Algebraic geometry/topology - ramification - sheaf cohomology

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I'm looking for some resources, introductory material, books, websites, etc into the general results of riemann surface theory - I was told to investigate algebraic geometry (and possibly algebraic topology) and maybe also sheaf cohomology...

(I've been reading Knopp's Theory of Functions - but I was hoping for something more advanced, and modern also)

I was hoping for something that wouldn't assume too much prereq knowledge - approximately something along the lines of graduate-level complex analysis and algebra, and undergraduate level topology - but some extra self study is of course expected. :)

Any pointers? :) Thanks.
 
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Alan Beardon is an excellent mathematician and teacher of mathematics. Here is his book

http://www.amazon.ca/Riemann-Surfaces-Alan-Beardon/dp/0521659620

but you may also be able to find notes on his personal website at Cambridge that help.

Whilst algebraic geometry is a wonderful subject, it is not a prerequisite for understanding anything about Riemann surfaces. You really need a little point set topology (I assume you know what a compact space is), and complex analysis - you're essentially trying to put a topology on the possible analytic continuations of Laurent series.

Of course, once you understand them, you may want to look at more complicated things like holomorphic forms and such which is more algebro-geometric. I don't know a good reference for this: I've never found one. Griffiths and Harris is comprehensive but too long, for example.
 
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Yeah, Beardon's good. If you want to get onto differential forms, sheaf cohomology etc. then your best bet is Lectures on Riemann Surfaces by Otto Forster (it got me through my dissertation). You do need the motivation to read it (or maybe that's because I pretty much slept with it in my bed for six months?) and will probably only need the first/second of the three sections.

Algebraic geometry is pretty huge and if you nail sheaves (in Forster) then you can read some pretty high level books. If you start anywhere you should start with complex algebraic curves (really just Riemann surfaces from a different viewpoint). Complex Algebraic Curves by Frances Kirwan is sort of algebraic-geometry-lite, but has some nice pictures and is easy to digest. The real deal is William Fulton's Algebraic Curves: An Introduction to Algebraic Geometry (compare how he and Kirwan treat Bezout's theorem for example).

For general algebraic geometry see Basic Algebraic Geometry 1 by I.R. Shafarevich.
 

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