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A lot of the books I've found assert that there is a threefold categorical equivalence between (1) compact Riemann surfaces, (2) smooth projective algebraic curves, and (3) function fields of transcendence degree one. None of them give many details though. I know how to embed compact Riemann surfaces in projective space and then use Chow's theorem to establish an analytic isomorphism between (1) and (2). Are there any good references which give a detailed argument for categorical equivalence though?

More specifically, I guess I want answers/references for the following:

  1. What are the arrows associated with each of these categories? (everybody is kind of vague about this)
  2. What are the functors which establish the equivalence? For Riemann surfaces and algebraic curves, is it just the isomorphism constructed from the embedding and Chow's theorem?
  3. How do I show that these functors are full and faithful?

I would also appreciate it if someone could direct me to a good treatment of the relation between function fields and the other two. Miranda, Kirwan, etc. don't cover this point too well.

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    $\begingroup$ The morphisms are 1) nonconstant holomorphic maps, 2) nonconstant algebraic maps, and 3) maps of extensions of $\mathbb{C}$. The easy functors are taking the function field (2 to 3 and also 1 to 3) and taking the analytification (2 to 1). I don't know how the proofs of equivalence go though. $\endgroup$ – Qiaochu Yuan Aug 17 '15 at 18:12
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The equivalence between the category of compact Riemann surfaces with nonconstant holomorphic maps and the category of function fields over the complex numbers of transcendence degree one with morphisms of complex algebras can be found in Chapter 1.3 of 'Introduction to Compact Riemann Surfaces and Dessins d’Enfants' by Ernesto Girondo and Gabino González-Diez.

The equivalence between the latter category of function fields and nonsingular projective algebraic curves with dominant morphisms over the complex numbers is more or less in the first chapter of Robin Hartshorne's book on algebraic geometry.

If you have any further questions, feel free to ask.

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