Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. References for Riemann surfaces Ask Question. Asked 11 months ago. Active 11 months ago. Viewed 1k times. Introduction to compact Riemann surfaces. Introduction to compact Riemann surfaces, Jacobians, and abelian varieties. Riemann surfaces. Farkas and Kra. Lectures on Riemann surfaces.
The Concept of a Riemann Surface | study by Weyl | gyqacyxaja.cf
Introduction to algebraic curves. Compact Riemann surfaces. Complex algebraic curves. Complex analysis on Riemann surfaces.
Riemann surfaces, dynamics and geometry. Algebraic curves and Riemann surfaces. Narasimhan and Nievergelt. Complex analysis in one variable. Quelques aspects des surfaces de Riemann. Introduction to Riemann surfaces. Riemann surfaces by way of complex analytic geometry. The concept of a Riemann surface. I can just use the accepted answer there : According to Ted Shifrin : It is extremely well-written, but definitely more analytic in flavor.
Are you interested only in books or also in lecture notes that are freely available online on authors' web pages? There are some very good ones. Oct 19 '18 at G: I was aiming for comments on the 'main' classical references, which I tried to all include in the list but let me know if I missed any major ones. I think that online lecture notes are fine as long as they are an important reference, eg McMullen's notes in my list. 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 :- 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. 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.
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It finishes with introducing line bundles. 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. 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.
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The first out of three part of the notes is dedicated to the KdV equation, while the third part deals with Baker-Akhiezer Functions. Tamara Grava - Riemann Surfaces lecture notes, : improved version based on Dubrovin's notes, but defines a Riemann Surface as a 1-dimensional complex-analytic manifold. 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 :- Eynard - Lectures on Compact Riemann Surfaces Farkas, Kra - Riemann Surfaces : 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.
Moreover, there is a proper sub chapter on intersection theory on Riemann Surfaces. Catherine's College. Oxford Academic. Google Scholar. Cite Citation. Permissions Icon Permissions. Article PDF first page preview.
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