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I just finished the NPTEL YouTube course on Riemann Surfaces, and I am looking for references where I might find complete proofs of the following four facts. I would like to emphasize the word complete, since many of the canonical sources, and all of the results of my initial Google searches, including those pointing to questions on this site, were brief or lacking in detail sufficient for my purposes.

First, I am looking for a proof that every $g$-holed torus, $g \geq 2$, has a Riemann Surface structure. I would prefer a proof that involves showing that these tori can be holomorphically covered by the upper half plane, since in the NPTEL course, it was shown that one can "push down" complex structures to quotient spaces via (holomorphic) universal covering maps. Any other sufficiently thorough proof is perfectly fine, however.

Second, I am looking for a complete proof that every dimension-$1$ projective variety is a compact Riemann Surface. Also, is some notion of nonsingularity required for the variety for this to be true? I have seen the statement both with and without that qualifier.

Third, I am looking for a complete proof that every compact Riemann Surface is a projective variety of dimension $1$ (again, do we need some kind of nonsingularity condition here?). I realize this is the most nontrivial request thus far, and I would respectfully ask that if the proof in question relies on highly nontrivial theorems (Chow, Kodaira), that references be given for complete proofs of those as well.

Finally, we showed that the moduli space of holomorphic isomorphism classes on complex genus-$1$ tori is isomorphic as a Riemann Surface to $\mathbb C$. Is there a source that might cover this same type of result in detail for tori with genus $2$ or greater?

KReiser
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Johnny Apple
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  • I think these facts are proved somewhere in Griffiths and Harris. For your last question, here's a start. – Viktor Vaughn Apr 02 '20 at 04:24
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    This is probably a bit too broad. Question 1 is the uniformization theorem, question 2 is basic algebraic geometry, question 3 is Chow's theorem, and question 4 is Teichmuller theory. Already you're looking at three different directions which would likely be covered in at least three different courses at the graduate level. – KReiser Apr 02 '20 at 06:36
  • The isomorphism classes of finite field extensions $F/\Bbb{C}(z)$ correspond to the isomorphism classes of compact Riemann surfaces. Concretely $F=Frac(\Bbb{C}[z,y]/(h(z,y))$ then the non-singular points of $h(z,y)=0$ form a Riemann surface, each point corresponds to a discrete valuation on $F$. The only points we are missing are the finitely many singular points and the finitely many points above $z=\infty$, they are found from $F=Frac(\Bbb{C}[1/z,u]/(g(1/z,u))$, gluing $g$ and $h$ together through the point-DVR correspondence. – reuns Apr 02 '20 at 06:41
  • I'd like to emphasize I'm merely looking for solid references with complete proofs. I'm certainly not asking anyone to replicate a full proof, or even a portion of one. A list of X books guaranteed to contain each of these proofs in its fullness would be an ideal answer. – Johnny Apple Apr 02 '20 at 07:03
  • @KReiser I'd also like to clarify for question 1: I already have seen that quotients of sphere, plane, and upper half plane are Riemann Surfaces. What was never shown to me, and what I can't seem to find in any book, is exactly why a $g$-torus IS a such a quotient. I've never seen an explicit Riemann Surfaces structure given on them, nor a holomorphic covering map from the upper half plane be laid out. – Johnny Apple Apr 02 '20 at 07:15
  • I do understand that it's a reference request - I think it's too broad even accounting for that. Anyways, the $g$-torus issue is straightforward: take a topological universal cover, use the natural covering map to pull back the complex structure to the universal cover, and now you have a simply connected complex manifold, which by the uniformization theorem is conformally equivalent to the sphere, the plane, or the hyperbolic plane. You can now decide which one it is by the fact that the covering map is a local isometry by construction and preserves curvature. – KReiser Apr 02 '20 at 07:25
  • @KReiser How do we know infer the complex structure on the $g$ torus from that? How do we know such an object is a Riemann Surface to begin with? It's intuitively clear, of course, but how do we sit down and write out the complex structure, either directly, or from the use of holomorphic covers? I have never even seen it verified that the $g$ torus is holomorphically covered by the upper half plane. Whereas I have seen explicit proofs that a regular old torus is covered by the complex plane, and that there is a corresponding holomorphic covering map. – Johnny Apple Apr 02 '20 at 07:46
  • @KReiser In other words, I know that if a $g$ torus was a Riemann Surface, it would be covered by the upper half plane. Then we could produce a covering map and push down the structure to the torus. But that's all already assuming it's a Riemann Surface, and to show that, you either need to explicitly write out the charts, or realize it as a holomorphic quotient of the upper half plane explicitly, or perhaps use some kind of gluing theorem. – Johnny Apple Apr 02 '20 at 07:50
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    Ah, I had your request the wrong way round. Here is one option: smoothly embed your surface of genus $g$ in to $\Bbb R^3$, pick a consistent choice of normal, then define an almost-complex structure on your surface by "rotation by $\pi/2$ from the perspective of the normal" (the right-hand rule). As all almost-complex structures are complex structures in real dimension 2, this gives a complex structure. – KReiser Apr 02 '20 at 08:07
  • For 1, even more is true: Every orientable surface admits a complex structure: https://math.stackexchange.com/questions/2510127/when-is-a-riemannian-manifold-a-riemann-surface?rq=1. Question 2: If and only if it is a nonsingular complex projective curve. (This also works for smooth complex quasiprojective curves.) There is not much to be proven here, it is a consequence of the holomorphic implicit function theorem. 3 will be found in any Riemann surfaces textbook. For 4, see https://mathoverflow.net/questions/10514/teichmuller-theory-introduction – Moishe Kohan Apr 03 '20 at 03:14
  • @MoisheKohan Thank you for the reply. For #1, I followed your link, only to notice the accepted answer was incomplete (it failed to use orientability). I also have seen explicit Riemann Surface structures on every other Riemann Surface except the $g$-torus, so I was actually hoping for something more specific. #2 may be easy, but I would still like a solid reference. I think Miranda proves this, but I want to make sure the statement is the same. #3 is not in any book I have. It's mentioned, but not shown. #4's Hubbard reference looks promising (and it contains a Uniformization Theorem proof!). – Johnny Apple Apr 03 '20 at 04:54

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