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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?

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I personally like the notes by Eugene Xia: Abelian and Non-Abelian Cohomology 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 and Moduli of representations of the fundamental group of a smooth projective variety II

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 by Goldman & Xia. Many of the general features of the theory are already present in this case.

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I recently attented a nice online talk by Pengfei Huang and he indicated two sources:

I am not sure to which extent they can be called complete references, but as general references on the subject, I think they certainly deserve to be mentionned.

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