de-Rham moduli space over a compact Riemann surface

Question

Let $X$ be a smooth projective curve over $\mathbb C$ and $M_{dR}$ denote the moduli space of stable $\Lambda$-connections of fixed rank and degree $0$. Is it known whether $M_{dR}$ is a smooth variety? If yes, then is there a simple argument like in the case of moduli of stable Higgs bundles?

Answer 1

For $\lambda \neq 0$, $\lambda$-connections are just $\lambda$ times a holomorphic connection. On a Riemann surface, every holomorphic connection is flat and thus corresponds to a unique local system, whose monodromy gives a linear representation of the fundamental group $\pi_1(X)$ modulo conjugation.

Therefore, the moduli space of semistable pairs $(E,\nabla)$ where $E$ is a holomorphic vector bundle of rank $r$ and $\nabla$ is a $\lambda$-connection on $E$ is isomorphic to the rank $r$ character variety of $\pi_1(X)$, i.e. the GIT quotient of the affine variety $\mathrm{Hom}(\pi_1(X), \mathrm{GL}(r,\mathbb C))$ under the conjugation action of $\mathrm{PGL}(r,\mathbb C)$, and the moduli space of stable pairs is the Zariski open subset consisting of conjugacy classes of irreducible representations.

There are several ways to prove that irreducible representations are smooth points of the character variety. One way would to first show that they are smooth points of $\mathrm{Hom}(\pi_1(X), \mathrm{GL}(r,\mathbb C))$ by an explicit computation, and then remark that $\mathrm{PGL(r,\mathbb C)}$ acts freely on the set of irreducible representations by Schur's lemma. Another way would be to compute the dimension of the tangent space to the character variety at $[\rho]$, which is isomorphic to the twisted cohomology group $H^1(\pi_1(X),\rho)$. When replacing $\mathrm{GL}(r,\mathbb C)$ by some other complex reductive group $G$, one has to be careful that the stable part of the character variety may have orbifold points, corresponding to irreducible representations whose centralizer in $G/Z(G)$ is finite but not trivial. More details can be found for instance in Labourie's Lectures on representations of surface groups: https://math.unice.fr/~labourie/preprints/pdf/surfaces.pdf

Note that the non-Abelian Hodge correspondance of Hitchin, Corlette, Simpson... gives a real analytic isomorphism between the moduli space of stable Higgs bundles of rank $r$ and degree $0$ and the moduli space of rank $r$ $\lambda$-connections, so whatever simple argument giving you the smoothness of the former gives you the smoothness of the latter. The structure of complex algebraic varieties do not agree though.