3
votes
1answer
74 views
+50

When functional equations determine a unique global analytic function?

I'm learning about analytic continuations and global analytic functions which were seen to be connected components of the sheaf of analytic germs. Sometimes we get problem sets in which we are ...
3
votes
1answer
34 views

How does Ahlfors define derivative on a Riemann Surface?

I'm reading a passage in Ahlfors (3rd Edition page 298) and he says the following: He has previously defined $G_0(f)$ to be the connected component of any germ generated by $f$. Then he wants to ...
3
votes
1answer
36 views

The number of holomorphic coverings (with given degree) of the punctured sphere is finite.

I'm looking for a proof of the following theorem: Fix a finite set $B=\{y_1,\ldots,y_k\}\subseteq \mathbb P^1(\mathbb C)$, then there is only a finite number of isomoprhism classes of ...
1
vote
0answers
29 views

Simple Branched covering over sphere.

A simple branched covering is a branched covering with branching points of degree at most 2, in some context, it is also required to have at most one branching point in each fiber. My question is ...
4
votes
0answers
73 views

universal covering of punctured plane and Poincaré metric

I want to prove the following result: Let $\Omega$ be the domain $\mathbb{C}\backslash \{a_1,a_2,a_3,a_4\}$. Its universal covering is the unit disk, and the standard Poincaré metric is pulled back to ...
0
votes
1answer
21 views

Real part of integral over holomorphic 1-form is zero implies the one-form is zero

Suppose we are working on a Riemann Surface $X$, assume of genus g $\geq$ 1. Let $\omega$ be a holomorphic 1-form on $X$, with $$ \textrm{Re} \int_\gamma \omega = 0$$ for every closed contour ...
1
vote
1answer
31 views

Base-point-free linear systems (elementary?) property

I'm having troubles solving exercise K on page 167 of the book "Algebraic curves and Riemann surfaces" of Miranda. The question is the following one : Let Q be a base-point-free linear system, let ...
2
votes
0answers
37 views

Is the image of the Riemann sphere a Riemann surface?

I am looking for conditions on $f$ such that $f:S^2 \to \mathbb C$ is such that the image, $f(S^2)$ is a Riemann surface. Must $f$ be analytic, or something stronger? Or are there no simple conditions ...
4
votes
2answers
60 views

Orientation on Riemann surfaces

$\mathcal{X}$ is a Riemann surface and $\mathcal{E}^{(2)}(\mathcal{X})$ is the $\mathbb{C}$-Vector space of all differentiable $2$-forms on $\mathcal{X}$. I want to define the orientation of ...
1
vote
0answers
7 views

affine curve. analysis

Can anyone explain what is written in book? $\Gamma(w,E)\equiv w^{2}-E^{n}+\sum_{i>0,j\ge 0, ni+2j<2n} g_{ij}w^{i}E^{j}=0, $ where $g_{1,0}$ doesn't equal to zero. "At infinity it compactified ...
1
vote
1answer
81 views

Why is the projection map proper?

It might be a silly question. I got stuck there. For the context, see S.K.Donaldson's Riemann Surfaces, chapter 4, section 4.2.3. Suppose $P(z,w)=a_0(z)+a_1(z)w+\dotsb+a_n(z)w^n\in\mathbb C[z,w]$ is ...
2
votes
2answers
58 views

Riemann surface associated with complete analytic function of $(z^2-1)^{1/3}$

I'm trying to define an analytic function on '$\mathbb{C}$' of the form $f(z)=(z^2-1)^{1/3}$, i.e. I first remove two semi-infinite rays $l_1$ and $l_2$, one going from $1$ to $\infty$ along the ...
1
vote
1answer
62 views

Constructing a meromorphic function

I need help with the following problem. "Let $C : y^2 = x^3 − 5x^2 + 6x$ be a cubic curve with the standard group law. Find a meromorphic function on $C$ having the pole of order two at ...
4
votes
2answers
57 views

Holomorphic functions on algebraic curves

I have been asked to solve the following problem, but I really need some help... How are the holomorphic functions $f:C\to D$, where $C,D$ are nonsingular algebraic curves of genus 1? I know that I ...
0
votes
0answers
38 views

Analytic Continuation of a Function Containing a Square Root to a Second Riemann Sheet

Consider the function $f(z) = g_1(z) + \sqrt{z} \, g_2(z)$, where $g_1(z)$ and $g_2(z)$ are entire functions, and we take the principal branch of the square root. $f$ is analytic on $\mathbb{C} / \{z ...
1
vote
0answers
50 views

Constructing Riemann surfaces

At the risk of asking a question that has been already answered, I have been trying to figure out how to construct the Riemann surface of slightly more complicated examples, but after reading examples ...
2
votes
0answers
49 views

Why does the space of germs construction correspond to the gluing construction of Riemann surfaces?

I know this might be too broad / vague a question, but still looking for somebody to write something meaningful about this. When constructing the Riemann surfaces, why does the space of germs ...
1
vote
2answers
86 views

holomorphic function $f: \mathbb{C}\setminus [-1,1]\longrightarrow \mathbb{C}$ such that $f^2(z)=z^2-1$.

Prove there does not exist a holomorphic function $f: \mathbb{C}\setminus [-1,1]\longrightarrow \mathbb{C}$ such that $f^2(z)=z^2-1$ for all $z\in \mathbb{C}\setminus [-1,1]$. I really do not know ...
4
votes
1answer
38 views

Extending a biholomorphic map between two Riemann surfaces

Consider the following problem: $X$ and $Y$ are two compact Riemann surfaces, $S$ is a finite subset of $Y$ and $f:X\longrightarrow Y$ is a holomorphic map whose set of branch points is $S$. Now ...
2
votes
1answer
97 views

What are the things I need to know to study Topology

I am looking to learn about Riemann Surfaces but I know that beforehand I need to study certain subjects like Metric and Topological Spaces, Complex/Real Analysis and Complex Functions. Can anyone ...
0
votes
2answers
47 views

Riemann Sphere/Surfaces Pre-Requisites

I have recently developed a large interest in everything to do with Riemann Sphere/Surfaces. I wish to understand the topic quite well but I know that I will need to read a good number of books on ...
1
vote
1answer
50 views

Biholomorphic maps from unit disc

Let $f$ be biholomorphic map from the unit disc onto some $D \subset \overline{\mathbb{C}}$ (considered as a Riemann sphere, so it is holomorphic) with $$f(z)=\frac{1}{z}+c_1z+c_2z^2+\cdots$$ What ...
5
votes
1answer
94 views

Is there an algebraic invariant for complex curves that's mapped to injectively?

Consider the functor $\pi_1: \text{Closed Surfaces} \rightarrow \textbf{Grp}$. This is homotopy invariant; every closed topological surface has a unique fundamental group. In the reverse direction, by ...
3
votes
0answers
65 views

Computing Riemann surfaces of a given algebraic function

I've never seen written in a book a way or an algorithm for computing Riemann surfaces of a given algebraic function. I would like to know how to construct such Riemann surface using intuitive cutting ...
1
vote
1answer
45 views

Why is an admissible function from a non-compact surface non-surjective?

I'm at the end of the proof of uniformization for simply connected manifolds in Farkas-Kra's Riemann Surfaces text. I feel like I'm missing something really obvious here. (The proof in question is on ...
1
vote
1answer
41 views

Extending maps on a Riemann surface

I came across the following definition for functions on a Riemann surface: A nonconstant analytic function on a Riemann surface, $f_{1}:X_{1} \rightarrow \mathbb{C}$ extends $f_{0}:X_{0} \rightarrow ...
3
votes
1answer
48 views

Hyperbolic Metric with respect to $\pi: \mathbb{H} \rightarrow {{\Delta}^{*}}$

I have the following problem: Find the unique metric $\rho=\rho(z)\left|dz\right|$ on the punctured unit disk $\Delta^{*}$ such that $\pi^{*}(\rho)=\left|dz\right|/(\mathrm{Im}(z))$ where $\pi: ...
7
votes
1answer
201 views

Projective closure of an algebraic curve as a compactification of Riemann surface

Assume $f \in \mathbb{C}[x,y]$ a polynomial such that the affine algebraic curve $X=V(f)$ has no singular points. Then there is a natural structure of non-compact Riemann surface on $X$, which can be ...
1
vote
0answers
33 views

Intuitively what is it if making a modification of a torus?

It is well-known that if we have a equivalence relation in $\mathbb{R}^2$:$(z_1,z_2)\sim (z_1',z_2')$ iff $$\begin{pmatrix} z_1'\\ z_2' \\ \end{pmatrix}=\begin{pmatrix} 1&0\\ 0&1 \\ ...
4
votes
0answers
64 views

Probably Riemann surface integral

Here is the integral: May you please suggest some beautiful idea on using Riemann surface, or some Gauss-Ostrogradsky at the beginning. Also, the initial integral looks really symmetric, so maybe ...
4
votes
2answers
86 views

A strange application of the Heine-Borel lemma (Ahlfors)

In Ahlfors' Complex Analysis text, pages 289-290, the author discusses analytic continuation along arcs. Among other things, he proves the equivalence with analytic continuation using a chain of ...
0
votes
2answers
135 views

Meromorphic and holomorphic functions on Riemann surfaces

Notation: If $X$ is a Riemann surface, $\mathscr O(X)$ is the ring of holomorphic functions on $X$ and $\mathscr M(X)$ is the field of meromorphic functions on $X$. If $X$ and $Y$ are two ...
2
votes
1answer
65 views

Prove that a particular holomorphic mapping is surjective

When I try to find the covering space of a punctured unit circle, I met some obstacles. The only one remain is given as follows: Suppose $H$ is the left half plane, $D^*$ is the punctured unit ...
1
vote
1answer
36 views

Generators of the first singular homology group of a Riemann surface

Let $X$ be a Riemann surface embedded in $\mathbb C^2$ with coordinates $(z,w)$ and let $\pi_z \colon X \to \mathbb C$ be the projection on first coordinate with property that that for cofinite number ...
6
votes
1answer
131 views

Deep reason why infinite sheet means logarithm while finite sheet means polynomial?

When reading the "Lectures on Riemann Surfaces" by Otto Forster on page 37, he claimed that Suppose $X$ is a Riemann surface and $f:X\to D^{*}$( $D^*$ is the punctured unit disk ...
1
vote
0answers
56 views

Number of fixed points of automorphism on Riemann Surface

I want to prove that the number of fixed points of a non-identity automorphism on a compact Riemann surface $X$ is at most 2g+2. Following hints given, I have considered the divisor $D = (g+1)P$, ...
0
votes
0answers
45 views

Why is the Chern class of a line bundle well defined?

Let $M$ be a compact Riemann surface with a finite open covering $$ M = \bigcup_{i=1}^{n} U_i $$ which has the property that every intersection is contractible (i.e. it is a good cover). To each two ...
2
votes
0answers
37 views

Reference request for an explicit description of the group of deck transformations acting on the universal cover of a Riemann Surface

Let $\Sigma_g$ be a geuns $g$ Riemann surface with $g \geq 2$. It can be thought of in the following way: it is the quotient space $$\mathbb{H}/\pi_1(\Sigma_g)$$ where an element of ...
0
votes
0answers
27 views

Generating Fuchsian Groups

Suppose $\Lambda$ is a finite (more than $2$ points) or countable discrete set in the unit disk $\mathbb{D}\subset\mathbb{C}$, how does one generate the Fuchsian group $\Gamma$ for which ...
1
vote
1answer
52 views

Finding local normal form of a holomorphic function

So I'm trying to find local coordinates to compute the local normal form of a holomorphic function. I have $f : \mathbb{P}^1 \to \mathbb{P}^1$ given by $f(z) = \frac{z}{(z-1)^2}$. Now we have a nice ...
4
votes
1answer
48 views

Dimension of a meromorphic differentials space

What is the dimension $d_{ \large k,n}$ of the space of the degree $k$ meromorphic differentials on the sphere with fixed residues ($\alpha_i$) at $n$ points $z_i$ ? The question is asked in this ...
3
votes
0answers
78 views

Explicitly realizing Riemann surfaces as a quotient of the upper-half plane

Let $\Sigma_g$ be a Riemann surface of genus $g \ge 2$. Then it is known that $\Sigma_g$ is (holomorphically) a quotient of the upper-half-plane (or unit disk) by a group $\Gamma$ of hyperbolic ...
2
votes
1answer
100 views

Image of holomorphic function on punctured compact Riemann Surface is Dense

Let $M$ be a compact Riemann surface and $S \subset M$ is discrete. Suppose $f: M \setminus S \to \mathbb{C}$ is analytic and nonconstant. Show that the image of $M \setminus S$ under $f$ is dense in ...
1
vote
1answer
94 views

Example 1.7 Girondo's Introduction to compact riemann surfaces

Consider first the algebraic equation $$y^{2}=\prod_{k=1}^{2g+1}(x-a_k)$$ where $\{a_k\}_{k=1}^{2g+1}$ is a collection of $2g+1$ distinct complex numbers, and let $$ S^\circ =\{(x,y) \in ...
4
votes
1answer
105 views

A complex structure on the tangent space

I am reading the book Riemann surface by Donaldson. I want to understand the following Lemma (p.74). Lemma. Let $X$ be a Riemann surface. There is a unique way to define a complex structure on ...
2
votes
0answers
76 views

Visualize a projective curve $X^3+Y^3=Z^3$ in $P_2(C)$ as a torus

Let $P_2(C)$ be the 2 dimensional complex projective space, I want to prove that the projective curve defined by $M=\{[X,Y,Z]\in P_2(C)|X^3+Y^3=Z^3\}$ is a torus. I know that there is a theorem saying ...
3
votes
0answers
74 views

Is a complex polynomial a regular covering? What is its group of deck tranformations?

We know that a complex polynomial $P$ of degree $n$ is an $n$-sheeted covering from $$\{\mathbb{C} - P^{-1}\{\text{critical values of }P\}\} \to \{\mathbb{C} - \{\text{critical values of }P\}\}. $$ ...
0
votes
1answer
257 views

The sum of the residues of a meromorphic function on a Riemann surface

How can one see that the sum of the residues of a meromorphic function on a Riemann surface $ \Sigma_g$ of positive genus is always zero? This is not true for the Riemann sphere $\mathbb{CP}^1$.
0
votes
1answer
73 views

What is complex conjugation in Hyperbolic Geometry

Suppose we are working in the hyperbolic plane, that is the set $$ \mathbb{H}^2 = \{ u + iv \colon v > 0\} \quad \text{with metric} \quad \frac{du^2 + dv^2}{v^2}\,. $$ Now, formally, I can rewrite ...
2
votes
1answer
56 views

Existence of a holomorphic map from Riemann Surface to an algebraic curve .

Let $C$ be an algebraic curve in $\mathbb P^2( \mathbb C)$ with singular points $p_i : \{1 \le i \le n \}$ . Then there exists a holomorphic map $\Phi : S \to C$ , where $S$ is a Riemann surface. ...