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 Gauß-Manin connection on the natural family of the smooth SL(2, ℂ)-representation varieties of the one-holed torus.

Algèbres simplicialesS1-équivariantes, théorie de de Rham et théorèmes HKR multiplicatifs

This work establishes a comparison between functions on derived loop spaces (Toën and Vezzosi,Chern character, loop spaces and derived algebraic geometry, inAlgebraic topology: the Abel symposium 2007, Abel Symposia, vol. 4, eds N. Baas, E. M. Friedlander, B. Jahren and P. A. Østvær (Springer, 2009), ISBN:978-3-642-01199-3) and de Rham theory. IfAis a smooth commutativek-algebra andkhas characteristic 0, we show that two objects,S1⊗Aand ϵ(A), determine one another, functorially inA. The objectS1⊗Ais theS1-equivariant simplicialk-algebra obtained by tensoringAby the simplicial groupS1:=Bℤ, while the object ϵ(A) is the de Rham algebra ofA, endowed with the de Rham differential, and viewed as aϵ-dg-algebra(see the main text). We define an equivalence φ between the homotopy theory of simplicial commutativeS1-equivariantk-algebras and the homotopy theory of ϵ-dg-algebras, and we show the existence of a functorial equivalence ϕ(S1⊗A)∼ϵ(A) . We deduce from this the comparison mentioned above, identifying theS1-equivariant functions on the derived loop spaceLXof a smoothk-schemeXwith the algebraic de Rham cohomology of X/k. As corollaries, we obtainfunctorialandmultiplicativeversions of decomposition theorems for Hochschild homology (in the spirit of Hochschild–Kostant–Rosenberg) for arbitrary semi-separatedk-schemes. By construction, these decompositions aremoreovercompatible with theS1-action on the Hochschild complex, on one hand, and with the de Rham differential, on the other hand.