Intersection cohomology of character varieties for punctured Riemann surfaces

TL;DR

This paper computes the intersection cohomology of character varieties for punctured Riemann surfaces with various monodromies, confirming a conjecture and extending previous results to more general cases.

Contribution

It extends Mellit's results to monodromies of any Jordan type, proving a conjecture by Letellier about the Poincaré polynomial of these varieties.

Findings

Computed intersection cohomology for all Jordan types

Confirmed the Poincaré polynomial specialization conjecture

Extended previous results to more general monodromies

Abstract

We study intersection cohomology of character varieties for punctured Riemann surfaces with prescribed monodromies around the punctures. Relying on previous result from Mellit for semisimple monodromies we compute the intersection cohomology of character varieties with monodromies of any Jordan type. This proves the Poincar\'e polynomial specialization of a conjecture from Letellier.