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I am starting a scholarship on geometry and the subject of research is going to be Riemann surfaces (we will focus on compact Riemann surfaces). I am finishing my undergraduate studies so my knowledge of mathematics is pretty humble: I have attended Geometry of Curves and Surfaces, Algebraic Topology and Complex Analysis undergraduate modules. Which books/notes would you recommend for me to start this topic? Thank you very much!

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I'm very fond of Forster's book Lectures on Riemann Surfaces. Check it out. There is also a lovely book by Phillip Griffiths called Introduction to Algebraic Curves.

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Riemann surfaces is a very standard topics in math, then you can find a lot of books talking about Riemann surfaces under different point of views.

I can suggest you:

-Compact Riemann surfaces - J.Jost,

-Riemann Surfaces - S.Donaldson,

-Riemann Surfaces - Farkas and Kra,

-Algebraic curves and Riemann surfaces - R.Miranda

The first is pretty analytical, he talks about sobolev spaces of function on Riemann surfaces besides geometric structures on Riemann surfaces. The second is more difficult respect the first, and he use a lot of basic algebraic geometry. I never read the others but I know that they are excellent books.

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