Arithmetic of singular character varieties and their $E$-polynomials

TL;DR

This paper computes the $E$-polynomials of various $SL_n(C)$ and $GL_n(C)$ character varieties of surfaces, introducing a new, simpler arithmetic method to handle singular cases and extending previous results.

Contribution

It introduces a novel arithmetic approach to compute $E$-polynomials of singular character varieties, extending existing methods to untwisted cases and higher genus surfaces.

Findings

Calculated $E$-polynomials for $SL_3(C)$ and $GL_3(C)$-character varieties of surfaces.

Provided a simpler computation method for $SL_2(C)$-character varieties.

Extended techniques to singular and untwisted character varieties.

Abstract

We calculate the $E$-polynomials of the $SL_3(\mathbb{C})$ and $GL_3(\mathbb{C})$-character varieties of compact oriented surfaces of any genus and the $E$-polynomials of the $SL_2(\mathbb{C})$ and $GL_2(\mathbb{C})$-character varieties of compact non-orientable surfaces of any Euler characteristic. Our methods also give a new and significantly simpler computation of the $E$-polynomials of the $SL_2(\mathbb{C})$-character varieties of compact orientable surfaces, which were computed by Logares, Mu\~noz and Newstead for genus $g=1,2$ and by Martinez and Mu\~noz for $g \ge 3$. Our technique is based on the arithmetic of character varieties over finite fields. More specifically, we show how to extend the approach of Hausel and Rodriguez-Villegas used for non-singular (twisted) character varieties to the singular (untwisted) case.