Arithmetic of singular character varieties and their E-polynomials

Abstract

We calculate the E-polynomials of the SL₃(C)- and GL₃(C)-character varieties of compact oriented surfaces of any genus and the E-polynomials of the SL₂(C)- and GL₂(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 SL₂(C)-character varieties of compact orientable surfaces, which were computed by Logares, Muñoz and Newstead for genus g = 1, 2 and by Martinez and Muñoz for g ⩾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.