4
votes
2answers
37 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
22 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
31 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
39 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
78 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
30 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
2answers
83 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
45 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
42 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
91 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
51 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
39 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
38 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
44 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
141 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
31 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
57 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
80 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
108 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
53 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
26 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
128 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
45 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
42 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
36 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
24 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
43 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
46 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
70 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
92 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
88 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
93 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
73 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
70 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
210 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
71 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
52 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. ...
1
vote
0answers
50 views

What exactly does it mean to say that “functions cannot be integrated on Riemann surfaces”?

I've seen statements of this sort used to motivate the introduction of differential forms, and I'm not sure exactly what's meant. Obviously if you start by defining differentiation as an operation ...
1
vote
0answers
148 views

Every non-constant holomorphic map of Riemann surfaces is a ramified covering

I'm reading "Riemann Surfaces" by Farkas and Kra. Section I.1.6 contains the following proposition: Let $f : M \to N$ be a non-constant holomorphic mapping between compact Riemann surfaces. ...
1
vote
1answer
86 views

Question on a theorem on Riemann surfaces

In the book "Lecture on Riemann surface" of Forster, in the page 23, there is a theorem as follows: Suppose $X$ and $Y$ are Riemann surfaces, $p: Y\rightarrow X$ is an unbranched holomorphic map ...
0
votes
1answer
94 views

Complex analysis knowledge that required to understand material in Riemann Surface

I have taken a course on complex analysis in university, at that time the instructor chose the book "Complex Analysis" by Serge Lang. Now I am participating a cemina on Riemann Surfaces which truly ...
1
vote
1answer
87 views

Existence of meromorphic function implies biholomorphic map onto the sphere.

Let $M$ be a closed simply connected Riemann surface, and let $f: M \to \overline{\mathbb C}$ be a meromorphic map with a simple pole in a point $p \in M$. Is it true that $f$ is injective? That $f$ ...
1
vote
0answers
113 views

Diagram of Riemann Surface for $\sqrt {z^2-1}$

I'm trying to discuss what $\sqrt {z^2-1}$ looks like. I'm trying to relate it to the square root function $z \mapsto \sqrt z$ but not getting anywhere. Any help would be greatly appreciated. Thanks
2
votes
1answer
114 views

Complete analytic function for $\sqrt{1+\sqrt{z}}$.

Let $U=\mathbb{C}\setminus(-\infty,0]$. There is an analytic function $\phi$ on $U$, such that $\phi(z)^2=z$. $\phi$ is the inverse to $z\mapsto z^2$ on the right half plane. The image of $U$ under ...
2
votes
2answers
132 views

Riemann surface arising as a quotient of the upper half-plane.

Let $H$ be the upper half-plane $\{z \in \mathbb C \mid \Im(z) > 0\}$. For a fixed real $\lambda > 0$, let be the automorphism $$d_\lambda : H \to H, z \mapsto \lambda z .$$ Denote $\Gamma$ the ...
1
vote
0answers
64 views

On the construction of hyperelliptic Riemann surfaces.

I have seen two ways to construct hyperelliptic curves, and it seems to me that the intuition behind the change of coordinate is not the same. I like better the second construction (which is pretty ...
1
vote
1answer
101 views

Is a meromorphic function on the Riemann Sphere completely determined modulo scalar by its zeroes and poles and their orders?

Let's say that all I know about a function $f$ is that it is meromorphic on $\mathbb{C}_\infty$, and that $\{z_i\}$ is the sets of zeroes of $f$, each one of order $n_i$ and $\{\lambda_j\}$ its set of ...
1
vote
1answer
145 views

How to construct a Riemann surface for the inverse of $k(z)=z+1/z$

The following is (a rephrasing of) problem 2, in Chapter 10 of Greene and Krantz's Function Theory of One Complex Variable: Construct Riemann surfaces for the (local) "inverse functions" of ...
1
vote
1answer
55 views

a special case of the fundamental normality theorem for Riemann surfaces

I am trying to prove the exercise 27.1 on page 213 from O. Forster's "Lectures on Riemann Surfaces." Please note that this is not homework, I am just trying the exercises. The question is as follows: ...
2
votes
0answers
53 views

Torus biholomorphic to smooth cubic curve?

I am trying to understand that all compact genus 1 Riemann surfaces are biholomorphic to a smooth cubic curve. ( assuming that $\dim H^{1,0} = \dim H^{0,1} = \frac{1}{2} \dim H^1$ ) I think I ...