The algebraic de rham cohomology of representation varieties

The SL(2, C)-representation varieties of punctured surfaces form natural families parameterized by monodromies at the punctures. In this paper, we 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 Gauβ-Manin connection on the natural family of the smooth SL(2, C)-representation varieties of the one-holed torus.