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

learn more… | top users | synonyms (1)

0
votes
0answers
10 views

Power series function on Riemann surfaces

I have some questions from Farkas-Kra's Riemann surface (see here for notations if needed). Below, main part of the book is attached as picture. The problem I am facing are following. (1) I am not ...
1
vote
0answers
22 views

On lifts of a trajectory of a quadratic differential

Let $X$ be a Riemann surface of genus $g\ge 2$ and $q$ a holomorphic quadratic differential on $X$. The differential $q$ defines a flat metric with conical singularities on $X$: if $q=f(z)dz^2$ ...
1
vote
0answers
62 views

Dimension of a sheaf cohomology group on a genus 1 curve

Let $\mathcal{M}_{g,1}$ be the moduli space of genus 1 curves with 1 puncture. For simplicity let's take $g > 1$. As usual, there is a natural fibration $C \rightarrow \mathcal{M}_{g,1} \rightarrow ...
1
vote
0answers
24 views

PDEs on higher genus Riemann surfaces, e.g. Klein Curve

I'm trying to solve a PDE on compact Riemann surfaces of genus g > 1. Since these can be obtained as quotients of the upper half plane $\mathbb{H}_2$ by some Fuchsian group $\Gamma$, I suppose it's ...
3
votes
0answers
19 views

Does a hyperelliptic Riemann surface $S$ with $\# Aut(S)=2$ exist?

If a Riemann surface $S$ has genus $g\geq 2$, its automotphisms group is finite. I was wondering if there exists a hyperelliptic Riemann surface $S$ with $\# Aut(S)=2$. In other words, I was wondering ...
2
votes
0answers
47 views

Homological description of the degree of a map to $\mathbb P^n$

Let $f \colon X \to \mathbb P^n$, $n \geq 2$, be a holomorphic map from a compact Riemann surface $X$ and whose image $f(X)$ is a smooth projective curve. There are two notions of degree for such a ...
1
vote
0answers
63 views
+100

Rational sections of invertible sheaves and hermitian inner products

Notations: Let $X$ be a $\mathbb C$-scheme of finite type, projective, integral and of dimension $1$ (i.e. an algebraic curve) and with function field $K(X)$. The set of closed points is $X(\mathbb ...
2
votes
1answer
18 views

non-equivalent Riemann surfaces of genus $1$

It is well known that all the compact orientable connected Hausdorff genus $1$ surfaces are homeomorphic, but they may have different complex structures. In fact, consider the following connected ...
2
votes
0answers
17 views

Groups acting on Riemann Surfaces and Automorphic Function

Consider the following paragraph from a book of Magnus on Combinatorial Group Theory. ... the simple group $G_{168}$ of order $168$ acts on a genus $3$ surface, is important for the theory of ...
1
vote
1answer
31 views

$g: (U \times U - D) \to \mathbb{R}$ is continuous, $D$ diagonal? [closed]

Do we have necessarily have that$$g: (U \times U - D) \to \mathbb{R},$$is continuous, where $D$ is the diagonal? Idea. Perhaps we want to apply the maximum-minimum principle to $G(z, z_0)$?
4
votes
4answers
124 views

Equation of a Riemann surface?

Intuitively in complex analysis I know what a Riemann surface is. It is a surface such that at every point on it the value of a function $f(z)$ is single-valued. However, how would I go about finding ...
1
vote
0answers
16 views

Counting zeroes of global sections

Let $X$ be a compact connected Riemann surface and let $\Phi:M\rightarrow N$ be an elliptic differential operator where $M$ and $N$ are two complex line bundles on $X$. Let $f$ be a $C^\infty$-global ...
4
votes
2answers
35 views

Upper half-plane $\overline{\mathbb{H}}$ with two boundary punctures

Consider $\overline{\mathbb H}$ with two puncture $P_1$ and $P_2$ on the real line, with coordinates $z = x_1$ and $z = x_2$, respectively. Consider another copy of $\overline{\mathbb H}$ with two ...
5
votes
1answer
40 views

Constant Curvature Metric and Biholomorphic Equivalence

This is probably a dumb question, but let's try it anyway. I know two versions of the uniformization theorem, and I don't understand their equivalence. The first says that every Riemann surface has a ...
1
vote
1answer
23 views

For $h$ an odd degree polynomial, $\{(z,w)\in\mathbb{C}^2\mid w^2=h(z)\}$ can be made into a compact Riemann surface by adding 1 point at inifinty

I want to prove that for $h$ an odd degree polynomial, $S=\{(z,w)\in\mathbb{C}^2\mid w^2=h(z)\}$ can be made into a compact Riemann surface by adding 1 point at inifinty. My problem is that I can ...
1
vote
1answer
73 views

How to prove the number of poles minus the number of zeros is $2-2g$?

I want to show that, for all differentials on the same Riemann surface S the number of poles minus the number of zeros, counting multiplicities, always equals $2-2g$. It says this can be deduced from ...
4
votes
2answers
49 views

Pullback of a complex $ 1$-form

Let $p = \operatorname{exp} : \mathbb{C} \to \mathbb{C}^*$ be a covering and $(U,z)$ a chart of $\mathbb{C}^*$ with $z = x + iy$. Let $\omega = dz/z$ be a one-form on $U$. Problem: Find the pullback ...
1
vote
0answers
14 views

Maximal analytic continuation gives rise to a covering

Suppose that $a$ is a point on a connected Riemann surface $X$ and $\varphi \in \mathcal{O}_a$ admits an analytic continuation along every curve in $X$ starting at $a$. Let $(Y, p, f, b)$ be the ...
3
votes
1answer
57 views

Euler characteristic singular surface

The setting is the one of algebraic curves over the complex numbers. It is known that in an irreducible nodal curve each node reduces the arithmetic genus by one: if $\tilde{C} \rightarrow C$ is the ...
4
votes
1answer
40 views

Linear fractional transformation of quadratic differentials on the Riemann sphere

Suppose I have the following quadratic differential on the Riemann sphere with four punctures: \begin{equation} q = -\frac{9 t \left(216+t^3\right)}{\left(-27+t^3\right)^2} dt^2 \end{equation} This ...
6
votes
2answers
54 views

Riemann surface of $f(z)=((z-1)(z-2)(z-3))^{2/3}$

I try to describe the Riemann surface of $f(z)=((z-1)(z-2)(z-3))^{2/3}$. I found the branch points 1,2, and 3 also realized $\infty$ is not a branch point. Since we take third root, I see three sheet. ...
6
votes
1answer
113 views

Riemann surface for square root function

Here, if we take a point $w$ with $w\ne 0$ from where blue colored part of sheet intersects with red one, i.e., from the intersecting 'line', is $f(w)$ unique? I think $f(w)$ takes two different ...
1
vote
0answers
28 views

Deck transformation of $p : Y \to X : z \mapsto z^3 - 3z$

Let $X = \mathbb{C} \setminus \{ \pm 2 \}$ and $Y = \mathbb{C} \setminus \{ \pm 1, \pm 2 \}$. The map $$ p : Y \to X : z \mapsto z^3 - 3z $$ is a 3-branched covering. Problem: Find ...
2
votes
1answer
24 views

Branch points of $f : \mathbb{C} \to \mathbb{P}^1 : z \mapsto \frac{1}{2}(z + \frac{1}{z})$

Problem: find the branch points of the function $$ f : \mathbb{C} \to \mathbb{P}^1 : z \mapsto \frac{1}{2}\bigg(z + \frac{1}{z}\bigg). $$ My try: The zeros are $i$ and $-i$, but I don't see why $f|V$ ...
0
votes
0answers
24 views

If $g: \mathbb{C} \rightarrow \mathbb{C} $ with $\Delta(g)=\Delta(f)=2 \pi u$ and $\lim_{z \rightarrow \infty} f(z)-g(z)=0.$ Then $g=f.$

Let $u: \mathbb{C} \rightarrow \mathbb{C}$ be a smooth function with compact support. Let $f=2 \pi \log * u $ where $*$ denotes the convolution product. (i.e $\int \log \vert y \vert u(x-y)d ...
4
votes
1answer
60 views

Serre duality explicitly on curves

Consider a Riemann surface $X$, with genus and marking so that a suitable moduli space exists. It's a well-known fact that the tangent space to that moduli space at $X$ (in other words, the space of ...
0
votes
0answers
42 views

Exercise of Rick Miranda is wrong? Actions over Riemann sphere

I'm studying the book Rick Miranda, Algebraic Curves and Riemann Surfaces and I have a question about the exercise H of page 84. The book says that $z \mapsto exp(2\pi i /r)z$ is an automorphism of ...
2
votes
1answer
18 views

genus of the quotient $g(X/G) \le g(X)$

Let $X$ be a Riemann Surface of genus $g(X)$ and $G$ a group acting holomorphically and effectively over $X$. I'm reading Miranda and he used twice the fact that the genus $g(X/G) \le g(X)$. He used ...
2
votes
0answers
26 views

Dihedral groups acting on Riemann surfaces

I'm studying the quotient riemann surface $X/G$. I'm looking for examples of dihedral groups $D_n$ acting on some riemann surfaces $X$ or at least acting on it's Jacobian JX. Does anybody knows some ...
0
votes
1answer
35 views

Formal definition of a Riemann surface?

How do we formally define a Riemann Surface in the context of complex analysis? I know the basic principal of what it is (the joining together along branch cuts of different branches of a function so ...
1
vote
1answer
25 views

For all $d \geq 1$ there exist a torus $X= \mathbb{C} / \Lambda$ and a holomorphic map $X \rightarrow X$ of degree $d.$

Prove that for all $d \geq 1$ there exist a torus $X= \mathbb{C} / \Lambda$ and a holomorphic map $X = \mathbb{C} / \Lambda\rightarrow X= \mathbb{C} / \Lambda$ of degree $d.$ Attempt: Let $\Lambda$ ...
4
votes
2answers
52 views

Evaluate square of first Chern class on K3 Surface

I want to let $X$ be a K3 surface, with $Y \subset X$ a smooth curve with genus $g$. Since $Y$ is a hypersurface, we have a line bundle $\mathcal{O}(Y)$ on $X$. I'm curious how to prove the ...
2
votes
1answer
38 views

Embed a bordered Riemann surface into punctured Riemann surfaces?

Let $U$ a bordered Riemann surface of genus $g$ with $n-1$ punctures and one hole (i.e., the border has one connected component). Is the following statement true: "For any punctured Riemann surface ...
1
vote
1answer
33 views

Step in the proof of Riemann Mapping Theorem

We have a simply connected domain $D$ in the Riemann sphere $\hat{\mathbb{C}}$ such that the complement $D \backslash \hat{\mathbb{C}}$ has more than one point. We use a Mobius transform $g$ to send ...
0
votes
1answer
37 views

Proof of equivalence of conformal and complex structures on a Riemann surface.

I am trying to understand the fundamentals of Riemann surface theory and so far I have the following: --Definition 1. A conformal structure on a Riemann surface $\Sigma$ is an equivalence class of ...
0
votes
0answers
46 views

Ramification of plane curve at infinity

Suppose I am given a smooth affine plane curve $f(x,y)=0$ that is singular at infinity (when put in projective coordinates). There is a projection map $\pi: (x, y) \mapsto x$, and ramification can be ...
1
vote
0answers
31 views

Riemann curvature tensor for geometry surfaces on $\mathbb{R}^3$

Let $M\subseteq \mathbb{R}^3$ be a regular surface and $p\in M$. The Riemann curvature tensor is defined by: $$\begin{array}{rcll} R_p:&T_pM\times T_pM\times T_pM&\longrightarrow &T_pM\\ ...
0
votes
1answer
62 views

Covering map in the context of Riemann Surfaces and Algebraic Topology

I am taking a course in Riemann surfaces and our lecturer has warned us that the definition of covering maps in the context of Riemann surfaces is strictly weaker than the ones used in Algebraic ...
2
votes
0answers
40 views

Harmonic maps between Riemann surfaces

In 'Compact Riemann surfaces' Jost defines harmonic maps between surfaces $S_1,S_2$, with local coordinates z on $S_1$ and metric $\rho^2|du\,d\overline{u}|$ on $S_2$ as $u\in C^2$ solving the ...
0
votes
0answers
29 views

Use differential form to prove meromorphic function on compact riemann surface has same zeros and poles

I am reading mine's modular form note, proposition 1.12 states that the sum of residues of a differential form on compact Riemman surface is 0. Then he states that applies this to $df/f$, then we can ...
0
votes
1answer
76 views

Using the Maximum Modulus Principle to prove that every holomorphic function on a compact Riemann surface is constant

I have read in a number of sources (including here) that a holomorphic function on a compact Riemann surface must be constant. The reason given has always been the Maximum Modulus principle, but ...
0
votes
0answers
23 views

Complex Atlas for Elliptic Curves over $\mathbb{C}$

I know that every elliptic curve over $\mathbb{C}$ is isomophic to a torus $\mathbb{C}/\Lambda$ in the sense of Riemann Surfaces, moreover $E(\Lambda)$ as topological subspace of ...
3
votes
2answers
107 views

On the Usual Orientation of Cubic Graphs in Random Construction of Riemann Surfaces

In "Random Construction of Riemann Surfaces", Robert Brooks and Eran Makover say : Definition 2.1 A left-hand turn path on $(\Gamma, \mathcal O)$ is a closed path on [the cubic graph] $\Gamma$ ...
1
vote
0answers
33 views

Fundamental domain of a Fuchsian group is non-compact if the group contains parabolic element

Suppose a discrete subgroup $\Gamma$ of $PSL(2,\mathbb{R})$ acts on $\mathbb{H}^2$. Why is the fundamental domain non-compact if $\Gamma$ contains a parabolic element? Thanks in advance.
0
votes
1answer
52 views

Choice of Fundamental Domain of Torus (Dehn Twists?)

So I would like to consider a lattice $\Lambda \subseteq \mathbb{C}$ generated by $(1,\tau)$ with $\tau$ in the upper-half complex plane. This lattice $\Lambda$ will remain fixed. If you choose the ...
2
votes
0answers
73 views

Construct the Riemann surface of the function $f(z)=((z-1)(z-2)(z-3))^{2/3}$.

Construct the Riemann surface of the function $f(z)=((z-1)(z-2)(z-3))^{2/3}$. 1,2 and 3 are branch points. But how can we determine $\infty$ is also a branch point. And how can we determine branch ...
2
votes
0answers
31 views

Subgroups of $\text{PSL}(2, \mathbb{R})$ Closed under Transposition

I am wondering, does anyone know if there is a classification of transposition-closed (Fuchsian) subgroups of $\text{PSL}(2, \mathbb{R})$? I can't read French, so for all I know it's sitting in the ...
4
votes
1answer
71 views

Divisors of degree $2g-2$ on a hyperelliptic curve of genus $g$

Suppose I have a divisor $D$ of degree $2g-2$ on a hyperelliptic curve of genus $g$. Then I can prove that either a) $K_C\otimes\mathcal{O}(-D)=\mathcal{O}_C$, that is $K_C\cong \mathcal{O}(D)$, or ...
5
votes
2answers
112 views

How do I get $ \int_0^1 \frac{dz}{\sqrt{z(z - 1\,)(z+1\,)}} = \frac{\sqrt{\pi}}{2} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{9}{4})}$?

While reading physics papers I found a very interesting integral so I decided to write it down. Let $p(z) = z^ 3 - 3\Lambda^ 2 z$ where $\Lambda$ could be any number. If you want $\Lambda = 1$ and ...
2
votes
0answers
68 views

Generators of commutator subgroup of fundamental group of genus-2 surface

Recall the fundamental group of a genus-2 surface: $$ \pi_1(\Sigma_2) = < a_1, b_1, a_2, b_2 \mid [a_1, b_1][a_2, b_2] = 1 > $$ By which I mean a free group of four variables, divided out by ...