# Lecture: Character varieties -- a symplectic perspective

RTG Lecture – Spring 2022

University of Heidelberg & KIT Karlsruhe

# Abstract

The goal of this minicourse is to introduce the notion of character variety. A character variety is, broadly speaking, a construction that associates a symplectic manifold to a surface and a Lie group. It comes with a natural action of the mapping class group of the surface that preserves its symplectic structure.

The minicourse will introduce all relevant notions and will, in particular, not assume any particular knowledge on symplectic geometry or mapping class groups. We will provide reminders on relevant notions of Lie theory, algebraic/analytic varieties and group cohomology.

# Lecture notes

Here is an extended version of the material that I will present in the lecture. These notes are a first draft and are meant to be updated in the future.

*Any comment or correction is always welcome!*

# Schedule

Lecture 1 (26.04.22): Short recap on Lie groups (introducing quadrable Lie groups) and surface groups. Definition of representation variety and its structure. Computation of Zariski tangent spaces.

Lecture 2 (10.05.22): Brief recap on group cohomology and symplectic geometry. Definition of the Goldman symplectic form.

Lecture 3 (21.06.22): Introduce mapping class groups and their action on character varieties.