So, I have come to a somewhat impasse concerning my class selection for next term, and I have exhausted all the 'biased' sources. So, I was wondering if anyone in this fantastic mathematical community has any input on the matter.
Next term I would like to take a course on Riemann surfaces. Initially I had intended to do an independent study with a fantastic teacher and use Otto Forsters book Lectures on Riemann Surfaces. But, recently I have come to learn that there is going to be a graduate course on Riemann surfaces taught as well. This class is taught by an expert in moduli spaces and will use Riemann Surfaces by Way of Complex Analytic Geometry by Dror Varolin (freely available on the author's website: here).
These books differ greatly in style–in the way that they approach the subject. So, of course it's important to decide which course I am going to take, and so it made sense to ask around about these different styles in the books.
Here is what I have gathered:
Forster's book is much more classical. It does things fairly sheaf-theoretically and involves quite a bit of algebra. Moreover, it seems to focus much more on topological algebraic considerations than anything else.
Varolin's book is much, much more analytic and PDEish. Most of the proofs seem to be calculations of sorts.
So, the issue is this. I am, at least historically, of a very algebraic persuasion. I eventually think I want to do something in algebraic geometry or algebraic number theory. This automatically makes me want to go more for Forster since I have a fair amount of experience with sheaves and cohomology. Moreover, I also have (only half-seriously) a dislike of very computational analysis [even though I know it's useful].
That said, I have been told by multiple people that Forster's approach to the subject is "dead"–no one does things like that any more. They tell me that Varolin's approach is much more focused on modern techniques, that it's closer to "the source".
So, the two things I was hoping someone could clear up for me is
1) Is it true that the analysis/PDE approach is much closer to what is actually important to learn about Riemann surfaces? Is that where the powerful theorems and techniques lie?
2) I am chiefly interested in Riemann surfaces so that I have a good geometric background for algebraic geometry. I do not want to be one of those people that can understand all of the algebra yet is clueless as to what is geometrically going on. Is Forster or Varolin's approach better suited to this goal?
Of course, any input about anything even slightly related to this that I did not ask, but you think would be helpful to know will be greatly appreciated.
Thanks again everyone!
NB: If you're answer is going to be "you should do both" (which I would imagine is both likely and correct) please, instead, indicate which you think would be preferable to do first. I know I will do both in tandem regardless, but I shall end up (inevitably) focusing on one approach over the other.
EDIT: I would like to make clear that one of the main reasons for this question is to basically figure out if people doing work/studying intensely in algebraic complex geometry or algebraic number theory (the more algebraic geometry side–like arithmetic geometry) feel that there is a reason for me to do the analysis part. Will it provide a prospective on things that will be elucidating, or helpful.
EDIT(2): I have decided to also post this on mathoverflow. While I know there is a considerable intersection between the participants here and there, I feel as though this question may be better suited for that website.
Best Answer
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.