singular irreducible holomorphic symplectic variety of type O'Grady 6.