The algebraic de Rham cohomology of representation varieties

TL;DR

This paper investigates the algebraic de Rham cohomology of SL(2,C)-representation varieties of punctured surfaces, identifying singular loci and computing cohomologies for specific cases, and explicitly describes the Gauss-Manin connection.

Contribution

It provides the first detailed analysis of the singularities and cohomologies of these representation varieties, including explicit connections, advancing understanding of their geometric structure.

Findings

Identified singular loci for one-holed and two-holed tori, and four-holed sphere.

Computed de Rham cohomologies for smooth cases of one-holed torus and four-holed sphere.

Explicitly described the Gauss-Manin connection for the one-holed torus family.

Abstract

The SL(2,C)-representation varieties of punctured surfaces form natural families parameterized by holonomies at the punctures. In this paper, we first compute the loci where these varieties are singular for the cases of one-holed and two-holed tori and the four-holed sphere. We then compute the de Rham cohomologies of these varieties of the one-holed torus and the four-holed sphere when the varieties are smooth via the Grothendieck theorem. Furthermore, we produce the explicit Gauss-Manin connection on the natural family of the smooth SL(2,C)-representation variety of the one-holed torus.