What are the best books to study riemann surfaces from

Question

I have taken a course in algebra , analysis.Which book should I start with at the very beginning(which would be easy enough to tackle) and what are the other prerequisites I need to know?

Answer 1

I recommend Rick Miranda's Algebraic Curves and Riemann Surfaces. Miranda starts from the basic definitions and includes a lot of examples which help the reader gain intuition about the definitions and theorems. The book has an algebraic bent in that Miranda is gradually building the reader toward basic algebraic geometry, as the last few chapters on sheaves indicate. He takes for granted a first course in complex analysis (other than comfort with basic abstract algebra, real analysis, and topology). However, I think you can get much out of the book without knowing too much complex analysis: for instance, at first you should know the basic properties of holomorphic/meromorphic functions; then as the book progresses you should know the basic theorems such as the open mapping theorem, max modulus, identity theorem, etc; coming to integration on Riemann surfaces, you should be aware of countour integration in $\mathbb{C}$ and the main results of this topic such as Cauchy's residue theorem. I am trying to indicate that you can very well study the required complex analysis side-by-side while studying Miranda; if you want to learn the material well, you should put the time into understanding the proofs of the basic theorems from complex analysis, because their proofs work with a little modification for their Riemann surfaces versions too. It goes without saying that you should do some exercises from each section.