I am considering reading one of 'Algebraic curves and Riemann Surfaces' by Rick Miranda or 'Lectures on Riemann Surfaces' by Otto Forster. Which one of these is more advanced and comprehensive ? What are the differences in the approaches of these two books ?
[Math] Good book for Riemann Surfaces
reference-requestriemann-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.
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:
-Riemann Surfaces - S.Donaldson,
-Riemann Surfaces - Farkas and Kra,
-Algebraic curves and Riemann surfaces - R.Miranda
-Lectures on Riemann Surfaces - Otto Forster
Donaldson's book is more difficult with respect to others t, and he use a lot of basic algebraic geometry. I have read Forster's book and have been pretty impressed by it. Another excellent analytic monograph from this point of view is the Princeton lecture notes on Riemann surfaces by Robert Gunning, which is also a good place to learn sheaf theory. His main result is that all compact complex one manifolds occur as the Riemann surface of an algebraic curve. Miranda's book contains more study of the geometry of algebraic curves.
Riemann himself, as I recall, took an intermediate view, showing the equivalence of the categories of (irreducible) algebraic curves with that of (connected) compact complex manifolds equipped with a finite holomorphic map to P^1. Another extremely nice book, a little more advanced than Miranda, is the China notes on algebraic curves by Phillip Griffiths. Mumford's book Complex projective varieties I, also has a terrific chapter on curves from the complex analytic point of view.
After you learn the basics, the book of Arbarello, Cornalba, Griffiths, Harris, is just amazing. Of course Riemann's thesis and followup paper on theory of abelian functions is rather incredible as well.
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Ted Shifrin notes above: