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

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Splitness of a short exact sequence on a curve

Let $C$ be a curve with genus $g > 1$. Consider the product $C \times C$, with natural projections $p_1$ and $p_2$ (from the first and second factor, respectively) to $C$. Consider the following ...
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1answer
27 views

Does there exist a conformal $\phi: D\rightarrow\Omega\cup\{\infty\}$?

Let $\gamma$ be a Jordan curve and $\Omega$ the unbounded connected component of $\mathbb{C}\setminus\gamma$. $\Omega$ is not simply connected in $\mathbb{C}$, but $\Omega\cup\{\infty\}$ is simply ...
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17 views

Are Fuchsian groups without elliptic and parabolic elements at most countable? [duplicate]

Let $G \subset PSL(2, \Bbb R)$ be a discrete subgroup without elliptic or parabolic elements. Does it follow that it is at most countable? Subgroups as above have the property that the quotients of ...
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25 views

Solution to Cauchy-Riemann Differential Equation of Compact Support

I'm working through Forster's $\textit{Lectures on Riemann Surfaces}$ and am struggling with the following problem: Suppose $g \in \mathcal{E}(\mathbb{C})$ is of compact support. Prove there is a ...
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1answer
49 views

Homology of $Z(x_0^2+x_1^2+x_2^2)\subset \mathbb{C}P^2$

I want to compute the homology of $M=Z(x_0^2+x_1^2+x_2^2)\subset \mathbb{C}P^2$. I think I have the answer, but I'm not sure how to make it precise. My approach is to consider the affine cover ...
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1answer
25 views

Preimage of $[0,1]$ under $f:\mathbb{C}\rightarrow\mathbb{C}, z\mapsto \frac{-27(1+\frac{1}{x^{3}-3})^{2}}{x^{3}-3}$

I want to find $$ f^{-1}([0,1]) $$ where $$f:\mathbb{C}\rightarrow\mathbb{C}, z\mapsto \frac{-27(1+\frac{1}{x^{3}-3})^{2}}{x^{3}-3}.$$ I have to do this in order to find a dessins d'enfant associated ...
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0answers
12 views

Hyperbolic half-planes are geodesically-convex

I'm trying to understand the concept of Dirichlet domains associated to the action of a Fuchsian group $G$ on $\Bbb H$ (the upper half-plane of $\Bbb R^2$ endowed with its usual hyperbolic metric). ...
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24 views

Computing the group of deck transformations w.r.t. a polynomial

Let $p: \mathbb{C}\backslash Y' \to \mathbb{C}\backslash X'$ be a polynomial where $Y'$ is the set of branch points and $X'$ is the image of $Y'$ under $p$. If $\deg p = n$ then $p$ is an unbranched ...
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12 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 ...
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26 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 $q=f(z)dz^2$ ...
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77 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 ...
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0answers
25 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 ...
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1answer
31 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 ...
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0answers
50 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 ...
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77 views

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 ...
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19 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 ...
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18 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 ...
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1answer
38 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 $G(z, z_0)$?
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126 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 ...
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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 ...
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2answers
42 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 ...
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1answer
43 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 ...
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1answer
24 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 ...
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1answer
77 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 ...
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2answers
51 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 ...
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16 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 ...
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1answer
61 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 ...
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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 ...
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2answers
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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. ...
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1answer
115 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 ...
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0answers
30 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 ...
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1answer
25 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 $f|V$ ...
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25 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
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1answer
63 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 ...
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0answers
43 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 ...
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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 ...
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0answers
27 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 ...
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1answer
38 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 ...
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1answer
26 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 $\Lambda$ ...
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2answers
56 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 ...
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1answer
41 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 ...
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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 ...
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2answers
48 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 ...
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0answers
51 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 ...
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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\\ ...
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1answer
63 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 ...
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41 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 ...
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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 ...
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1answer
83 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 ...
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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 ...