Complex Analysis and Riemann Surfaces

Uniformization and quotients

• MSE for complex plane

• Donaldson p. 84 Theorem 3: Holomorphic 1-form $\alpha$ on $X$ compact Riemann surface with no zeros allows us to construct isomorphism $\mathbb{C}/\Lambda\to X$ (simple case of uniformization...) pulling $\alpha$ back to $du$ where $u\colon \mathbb{C}\to \mathbb{C}$ is identity (invariant differential). Proof is by using universal cover of $X$ to study integration of $\alpha$ along different paths in $X$... get an actual map $\widetilde{X}\to \mathbb{C}$ this way, with $du$ essentially corresponding to $\alpha$. Of course, in the final general theory this is a consequence of $2g-2=0$, and for $g=1$ the global holomorphic 1-forms span dimension one space, which must be generated by this invariant differential $du$ (once we've constructed it).

• (Global) holomorphic function on compact Riemann surface must be trivial (constant). Allowing meromorphic with one pole of order one allows us to construct isomorphism $X\to \mathbb{P}^1$. In Weierstrass construction for $\mathbb{C}/\Lambda$, the next simplest case, note that $\wp$ has a single pole, which is a double pole!

• The first point arose from $w^2 = f(z)$ (for $f$ degree 3 or 4 say) with invariant differential $\frac{dz}{w}$. This is related to second perspective, where $\frac{d\wp}{\wp'} = du$ so take $z=\wp(u)$ and $w=\wp'(u)$. By the proof of first point applied to $X$ cut out by $w^2 = f(z)$ (construct compact Riemann surface from this affine chart in the standard way, thinking about monodromy to decide how many patches to add at infinity... if we don't do this carefully we won't get a smooth thing; cf. blowup / normalization / resolve singularities story), we see that what Weierstrass $\wp$ has really done is explicitly construct the uniformizing isomorphism $\mathbb{C}/\Lambda\to X$.

Curve theory references

Standard references: Hartshorne, Qing Liu, Vakil. Stichtenoth is a good reference for the function field perspective, including a simple argument for asymptotic Riemann Roch.

See Roy Smith's answers on MO and MSE. Other users to read?

MO: Elementary Proof of Riemann-Roch for Compact Riemann Surfaces; see Roy Smith and David Speyer answers in particular. Terry Tao blog post.

Joe Harris course notes; see e.g. Atanas Atanasov notes.

Mohan: $\operatorname{Sym}^d C \to \operatorname{Pic}^d C$ (dimension $d$ mapping to dimension $g$) supposed to give heuristic RR dimension count (even with corrections?). Observe that $\operatorname{Sym}^d C$ corresponds to effective divisors, so a fiber of this map will contain all effective divisors of the form $(f)+D$ with $(f)$ principal and $D$ fixed in the divisor class $[D]\in \operatorname{Pic}^d C$ being considered (so $D$ need not be an effective representative, which indeed might not exist; the only effectiveness restriction is on the $(f)+D \geq 0$ points in the fiber, which will be empty if the class $[D]$ has no effective representative).

Also Mohan: some direct way of seeing $\chi(D+p) = \chi(D)+1$, with an underlying geometric result (gluing line bundles? or sections? or total spaces?) of independent (perhaps greater) interest: maybe inquire about details. What is the geometric intuition for Euler characteristics? See MSE: Meaning of holomorphic Euler characteristics?; MO: Why is the Euler characteristic of powers of a line bundle a polynomial in the power?; MO: Effect of tensor product on euler characteristic of line bundles; MO: Euler characteristic of a curve;

MO: Intuition behind Riemann's bilinear relations

MO: What does the Jacobian of a curve tell us about the curve?

MO: How do you see the genus of a curve, just looking at its function field?

MO: Elliptic Curves with CM and Class Field Theory

MSE: Constructing one-forms on a Riemann surface using the uniformization theorem and MO: The existence of meromorphic functions on Riemann surfaces. See Tsimerman generals.

Theta functions:

• See Donaldson p. 88, Section 6.3 nice Fourier perspective (everything is analytic, should be OK to Fourier expand) after starting from holomorphic functions with natural period-based transformation properties (which by argument principle allows easily to count zeros).

• Equidistribution results (Weyl style) using this? See MO. How does this differ from van der Corput method? Or the ergodic method Dylan mentioned?

Collect uses of properness in Donaldson and elsewhere, e.g. proof that degree is constant, p. 44-45, Proposition 7.

Compare with algebraic geometry uses and analogs.

Understand this comment of KConrad:

For a finite extension of number fields, the story is similar: ramification indices of all prime ideals over the same prime must be the same. Also residue field degrees of all prime ideals over the same prime must be the same. Moreover, there is a converse result in this case: if, for each prime downstairs, all prime ideals above it have equal ramification indices and equal residue field degrees then the extension is Galois, and the hypothesis on ramification is actually not needed to get the conclusion. Over C residue field degrees are all 1, so no information is stored there.

I accidentally found the originating MO question by wondering how unramified maps of Riemann surfaces relate to notion of Galois cover. (Riemann-Hurwitz says...)

Maybe related:

• MO: densities-of-different-splittings-in-non-galois-extension
• MWood: splitting in non-Galois extensions

Sheaves of differentials

Nice to have different perspectives. See even Neukirch for "arithmetic Riemann-Roch theory" (including a notion of genus of number field, and so on).

Want to merge classical and modern perspectives on calculus, tangent/cotangent spaces, etc. See Liu and Vakil for examples, intuition, and pictures. At the beginning, Vakil's "Key fact 21.2.3." on generators and relations of module of differentials (see November 18, 2017 version) is quite nice, with a tight connection to the affine conormal exact sequence:

21.2.D. EXERCISE. Verify Key fact 21.2.3. (If you wish, use the affine conormal exact sequence, Theorem 21.2.12, to verify it; different people prefer to work through the theory in different orders. Just take care not to make circular arguments.)

Qing Liu Chapter 6, Proposition 1.8, relating differentials on towers of fiber spaces (i.e. "affine cotangent sequence" and "affine conormal sequence"). It's interesting that while differentials are (at least in some special cases of interest) easy to compute directly and explicitly by hand (at least intuitively, though checking the universal property may be boring), the proof goes more geometrically through derivations (Yoneda-esque dual perspective, using universal property of the module of relative differential forms). Reflect more deeply on this?

[EDIT after first reading: See also Vakil, "relative cotangent sequence, affine version" and "conormal exact sequence, affine version" for other discussion.]

Could perhaps be proved directly with more yoga, but the derivations sequence is much more transparent than the originals. I suppose it's not too honest to call derivations "purely geometric", but at least elements of the derivation sets are more or less tangent vectors, and the restriction and inducing of derivations can be understood geometrically in that way.

(Technical note: see the algebraic fact that let us conclude exactness from dual exactness: MSE: Dual sequence and its exactness; MSE: Verifying statement of exact sequences; MSE: Hom functors and exactness; MSE: Confusion about a proof: exactness of modules.)