# Perspectives on Riemann Surfaces

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.

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I was thinking about taking a look at Forster with similar motivation. I look forward to hearing what experiences people have had with it. –  Mike B Apr 22 '12 at 2:47
About these "multiple people" telling you Forster's text has an outdated approach: what area of math are they in?? –  KCd Apr 22 '12 at 3:17
@KCd The guy that actually told me does Spectral geometry. That said, I am pretty sure he told me a complex geometer told him this (one that is going to be coming to my school next year and teaching a follow-up course). I don't want to misquote this man though, he is very good, I may have just misunderstood him. –  Alex Youcis Apr 22 '12 at 3:22
@KCd I think he may also work heavily with moduli spaces. –  Alex Youcis Apr 22 '12 at 3:26

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.

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Thank you very much for your response! I edited my question because I'm not sure I made this clear. Do you think that tool-kit aside, learning the analysisy approach will make the geoemtry of the situation more clear when I start to do the sheafy/algebraic geometry approach? Thanks again! –  Alex Youcis Apr 22 '12 at 3:04
@AlexYoucis : I edited my post to say something about this (while you were making your comment). The analytic tools certainly say something about the geometry and it's great to know them, but given that there is an infinite amount of stuff to learn and a finite amount of time to learn it in, you're better off focusing on other things. –  Adam Smith Apr 22 '12 at 3:10
Once again, thank you for taking the time to answer! And, once again I feel like I must ask for some more clarifications. I agree that the analysis is probably not going to be extremely helpful in Grothendieck-style algebraic geometry (which is what I want to do) but I was more interested if the approach taken by Varolin will give me better intuition about, say, line bundles than Forster. –  Alex Youcis Apr 22 '12 at 3:24
@AlexYoucis : I doubt it. Certainly my intuition for eg line bundles on Riemann surfaces comes from algebraic geometry and not analysis; however, in some sense what works for you is kind of personal. –  Adam Smith Apr 22 '12 at 4:01

Here is my 2-cents answer, I'm not an algebraic geometry expert, nor a Riemann surfaces connaisseur, but rather an end-user (singularities, algorithms), nonetheless I often regret not to know more about complex geometry, so here are some reasons why I think that analytic methods in algebraic geometry should not be underestimated, without even talking about strategy for your future.

Complex geometry and algebraic geometry are sisters — or brothers, I don't know, in French geometry is feminine —, and complex geometry is the older one. I think you cannot pretend being an algebraic geometer without knowing a bit of complex geometry :

• algebraic geometry constantly imports complex methods, and sometimes you cannot really understand them without understanding the complex part ;
• a big and interesting part of the algebraic geometry literature, old and modern, uses the complex language : Dolbeault cohomology, homology, complex residues, etc ;
• even Grothendieck uses complex geometry ;)

In the end you will know both sides, so I think the real question is in what order will you learn them ? Geometry is hard, you should choose the course you think you will understand better. Modern approach to geometry (analytic or algebraic) all have a serious pedagogical drawback : they kind of hide the geometry, and you can't always rely on the teacher to make it apparent, there is a big gap between the definition understanding, and the deep understanding. Choosing the right order can ease to fill the gap.

“students who have sat through courses on differential and algebraic geometry (read by respected mathematicians) turned out to be acquainted neither with the Riemann surface of an elliptic curve $y^2 = x^3 + ax + b$ nor, in fact, with the topological classification of surfaces (not even mentioning elliptic integrals of first kind and the group property of an elliptic curve, that is, the Euler-Abel addition theorem). They were only taught Hodge structures and Jacobi varieties!” — V. I. Arnold

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This is exactly what I'm afraid of! I LOVE abstract nonsense, I really do. Pure algebra makes me happy. That said, I do NOT want to be someone like Arnold describes. I really want to understand what I am doing, and I want to take the subjects (in the correct order) that best help me do this. Thanks for your comment! I hope I get more contradictory answers like this :) –  Alex Youcis Apr 22 '12 at 4:07
@AlexYoucis : You can be just as ignorant of geometry if you are educated in the complex analytic school. The real key I think is learning large numbers of examples. –  Adam Smith Apr 22 '12 at 4:11

I would recommend you to read Curtis Mcmullen's notes and course outline. I found them to be very useful even after I finished my Riemann Surfaces class in sophomore year(during which I used Miranda's book very often). For Riemann Surfaces with analysis, dynamics, etc you can of course again consult his lecture or Donaldson's book. I would also recommend the book "Lecutres on Riemann Surfaces" by Gunning, in case you are willing to read old books and have the patience to understand it carefully. The classical reference for researchers should be this.

Generally speaking even randomly picking a book and read is not so bad. I do not understand why you necessarily have to worry about one book is better or worse than the other because of "algebraic" "analytic" - you will find a synthesis! Every time you learn something by reading other's treatment of a subject you do not understand well enough. Even after I finished the Mcmullen's notes, I still find reading Lang's book to be very helpful. Also, Riemann Surfaces is a huge subject, with on going research in multiple fronts in different levels. Hoping to achieve a deep understanding of it in one semester might be difficult - because you have to do a lot of problems and usually they are not easy...If Lang resurrected and ask me a random question I am sure I will not be able to answer it...

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"Generally speaking even randomly picking a book and read is not so bad." I cannot disagree more with this sentiment. The right book has often made all the difference in the world for my understanding. That said, the subject of Riemann surfaces is blessed with a wide variety of excellent texts. In my humble opinion, not all subjects are as yet this way. –  Jesse Madnick Feb 9 '13 at 4:19
@JesseMadnick: picking up a random math book to read is the best way to know new mathematics. If one always follow along so call "good books" recommended by others, then one inevitably ends living in someone else's mental world. And this is the worst situation for a mathematican. Randomness is the source of diversity and innovation. While it has its down side, it is often not that difficult to tell if a book is badly written, therefore I believe the possibility of picking up the "wrong book" for an actual reader is quite small. –  Kerry Feb 13 '13 at 0:08
@JesseMadnick:So here is the recommendation: spend one day in the O-P section of the library read some books you might find interesting! –  Kerry Feb 13 '13 at 0:13
@JesseMadnick: I think this is a common misunderstanding due to impatience.A lot of good books are not that easy to read and are not "crystal clear" to the wrong audience.But they offer in depth discussion on topics unavailable or discussed superficially in other places. While we can all blame the authors for their poor writing style does not fit our learning style,sometimes it is worth to think about the bigger picture behind a seemingly poorly written book and judge what the author is thinking about. Luckily in math we can all prove things ourselves and tell good and bad is not so difficult. –  Kerry Feb 16 '13 at 1:43
@ChangweiZhou: I'm not "blaming" any of the authors, or saying that anyone has "poor writing style." I'm just saying that many books out there don't fit everyone's learning style, and some areas are deficient in a variety of books suitable for beginners, who are unable to prove things themselves. Finally -- and perhaps I'm just being impatient here, but -- I find your tone to be very patronizing. –  Jesse Madnick Feb 16 '13 at 2:02

I would recommend to go with varolin's approach. You may keep forster along side but varolin i think gives the best introduction to fascinating program started by siu,demailly. As far as complex numbers are concerned, I think hard analytic methods rule!

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