Arithmetic geometry of character varieties with regular monodromy

TL;DR

This paper investigates the geometric and topological properties of character varieties associated with surface group representations into reductive groups, including smoothness, point counting, and invariants, with applications to rigidity phenomena.

Contribution

It provides new results on smoothness conditions, explicit point counts, and topological invariants of character varieties with specific monodromy types, advancing understanding of their structure.

Findings

Character varieties are smooth and equidimensional under certain conditions.

Explicit formulas for point counts and E-polynomials of these varieties.

Determination of topological invariants like Euler characteristic and connected components.

Abstract

We study character varieties arising as moduli of representations of an orientable surface group into a reductive group $G$. We first show that if $G/Z$ acts freely on the representation variety, then both the representation variety and the character variety are smooth and equidimensional. Next, we count points on a family of smooth character varieties; namely, those involving both regular semisimple and regular unipotent monodromy. In particular, we show that these varieties are polynomial count and obtain an explicit expression for their $E$-polynomials. Finally, by analysing the $E$-polynomial, we determine certain topological invariants of these varieties such as the Euler characteristic and the number of connected components. As an application, we give an example of a cohomologically rigid representation which is not physically rigid.