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!
[Math] Books on Riemann Surfaces
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It's simply not true that Forster's approach to the subject is "dead".
If you are interested in algebraic geometry, then you'll be mostly interested in compact Riemann surfaces (which are basically the same as smooth projective algebraic curves over $\mathbb{C}$). The main theorems about these that a first course should cover are Riemann-Roch and Abel's theorem. Here Forster's treatment (eg stating Riemann-Roch in terms of sheaf cohomology and deriving it from Serre duality) is the standard modern treatment. It might not be my first choice of textbook (the subject is blessed with many good books), but it certainly would prepare you for algebraic geometry better than a course that is focused in analytic topics.
I have not looked at the other book you mention, but I would guess that it focuses more on open Riemann surfaces. While the third part of Forster's book covers these, it would not surprise me if the analysis people consider his treatment dated.
EDIT : In reply to your edit, I'm not exactly an algebraic geometer, but I'm a heavy user of Grothendieck-style algebraic algebraic geometry. Certainly there are people using hard analytic tools to prove things in algebraic geometry (eg the Siu school), but my feeling is that most people in the subject do not use them. Given the choices you have, you would probably profit more from a course using Forster's book.
Narasimhan-Nievergelt's Complex Analysis in One Variable is exactly the book you want.
It is completely geometric and will introduce you, starting from scratch, not only to Riemann surfaces but also to the theory or holomorphic functions of several variables, covering spaces, cohomology,...
This unique book emphasizes how little you have to know of the classical function of one complex variable: just the forty pages of Chapter 1, aptly named Elementary Theory of Holomorphic Functions.
A book with a similar philosophy is Analyse Complexe by Dolbeault, he of the Dolbeault cohomology, which has the drawback of being in French (albeit in mathematical French, which is a far cry from Mallarmé or Proust French...)
It is an underappreciated fact, displayed in both these books, that most of the material found in books on complex analysis of one variable is useless for the study of Riemann surfaces and more generally complex manifolds.
For example all the clever computations of real integrals by residue calculus, evaluation of convergence radius of power series, asymptotic methods, Weierstraß products, Schwarz-Christoffel transformations, ... are irrelevant in complex analytic geometry: I challenge anyone to find the slightest trace of these in the work of the recently deceased H. Grauert, arguably the greatest 20th century specialist in the geometry of complex analytic spaces.
Best Answer
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.