For questions about Riemann surfaces, that is compact analytic manifolds of (complex) dimension 1, and related topics.

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3
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31 views

About an example of normal bundle of a curve over a surface

I know the definition of the normal bundle $N_{C/S}$ of a curve $C$ over a surface $S$ as the cokernel of the injection $T_C \subset T_S|_C$ where $T$ is the tangent bundle. I would like to exhibit an ...
2
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1answer
42 views

Proof that $H^{0,1} \oplus H^{1,0} = H_{DR}^1$

I am struggling with a proof from Donaldson's Riemann Surfaces which he leaves as an exercise. we want to construct an isomorphism from the direct sum of $H^{1,0}(X)$, the set of holomorphic 1-forms ...
5
votes
1answer
39 views

Riemann-Roch analysis of point divisor ring on smooth genus 3 Riemann surface

Let $C=C_4\subset\mathbb{P}^2$ be the smooth genus 3 Riemann surface given by a quartic curve. Let $P\in C$ be a point, and $D=P$ the divisor given by the point $P$. Let ...
2
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0answers
50 views

Proofs of Hodge duality: $H^{0,1}(X) = H^{1,0}(X)^*$

I am looking for a proof of this fact, where $H^{1,0} = Ker(d: \mathscr{E}^{1,0} \rightarrow \mathscr{E}^{2})$ and $H^{0,1} = Coker(\overline{\partial}: \mathscr{E} \rightarrow \mathscr{E}^{0,1}$, ...
4
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0answers
43 views

Proof of Serre duality for $D=0$

I have been working through a proof of Serre duality, which proceeds by induction on the divisor $D$, but I am having trouble with the base-case. How can I prove that on a riemann surface X, $H^0(X, ...
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0answers
19 views

Riemann Surface of Genus 1

Let $X$ be a compact Riemann Surface of genus 1. Let $Cl_0(X) := \frac{Div(X)}{PDiv(X)}$, where $PDiv(X)$ is the subgroup of principal divisors on $X$. Let $P \in X$ be a fixed point. We have a ...
0
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0answers
21 views

Help to understand the proof of the Riemann Munford relation

Here i post a file where from page 617 to 618 there is the proof of the Riemann mumford relation that is the theorem 1.13. My problem is to understand the beginning of that proof. In particular ...
5
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1answer
69 views

References for the threefold categorical equivalence of compact Riemann surfaces?

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 ...
6
votes
1answer
85 views

First sheaf cohomology $H^1(\mathscr{O}_D, \mathbb{D})=0$

Can I get a hint on this problem? Given a finite divisor $D=p_1+\dots +p_m -q_1 -\dots -q_n$ on the unit disk $\mathbb{D}$, how do I show that the first sheaf cohomology group $H^1(\mathscr{O}_D, ...
1
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1answer
44 views

Show that Weierstrass function is elliptic function.

Prove that Weierstrass function is periodic with respect to lattice $L (L\subset \mathbb{C})$ .i-e $f(z+w,L)=f(z,L)$ ($w\in L$). $f(z,L)=\frac{1}{z^2}+\sum_{0\ne w\in ...
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0answers
12 views

Riemann surface for $w = f(z) = \sqrt{z^2}$

To obtain Riemann surface for $w = f(z) = \sqrt{z}$ we get two copies of $z$-planes with cuts. After they are joined $f(z)$ gives us a one-to-one correspondence between this Riemann surface and ...
4
votes
0answers
60 views

Branch points and branching number on Riemann surface

I am attempting to analyze the Riemann surface of the algebraic function $w=(\sqrt{z} - 1)^{1/4}$. To do this, I started out by writing as a polynomial, $P(w,z) = w^8 +2w^4 + 1 -z = 0$. Next, I want ...
0
votes
0answers
25 views

To use topology and Riemann surface in number theory.

I would like to learn some rudiment topology and Riemann surface in order to apply in number theory. I already know some algebraic topology, like covering space and fundamental group, singular ...
0
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3answers
84 views

Show that a function from a Riemann Surface $g:Y\to\mathbb{C}$ is holomorphic iff its composition with a proper holomorphic map is holomorphic.

I'm trying to show the following: Let $f:X\to Y$ be a proper holomorphic map between connected, non-empty Riemann Surfaces. Show that a map $g:Y\to\mathbb{C}$ is holomorphic if and only if its ...
2
votes
1answer
61 views

Riemann-Roch Theorem and Ideals of a Ring

I found in some Math book a comment stating that the study of Ideals in ring theory à la Dedekind (all kinds of ideals? only one-sided ideals?) could be transferred to other areas (specifically, ...
2
votes
1answer
38 views

Attempt at understanding Weierstrass points

I'm reading through Springer - Riemann surfaces and Farkas and Kra - Riemann surfaces and theta functions. I'm attempting to get an understanding of Weierstrass points. I've come up with a (hopefully) ...
2
votes
0answers
66 views

Understanding the “shape” of a singular Riemann surface

Consider the singular Riemann surface given by the following expression: $$z^d w^d-z^d-w^d+t=0\ ,$$ where $t$ is a parameter in $(-1,1)$ and $d$ is a positive integer greater than 2. For $t\neq0$ the ...
1
vote
2answers
26 views

Quotient of space and a group of maps, Riemann surfaces

I've been attempting to study Riemann surfaces, and I have continuously run into this notion which eludes me. I see people write things like $ \mathbb H / <z\mapsto z+1>$ or $\mathbb D / PSL$. I ...
0
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0answers
20 views

Lemma in Farkas/Kra: Riemann surfaces on construction of domain satisfying certain properties

I'm having some trouble understanding the proof of this lemma. I can follow the construction of $u$ and $D$, however the final step in the proof seems to be without justification. Specifically why ...
2
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0answers
24 views

Corollary of Wolpert lemma

Recall Wolpert Lemma. Let $S$ be a surface with genus greater than 2, let $[X,f]$ and $[Y,g]$ two points of $T(S)$ (Teichmüller space) and let $\phi \colon X \to Y$ a $K$ quasi conformal homeo. Then ...
2
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0answers
15 views

Determining if an equation represents (?) a Riemann surface

This is my first exposure to Riemann surfaces. I have studied complex analysis in an introductory course, and spent the last few weeks learning a little bit of deeper theory with Conway's Functions of ...
6
votes
1answer
79 views

Animation of Weierstrass $\wp$-function as a map from a torus to the sphere?

I am wondering if there exists somewhere an "animation" of one such map (for some lattice / torus), in the style of the kind of $z \mapsto z^2$ maps one encounters in complex analysis classes (one can ...
0
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0answers
18 views

Is the structure constant additive on connected components?

Definition of the Structure Constant Let $M$ be a Riemann surface and $\mu$ a smooth metric on it; let $\Delta_{\mu,\,M}$ be the Laplacian on $M$ induced by $\mu$ and ...
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0answers
62 views

Meromorphic functions with two poles

It is well known that a Riemann surface $C$ such that there exists a meromorphic function with just one simple pole on it, then $C $ is the Riemann sphere. What can be said if there exists a ...
1
vote
1answer
55 views

Correspondence between bitangents of a quartic and odd theta characteristics

Let $C$ be a Riemann surface of genus $g=3$. I can't understand why the following statements are true: If $C$ is not hyperelliptic, then the canonical series $|K|$ embeds $C$ as a smooth quartic in ...
1
vote
1answer
22 views

Extending functions on proper subsets of $\mathbb C$ to functions on proper subsets of $S^2$.

There are a number of nice results about extending holomorphic and meromorphic functions from the complex plane $\mathbb C$ to the Riemann sphere $S^2$. See for instance Does entire function extend ...
2
votes
0answers
33 views

Base of homology on a Riemann surface and holomorphic differentials

I have two questions: 1) Given a Riemann surface $X$ of genus $g$ and an holomorphic differential $\omega$ on $X$, is it always possible to find a base $\{\delta_i\}_{i=1,\dots 2g}$ of ...
0
votes
1answer
19 views

Confusion about definition of ramification points in Forster

In the book "Riemann surfaces" by Forster a ramification point of $f:Y\to X$ is defined as a point $y\in Y$ such that there is no neighborhood $V$ of $y$ such that $f|_V$ is injective. On the other ...
8
votes
0answers
126 views

$\tau$ structure of the sixth Painlevé equation

I am studying the isomonodromic deformations theory, which leads in the case of a $\mathcal{C}_{0,4}$ Riemann surface to the sixth Painlevé equation. I read that this equation had a ...
0
votes
1answer
40 views

Riemann surfaces from a number theoretic point of view

I need to learn the basic theory of Riemann surfaces and would like to pick a book which is most relevant to algebraic number theory. I have a good understanding of all underdraduate algebra and the ...
1
vote
0answers
10 views

Finiteness of valuation function defined in Levelt filtration.

I'm studying the Levelt filtration and a certain valuation function comes up and I'm trying to understand when (and why) it is finite. Let $S$ be a disk in the complex plane centred at $0$, $S' = S ...
2
votes
1answer
38 views

Proof of Hopf's theorem using Liouville

Hopf Theorem A topological sphere immersed as a constant mean curvature surface in $\mathbb{E}^3$ is a round sphere In Heinz Hopf's Differential Geometry in the Large, a proof is given of the ...
1
vote
1answer
50 views

Why the flat torus cannot be immersed in euclidean plane?

I am trying to prove the following claim: The flat $2$-dimensional torus cannot be isometrically immersed into $\mathbb{R}^2$ with the standard metric. That is, there is no immersion $f:T^2 ...
2
votes
1answer
32 views

Question about two definitions of Teichmuller space for a surface of genus $g$

There are many equivalent definitions of Teichmuller space for a surface of genus $g\ge 2$. One of them concerns the complex structure: the Teichmuller space $\mathcal{T}(g)$ is the set of the ...
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0answers
9 views

Convex standard fundamental polygon

The section of the wikipedia page "Fundamental Polygon" claims that one can construct a convex standard fundamental polygon from a metric fundamental polygon. I do not know a construction which ...
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0answers
20 views

Equivalence of Definitions of Quasiconformal Surfaces?

I have been reading John H. Hubbard's book $\textit{Teichmüller Theory vol. 1}$ and I am a little bit concerned with his definition of Quasiconformal Surface. Definition: A Quasiconformal surface ...
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1answer
89 views

The Weierstrass-Enneper representation, the Gauss map

Lemma: Let $x:S\to\mathbb{R}^3$ be a conformal minimal immersion of a Riemann surface. The 1-forms $f_k=(x_{k,u}-ix_{k,v})dz$ satisfy: $$ \sum_kf_k^2=0\qquad (1)\qquad \&\qquad ...
1
vote
1answer
28 views

Norm of a complex cross product

Let $c=(c_1,c_2,c_3)$ be a complex vector. How can we see that $\|c\|^2=\|c\times \bar{c}\|$? Here the bar means component wise complex conjugation, the norm is the Hermitian norm, and the cross ...
1
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1answer
26 views

Finding the Riemann surface of $w = z^{1/2}$

I'm trying to find the Euler characteristic of $R = \{(z,w) : f(w,z) = w^2 - z = 0\}$. To do this I'm using the Riemann-Hurwitz theorem with the projection $\Pi: R \to \mathbb{C}P^1$. Now in local ...
0
votes
0answers
26 views

Inverse of a constant function on an open set

I was working on holomorphic functions and Riemann surfaces, and I was wondering about the inverse of a constant function: Let $f:U\rightarrow V$ be a holomorphic function between two Riemann ...
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0answers
77 views

2 Definitions of Holomorphic functions on Riemann surfaces

In a lecture that I currently attend we defined Riemann surfaces and holomorphic mappings on it somewhat different than in another lecture that I attended a year ago. My question is: Are these ...
2
votes
0answers
23 views

Equation of the curve corresponding to a principal polarization

Let $\mathbb{C}^2/\Lambda$ be a principally polarized abelian surface. I think it is well-known how to write down the equation of the divisor (Riemann surface) corresponding to the polarization, in ...
2
votes
0answers
21 views

Proving Riemann-Hurwitz formula for riemann sphere

Given a rational map $f:\hat{\mathbb{C}} \to \hat{\mathbb{C}}$, where $\hat{\mathbb{C}}$ is the Riemann sphere, I need to prove that $2\deg(f) - 2 = \sum (v_f(p)-1)$, i.e. prove the Riemann-Hurwitz ...
4
votes
1answer
71 views

Link between Riemann surfaces and Galois theory

In my notes for a Geometry of Surfaces course that I'm studying, there is the following quote: (For those of you who like algebra and Galois theory) Studying compact connected Riemann surfaces is ...
0
votes
0answers
15 views

relation between degree and residues

Let $C$ a compact riemann surface of positive genus and $\omega_C$ the canonical divisor over $C$ with standard degree $2g-2$. Take on $C$ a divisor of positive degree $d$ and set ...
8
votes
2answers
99 views

$\operatorname{Gal}(\overline{K}/K)=\mathbb{Z}$ possible?

Is it possible to have $\operatorname{Gal}(\overline{K}/K)=\mathbb{Z}$? My question comes from the link beetween covering and field extensions. For covering the simplest example is ...
2
votes
1answer
67 views

Finding holomorphic map on Riemann surface from a map between two Riemann surfaces

I have a non-constant degree two map between Riemann surfaces $R$ and $S$, $f: R \to S$. I'm trying to find a holomorphic homeomorphism $\tau: R \to R$ such that $f(\tau) = f$ and $\tau^2$ is the ...
2
votes
1answer
67 views

Sheaf, étalé space with Riemann surfaces.

Let $f:X\rightarrow Y$ be an holomorphic map betwen two Riemann surfaces and let: $\Gamma:=${ $(x,y)\in X\times Y|y=f(x)$ } $\subset X\times Y$ be the graph of $f$. I have to show that ...
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vote
1answer
78 views

Riemann Roch Meromorphic section on a line bundle.

Let $g:\mathbb{C}\times \mathbb{C^*}\rightarrow \mathbb{C}\times\mathbb{C^*}$ defined by $g(z,w)=(w^n z,\alpha w)$ where $0<|\alpha|<1$. Let $G$ be the cyclic group spanned by $g$ and $A$ the ...
2
votes
1answer
42 views

How to prove that the flat torus is indeed flat?

The $n$-dimensional torus can be obtained as a quotient: $T^n=\mathbb{R}^n/\mathbb{Z}^n$. As pointed out here, the standard metric on $\mathbb{R}^n$ is invariant under translation by the elements of ...