87
votes

Accepted

### Is there a complex surface into which every Riemann surface embeds?

The answer is negative. Suppose for contradiction that $S$ is such a surface, and let me first assume that it is smooth and projective.
Fix $g\geq 24$. Then the coarse moduli space of genus $g$ ...

39
votes

Accepted

### Elementary Proof of Riemann-Roch for Compact Riemann Surfaces

RRT
There is a big difference in difficulty between the compact Riemann surface case and the projective curve case, for reasons already mentioned. Namely a projective curve comes equipped with a ...

32
votes

Accepted

### How did Riemann prove that the moduli space of compact Riemann surfaces of genus $g>1$ has dimension $3g-3$?

Riemann combines what is called Riemann-Roch and Riemann-Hurwitz nowadays.
He considers the dimension of the space of holomorphic maps of degree $d$ from the Riemann surface of genus $g$ to the sphere....

24
votes

Accepted

### Why are Green functions involved in intersection theory?

The Green's function is used not to measure distances in the surface but to measure distances in the line bundle. A Green's function on $X_{\mathbb C}$ that blows up at $D$ can be used to measure ...

23
votes

Accepted

### Is a one-dimensional compact complex analytic space necessarily projective?

Yes, every proper 1-dimensional complex-analytic space $X$ admits a closed immersion (in the sense of locally ringed spaces over $\mathbf{C}$) into an analytic projective space and more specifically ...

23
votes

Accepted

### Square root of the determinant line

There is no such isomorphism (at least for $g \geq 9$).
In
O. Randal-Williams, The Picard group of the moduli space of r-Spin Riemann surfaces. Advances in Mathematics 231 (1) (2012) 482-515.
I ...

22
votes

Accepted

### Poincaré metric on the Riemann sphere minus more than two points

Yes. The density of the Poincare metric with respect to the spherical metric is
a positive continuous function which tends to infinity at the punctures. Thus it
is bounded from below by some positive ...

20
votes

### Are mapping class groups of orientable surfaces good in the sense of Serre?

The braid groups are good (which are mapping class groups of punctured disks) by Proposition 3.5 of
Grunewald, F.; Jaikin-Zapirain, A.; Zalesskii, P. A., Cohomological goodness and the profinite ...

20
votes

### How to visualize the Riemann-Roch theorem from complex analysis or geometric topology considerations?

I'll try to give this a shot, but I'm coming from an algebraic geometry perspective. In the case of curves, it's not so clear to me what the line between algebraic geometry and complex geometry is, so ...

20
votes

### When does a group act effectively and holomorphically on some Riemann surface?

Donu's answer is correct but amounts to killing a fly with a gun shot: Greenberg proves a harder result than the one needed for the problem.
Theorem. Let $G$ be a countable group. Then there exists a ...

19
votes

Accepted

### Riemann surfaces with an atlas all of whose open sets are biholomorphic to $\mathbb{C}$?

Using the uniformization theorem, we can prove that the only compact connected Riemann surface admitting an open set biholomorphic to $\mathbb{C}$ is the Riemann sphere $\mathbb{P}^1(\mathbb{C})$. ...

19
votes

### References for Riemann surfaces

As it is evident from your bibliography list, there are two aspects of the theory: Riemann surfaces in the sense of 1-dimensional complex manifolds (which are not necessarily algebraic) and Complex ...

18
votes

Accepted

### The Teichmüller space $T_g$ of a closed riemann surface $S_g$ of genus $g \geq 2$ can't be parametrized by $6g−6$ geodesic length functions

I think Scott's argument is that the lengths of $6g-6$ curves can't form coordinates for Teichmuller space. If one has $6g-6$ geodesics which parameterize, then they must be filling (they meet every ...

18
votes

Accepted

### Are mapping class groups of orientable surfaces good in the sense of Serre?

This is an open and probably very difficult question. There have been purported proofs (for instance, this one), but they have all had fatal flaws.
The mapping class group is definitely not ...

18
votes

### Conceptual proof of classification of surfaces?

I guess the most conceptual proof is the one using Morse theory:
Take a Morse function on the (closed, orientable) surface S. If it has no saddle points, then (using the gradient flow) $S\cong S^2$. ...

18
votes

Accepted

### Two non constant meromorphic functions over a connected compact Riemann surface, could not be algebraically independent

Let $F$ be a polynomial of degree at most $n$.
For a point $x$ where $f$ has a pole of order $a$ and $g$ has a pole of order $b$, $F(f,g)$ has a pole of order at most $n\max(a,b)$. Locally near $x$, ...

18
votes

Accepted

### When does a group act effectively and holomorphically on some Riemann surface?

In fact:
Theorem: Any finite group $G$ is the automorphism group of a compact Riemann surface, and more generally a smooth projective algebraic curve over any algebraically closed field.
The Riemann ...

16
votes

### Embed a bordered Riemann surface into punctured Riemann surfaces?

The answer is no. For $g=0$, the problem is equivalent to the following: is there a univalent function in the unit disk which takes given values at finitely many given points. It is well known that ...

16
votes

### How to visualize the Riemann-Roch theorem from complex analysis or geometric topology considerations?

Complex geometric view of Riemann Roch for a curve $C$:
The essential Riemann Roch problem is the computation of the dimension of the vector space $H^0(D)$ where $D$ is an effective divisor on $C$. ...

16
votes

### Equations defining hyperbolic geodesics in $\mathbb C \setminus\{0,1\}$

I wondered this myself, I made some similar pictures to approximate the stable lamination of a pseudo-Anosov map (the blue curve is a geodesic, the other colors horocycles).
Every geodesic on the ...

15
votes

Accepted

### Does every Riemann surface with boundary immerse in C?

A more general result is proven in
Gunning, R. C., Narasimhan, R., Immersion of open Riemann surfaces. Math. Ann. 174, 103–108 (1967).
As for compact surfaces with boundary, it is essentially a ...

15
votes

### How to visualize the Riemann-Roch theorem from complex analysis or geometric topology considerations?

Here is my "visual" coming from the complex (and functional) analysis perspective, due to a novel proof by Taubes. For a holomorphic line bundle $E$ over a genus $g$ surface $C$, the RR theorem states ...

15
votes

### How did Riemann prove that the moduli space of compact Riemann surfaces of genus $g>1$ has dimension $3g-3$?

The original paper of Riemann is his celebrated "Theorie der Abel'schen Functionen" in Crelle's Journal of 1854. This paper can be found online at https://www.maths.tcd.ie/pub/HistMath/People/Riemann/...

15
votes

### Area of a smooth complex projective curve

A much more general result is given by Mumford, in Projective varieties I, Theorem 5.22: the volume of any $r$-dimensional smooth projective variety in $\Bbb{P}^N$ (the area in your case) is its ...

14
votes

Accepted

### Quotients of curves of genus $4$ by a free $\mathbb{Z}/ 3 \mathbb{Z}$-action

Yes. Start from a genus 2 curve $C_2$, and choose a point of order 3 in $JC_2$, giving rise to an étale $\mathbb{Z}/3$-covering $C_4\rightarrow C_2$.
Then $C_4$ cannot be hyperelliptic: a $g^1_2$ on ...

14
votes

Accepted

### Original reference for Riemann's inequality

B. Riemann, Theorie der Abel'schen Functionen, Journal für die reine und angewandte Mathematik 54, 101–155 (1857).
Here is a description of this contribution, by Jeremy Gray:
In this 1857 paper ...

14
votes

Accepted

### Do $\mathbb{A}^1-S$ and $\mathbb{A}^1-\{0,1\}$ have a finite etale cover in common?

The answer is positive if and only if $\mathbb{A}^1\setminus S$ is an arithmetic curve, i.e., $\pi_1(\mathbb{A}^1\setminus S)\subset \mathrm{Aut}(\mathbb{H}) = PSL_2(\mathbb{R})$ is an arithmetic ...

13
votes

### Elementary Proof of Riemann-Roch for Compact Riemann Surfaces

Joe Harris, as recorded in his course notes here, gives the following slick proof when both $D$ and $K-D$ are effective; it has the advantage of never mentioning $H^1$. See lecture 1 for this argument,...

13
votes

### Why are Green functions involved in intersection theory?

It is a good idea to restrict first to understand the case of arithmetic surfaces, which is much easier than the general case. For example, you do not have to care about Green currents (only Green ...

13
votes

Accepted

### Gluing Riemann surfaces

The answer to the first question is "yes". This is called conformal gluing, and the proof is based on the following lemma due to Lavrentiev: Let $\phi$ be an increasing diffeomorphism of $[-1,1]$ onto ...

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