Cohomology rings of character varieties

TL;DR

This paper discusses recent advances in understanding the cohomology rings of character varieties of Riemann surfaces, including proofs of conjectures and explicit descriptions linking to Hilbert schemes and symplectic geometry.

Contribution

It presents new results on the cohomology rings of character varieties, including the proof of the $P=W$ conjecture and explicit descriptions for punctured spheres.

Findings

Proof of the $P=W$ conjecture.

Explicit description of cohomology rings for punctured spheres.

Relation of the general case to symplectic geometry of Hilbert schemes.

Abstract

In this talk I give an introduction and present some recent progress towards understanding the cohomology rings of character varieties of Riemann surfaces, such as the proof of the $P=W$ conjecture and the computation of the zero-dimensional COHA. In the case of punctured sphere I present an explicit description relating the cohomology rings to the Hilbert scheme of $\mathbb{C}^2$, refining conjectures of Hausel-Letellier-Rodriguez-Villegas and Chuang-Diaconescu-Donagi-Pantev. I explain how the general case should be related to the symplectic geometry of the Hilbert scheme.