[Math] References for Riemann surfaces

complex-geometryreference-requestriemann-surfaces

I know this question has been asked before on MO and MSE (here, here, here, here) but the answers that were given were only partially helpful to me, and I suspect that I am not the only one.

I am about to teach a first course on Riemann surfaces, and I am trying to get a fairly comprehensive view of the main references, as a support for both myself and students.

I compiled a list, here goes in alphabetical order. Of course, it is necessarily subjective. For more detailed entries, I made a bibliography using the bibtex entries from MathSciNet: click here.

  1. Bobenko. Introduction to compact Riemann surfaces.
  2. Bost. Introduction to compact Riemann surfaces, Jacobians, and abelian varieties.
  3. de Saint-Gervais. Uniformisation des surfaces de Riemann: retour sur un théorème centenaire.
  4. Donaldson. Riemann surfaces.
  5. Farkas and Kra. Riemann surfaces.
  6. Forster. Lectures on Riemann surfaces.
  7. Griffiths. Introduction to algebraic curves.
  8. Gunning. Lectures on Riemann surfaces.
  9. Jost. Compact Riemann surfaces.
  10. Kirwan. Complex algebraic curves.
  11. McMullen. Complex analysis on Riemann surfaces.
  12. McMullen. Riemann surfaces, dynamics and geometry.
  13. Miranda. Algebraic curves and Riemann surfaces.
  14. Narasimhan. Compact Riemann surfaces.
  15. Narasimhan and Nievergelt. Complex analysis in one variable.
  16. Reyssat. Quelques aspects des surfaces de Riemann.
  17. Springer. Introduction to Riemann surfaces.
  18. Varolin. Riemann surfaces by way of complex analytic geometry.
  19. Weyl. The concept of a Riemann surface.

Having a good sense of what each of these books does, beyond a superficial first impression, is quite a colossal task (at least for me).

What I'm hoping is that if you know very well such or such reference in the list, you can give a short description of it: where it stands in the existing literature, what approach/viewpoint is adopted, what are its benefits and pitfalls. Of course, I am also happy to update the list with new references, especially if I missed some major ones.

As an example, for Forster's book (5.) I can just use the accepted answer there: According to Ted Shifrin:

It is extremely well-written, but definitely more analytic in flavor.
In particular, it includes pretty much all the analysis to prove
finite-dimensionality of sheaf cohomology on a compact Riemann
surface. It also deals quite a bit with non-compact Riemann surfaces,
but does include standard material on Abel's Theorem, the Abel-Jacobi
map, etc.

Best Answer

As it is evident from your bibliography list, there are two aspects of the theory: Riemann surfaces in the sense of 1-dimensional complex manifolds (which are not necessarily algebraic) and Complex Algebraic Curves (which are not necessarily smooth). It should be pointed out that some authors (old-school?) still use the term Riemann Surface to mean a Complex Algebraic Curve, regardless of whether it is smooth or not, thus also excluding the non-compact case.

I will now make a list of additional sources on Riemann Surfaces and Complex Algebraic Curves not present in your list and that focus exclusively on one or both of these two topics and then will edit my answer to add some information on each of them. There are many more references that include Riemann Surfaces and Complex Algebraic Curves as subsets of, for example, bigger text on Complex Geometry - for the moment I won't be mentioning them, but let me know if you are interested, they can be good sources too for some topic.

Legend: italicized references are present in OP's original list

  1. Arbarello, Cornalba, Griffiths, Harris - Geometry of Algebraic Curves Vol. I & II (1985,2011): As comprehensive as it is, this is not a first course on Complex Algebraic Curves, but rather reflects the state of the art at the time of writing. Notice the big difference between the years of the first and the second volume. The central topic of the first volume is Linear Series, while the second volume deals with all kinds of moduli spaces of curves. In the introduction of the first volume the authors write that the reader should have a working knowledge of algebraic geometry in the amount of the first chapter of Hartshorne's, but I don't think this actually suffices, perhaps they actually meant the second and third chapter of Hartshorne's. The second volume is above my paygrade to comment on :-)
  2. Bertola - Riemann Surfaces and Theta Functions (lecture notes): it has a completely analytic approach, focusing mostly on the compact case after introducing the initial generalities. It contains a nice discussion of the three kinds of abelian differentials by means of the theta divisor and introduces bidifferentials.
  3. Bobenko - Compact Riemann Surfaces: (obviously) it deals only with smooth complex algebraic curves, but it takes an analytic approach. It does not use sheaves. It contains a proof of Riemann-Roch (not all of them do). While it introduces all three kinds of abelian differentials, it does not discuss any of the reciprocity laws. It finishes with introducing line bundles.
  4. Brieskorn, Knörrer - Plane Algebraic Curves (1986):
  5. Cavalieri, Miles - Riemann Surfaces and Algebraic Curves, A First Course in Hurwitz Theory (2016): as the title suggests, it is an approach to Complex Algebraic Curves with strong focus on Hurwitz Theory. The basics of Riemann surfaces are layed out and then the author moves on to the counting. IMO the book is suitable for an undergraduate course since the prerequisites are low. However, singular complex algebraic curves are barely touched upon.
  6. Clemens - A Scrapbook of Complex Curve Theory (2ed.,2003)
  7. Dubrovin - Integrable Systems and Riemann Surfaces (lecture notes,2009): see the next reference. For Dubrovin Riemann Surfaces are complex algebraic curves. The notes are based on his book in Russian. However, what is not included in the next reference, is the connection with differential equations. The first (out of three) part of the notes is dedicated to the KdV equation, while the third part deals with Baker-Akhiezer Functions.
  8. Tamara Grava - Riemann Surfaces (lecture notes,2014): improved version based on Dubrovin's notes, but defines a Riemann Surface as a 1-dimensional complex-analytic manifold. It does not use sheaves. It deals almost only with compact Riemann Surfaces via an analytic approach, but also gives a discussion of resolution of singularities for complex algebraic curves. It includes a proof of Riemann-Roch. It does not mention line bundles at all. The chapter on divisors should be read with extra care as there might one or two hasty statements :-)
  9. Eynard - Lectures on Compact Riemann Surfaces (2018)
  10. Farkas, Kra - Riemann Surfaces (1980): it includes both the non-compact and the compact case and the treatment is analytic. It uses no sheaves (though IIRC the sheaf of holomorphic functions is given a definition somewhere). I am not fond of their proof of the reciprocity between abelian differentials of the third kind: IMO it is more elegant to introduce only one cut in the fundamental polygon, namely between the points $P$ and $Q$ instead of 4 cuts from a fixed origin point on the boundary of the fundamental polygon $O$ to $P$ and $O$ to $Q$ and then backwards (I could include a proof sketch for the $PQ$ cut if anyone is interested). Moreover, there is a proper (sub)chapter on intersection theory on Riemann Surfaces.
  11. Fulton - Algebraic Curves, Introduction to Algebraic Geometry (2008): it is a standard algebraic geometry introduction to algebraic curves over an algebraically closed field in a classical way, i.e. without sheaves and schemes.
  12. Gibson - Elementary Geometry of Algebraic Curves (1998): it is similar in spirit to Fulton's book, but it is probably (even) more visual and example-oriented.
  13. Gunning - Riemann Surfaces and 2nd Order Theta Functions
  14. Gunning - Some Topics in the Function Theory of Compact Riemann Surfaces (draft ver July 2015): definitely not recommended as a first read. It discusses standard topics of Riemann Surfaces like Holomorphic and Meromorphic Differentials etc. from a more advanced POV, definitely sheaf-theoretic. IMO, proofs can be sometimes a little terse to follow, but after all it is only a draft and not meant as an introductiory course for undergraduates.
  15. Griffiths - Introduction to Algebraic Curves (revised,1985): analytic approach without sheaf theory and sheaf cohomology. However, it is the only book on Riemann Surfaces (in a broad sense) I know of that discusses normalization in detail!
  16. Harris - Geometry of Algebraic Curves (lecture notes from Harvard,2015)
  17. Kirwan - Complex Algebraic Curves (1992)
  18. Kunz - Introduction to Plane Algebraic Curves (2005)
  19. Lang - Introduction to Algebraic and Abelian Functions (2ed.,1982)
  20. Miranda - Algebraic Curves and Riemann Surfaces (1995)
  21. Mumford - Curves and Their Jacobians (1999)
  22. Narasimhan - Compact Riemann Surfaces (1992,reprint,1996)
  23. Perutz - Riemann Surfaces (lecture notes,2016)
  24. Springer - Introduction to Riemann Surfaces (1957)
  25. Teleman - Riemann Surfaces (lecture notes,2003): though short (69 pages only), personally I found it to be very illuminating on many points and contains several nice, albeit hand-drawn, pictures!
  26. Varolin - Riemann Surfaces By Way of Analytic Geometry: it completely avoids sheaves despite being very detailed.
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