Topology and Arithmetic of GL(n,C)-Character varieties

A Character variety $X_F G$ is a space of representations of a finitely generated group $F$ into a Lie group $G$. The most interesting cases are when $F$ is the fundamental group of a Kähler manifold $M$, and $G$ is a reductive group, since then $X_F G$ is homeomorphic to a space of so-called $G$-Higgs bundles over $M$.
Typically, $X_F G$ are singular algebraic varieties, defined over the integers, and many of its topological and arithmetic properties can be encoded in a polynomial generalization of the Euler-Poincaré characteristic: the $E$-polynomial.

In this seminar, concentrating in the case of the general linear group $G=GL(n,\mathbb{C})$, we present a remarkable relation between the $E$-polynomials of $X_F G$ and those of $X_F^{irr} G$, the locus of irreducible representations of $F$ into $G$. All concepts will be motivated with several examples, and we will give an overview of known explicit computations of $E$-polynomials, as well as some conjectures and open problems.

This is joint work with A. Nozad, J. Silva and A. Zamora.


Date and Venue

Start Date
Room 1.09


Carlos Florentino

Speaker's Institution

Universidade de Lisboa, CMAFcIO



Geometry and Topology