Let $X$ be a compact Riemann surface. I would like to find a somehow complete reference for the proof of the so called non-Abelian Hodge correspondence relating Dolbeaut, Betti and Higgs bundle moduli spaces.


I've tried to read the original articles by Hitchin (1987) or Simpson (1990) but it seems to me that I've not found somehow a complete reference showing a precise proof of all the statements involved (for the case of a curve).


Does anyone know a book or notes related to this?



Non-Abelian Hodge theory
I personally like the notes by Eugene Xia: [Abelian and Non-Abelian Cohomology](https://arxiv.org/pdf/1404.5025.pdf) to build intuition.


But for a definitive source, I would read Simpson: [Moduli of representations of the fundamental group of a smooth projective variety I](http://www.numdam.org/article/PMIHES_1994__79__47_0.pdf) and [Moduli of representations of the fundamental group of a smooth projective variety II](http://www.numdam.org/article/PMIHES_1994__80__5_0.pdf)


True, the above results of Simpson assume the Lie group is $\mathrm{GL}(n,\mathbb{C})$ and the surface is closed for genus at least 2, but understanding those cases of the full non-Abelian Hodge theorem goes a long way. After that, you can navigate the current literature to find that the correspondence holds in the parabolic setting for arbitrary reductive Lie groups $G$ (look at writings by Oscar Garcia Prada or Peter Gothen for example).


I hope that helps!


P.S. For a well-written detailed exposition of the correspondence in the simplest case see:
*[Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces](https://arxiv.org/pdf/math/0402429.pdf)* by Goldman & Xia. Many of the general features of the theory are already present in this case.


