I've never fully understood the connection between Riemann surfaces and algebraic varieties. I'm particularly interested in the case of the modular curve of level N–I know how the Riemann surface is constructed by taking a quotient of the upper half-plane by the action of a congruence subgroup of the modular group, but not how the resulting manifold translates into a curve. From what I've read, it appears that the associated curve is defined by equations satisfied by functions defined on the manifold, but I don't understand which functions are involved in these equations. What exactly is the relation between the two types of objects?
Algebraic Geometry – Why are Riemann Surfaces Algebraic Curves?
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The Kodaira embedding theorem may be of interest. "It says (which) precisely complex manifolds are defined by homogeneous polynomials" c.f. http://en.wikipedia.org/wiki/Kodaira_embedding_theorem
Also the book Griffiths & Harris "Principles of Algebraic Geometry" does algebraic geometry from the complex viewpoint.
Let me try to answer your question to some extent; given the vague nature of the question, there will be no canonical answer.
You should think in terms of three different areas of mathematics:
Classical projective geometry (PG)
Riemannian geometry (RG)
Algebraic geometry (AG)
These areas have different agendas, different tools, different "favorite toys." These areas, however, share some common toys, such as projective spaces of various dimensions and their projective subspaces.
For instance, the real-projective space ${\mathbb R}P^n$ (and its complex analogue ${\mathbb C}P^n$) appears as one of these common toys. In order to become one, some further choices have to be made in RG and AG. From RG viewpoint, one needs to choose a Riemannian metric, the most important one is the Fubini-Study metric. From the AG viewpoint, one needs to make the projective space $P$ into an algebraic variety. The standard choice is to identify $P$ with ${\mathbb R}P^n$ (resp. ${\mathbb C}P^n$). But other important choices are Veronese embeddings. They are all isomorphic to the "standard" projective space but the choice of an embedding is important.
Moreover, from the AG viewpoint, it is also important to work with other fields and rings, e.g. say with the projective spaces over $p$-adic numbers, over finite fields, over rings of polynomials, etc. These will have increasingly less and less in common with PG and RG.
As a projective geometer, you can think of all projective lines in the projective plane (or space) as congruent, but images of the Veronese embeddings of the projective line in the projective planes are genuinely different ones. From the AG viewpoint, projective lines in the projective plane are the degree 1 rational curves. Veronese curves/surfaces have higher degree.
From PG viewpoint, "everything takes place inside of a fixed projective space." To some extent, this is also true from the AG viewpoint, except you may have to work with different projective embeddings and some (or many, depending on what you do) of the objects of AG are not projective varieties and do not embed as subsets in a projective space. But from the RG viewpoint, one usually considers Riemannian manifolds abstractly and not as embedded isometrically in a projective space. (Caveat: There are many exceptions to this rule, for instance, much of the theory of minimal surfaces deals with surfaces in the Euclidean 3-space.)
I did not even get started on Kahler geometry that links (complex) AG and RG, but, again, AG and RG agendas are different here.
As for some of your more specific questions:
i. A question in a comment
"Why do the projective straight lines (which are primitive notions in the axiomatic treatment of projective geometry) just happen to agree with the geodesics in the Euclidean plane?"
the answer is that this is not quite true, as Tabes says. For one thing, you have to work "over the real numbers" (i.e. with real projective spaces). Secondly, you have to remove a point from the projective line to make it an affine line. Or, if you like, you remove the line at infinity from the projective plane, which will make it into an affine plane and then affine lines become complete Euclidean geodesics (after you choose a metric!). With these caveats, the fact that you have mention is true but is an accident of geometry of a space of constant curvature. "Most" Riemannian manifolds do not have constant sectional curvature and do not admit even a local a "model" where their geodesics are (pieces of) Euclidean straight lines. (OK, I am lying a bit here, this is not a complete accident and spherical, Euclidean and hyperbolic geometries were and continue to be a source of inspiration of RG.)
ii.
"Do smooth projective varieties (at least over the real numbers) have a natural projective connection that somehow agrees with the variety structure?"
No, in general they do not. Once you fix an embedding in a projective space then, yes, you can induce the ambient connection from the FS metric. But there are many different projective embeddings of the same variety, so you do not have a canonical choice. As for "somehow agrees with the variety structure," I do not even know how to interpret this. Connections, curvature, Chern classes, etc, play important role in AG, but one should definitely not limit oneself to the tangent bundle here. Frequently, other bundles and their connections are more informative. For instance, sometimes you work with the canonical or anticanonical line bundle, sometimes you work with all line bundles simultaneously, or work with all stable vector bundles, etc.
iii.
"Which concepts from projective geometry and from Riemannian geometry have natural analogs in the intrinsic geometry of projective varieties?"
I would say that the curvature sign (more precisely, the sign of one of your favorite curvatures or even positivity of the curvature operator) and the notion of "positivity" in algebraic geometry which manifests itself in different ways. Assuming that a particular curvature is positive/negative, semipositive, etc., usually makes it possible to prove interesting theorems in RG. Ditto in AG:
Lazarsfeld, Robert, Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 48. Berlin: Springer (ISBN 3-540-22533-1/hbk). xviii, 387 p. (2004). ZBL1093.14501.
Lazarsfeld, Robert, Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 49. Berlin: Springer (ISBN 3-540-22534-X/hbk). xviii, 385 p. (2004). ZBL1093.14500.
Also the key dichotomy: Positive curvature-negative curvature (RG), rational variety-general type variety (AG).
iv.
"As raised above, what about straight lines? Do these have a generalization to projective surfaces/varieties?"
As far as I am concerned, this is too vague to be answerable. One can say that these are rational curves. (Which play an important role in huge chunk of AG.) But in many cases one works with varieties that simply do not contain rational curves. Then you consider curves of higher genus: Thinking of all these curves at once is frequently quite useful. Or, you become a complex-analyst and work with the Kobayashi metric; the Kobayashi-extremal disks can be regarded as analogues of straight lines.
I declare myself done here....
The bottom line is: Treat the three geometries as separate areas of math. Occasionally, they meet in one place and share their toys, tools and theorems.
Best Answer
As Akhil writes in his comment, to see that some given Riemann surface is an algebraic curve, you usually need Riemann-Roch. A prototypical example of its use is the usual proof of existence of a Weierstrass equation for an elliptic curves (see e.g. Silverman).
But for the modular curves, things are actually simpler. It is easy to see that a modular curve is a covering of $\Gamma(1)\backslash \mathbb{H}^*$. Now, the latter is the simplest Riemann surface there is, the Riemann sphere, and that is clearly an algebraic curve. There are lots of ways of seeing that $\Gamma(1)\backslash \mathbb{H}^*$ is the Riemann sphere, some of which are sketched in Milne's notes, Proposition 2.21. An explicit isomorphism of $\Gamma(1)\backslash \mathbb{H}$ with $\mathbb{C}$ is provided by the $j$-function.
You can now use the covering to obtain an equation for a modular curve. For the minute details of this calculation for the case $\Gamma_0(N)$, see Milne's notes, Theorem 6.1. This does not need Riemann-Roch, and only relies on explicit computations with the $j$-function.
A more powerful approach to modular curves, which will also give you more information about fields of definition, is by interpreting the modular curves are moduli varieties. Milne also sketches some of this in section 8, and much more can be found in his notes on Shimura varieties.