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

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29 views

Integral of $\omega\wedge\overline{\omega}$ on Riemann surface

Let $X$ be a Riemann surface of genus $g$ and $\omega$ a meromorphic 1-form on it. I've read that if $\omega$ has just a simple pole in $x\in X$ (and is holomorphic on $X\setminus\{x\}$) then the ...
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32 views

Hyperbolic metric geodesically complete

Consider the upper half plane model of the hyperbolic space ($\mathbb{H}$ with the riemannian metric $g=\frac{dx^2+dy^2}{y^2}$). It is known that $(\mathbb{H},g)$ is geodesically complete, which means ...
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1answer
17 views

Branch points and Ramification points of a meromorphic map between Riemann Surfaces

Let be $f(z)=\frac{z^3}{(1-z^2)}$ be considered as a meromorphic function on the Riemann Sphere $\mathbb C_{\infty}.$ Consider the affiliated holomoprhic map $F:\mathbb C_{\infty}\rightarrow \mathbb ...
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35 views

Basic question: Curvature transforms under Complexified Gauge Transformation

Let $E$ be a holomorphic vector bundle over a Riemann surface $M$ equipped with a Hermitian metric. Let $\nabla$ be the compatible connection on $E$ amd $g$ is a self adjoint complexified gauge ...
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1answer
12 views

A question about the gluing of Riemann surfaces

Often, the definition of Riemann surfaces is motivated by the example of the multi-valued function $f(z)=\sqrt{z}$. Every point $z\in \Bbb{C}$ has two images. Hence, this function has two "branches"; $...
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14 views

Argument principle for meromorphic forms on Riemann surfaces

Let $X$ be a compact Riemann surface and $D \subset X$ a compact domain with boundary $\partial D$. Let $\omega$ be a meromorphic $1$-form in a neighborhood of $D$ which does not have neither zeros, ...
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2answers
32 views

Zeros and poles of some meromorphic 1-forms on the riemann sphere

Let $X=\mathbb C_{\infty}$ be the Riemann sphere with the local coordinates $\{z\ ,1/z\}$. I want to show the following two statements: i) There does not exist any non-vanishing holomorphic 1-form on ...
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24 views

A non-constant holomorphic map $F$ between riemann-surfaces is an isomorphism

I want to show the following: Let $F:X\rightarrow Y$ be a non-constant and holomorphic map between compact riemann surfaces with $genus(X)=genus(Y)\geq 2$. In the above it holds that $F$ is an ...
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1answer
51 views

All compact genus 0 Riemann surfaces are isomorphic to a sphere

Where can I read the proof that all Riemann surfaces which are homeomorphic to a sphere are also isomorphic ?
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1answer
58 views

Two tori $\mathbb C/L$ and $\mathbb C/L'$ are isomorph if $L=L'$

Let be two lattices $L$ and $L'$ given such that $L\subseteq L'$. Consider the canonical map from $\mathbb C/L$ to $\mathbb C/L'$. Now I want to show that this map is an isomorphism (biholomorph) if ...
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28 views

Beginning master's student with gaps - References for Riemann Surfaces

I currently have Jost (as well as a few other texts), and have been working through it - I am a master's student who is trying to prepare for thesis work in closely related areas. However, it is far ...
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125 views

Differential Forms on the Riemann Sphere

I am struggling with the following exercise of Rick Miranda's "Algebraic Curves and Riemann Surfaces" (page 111): Let $X$ be the Riemann Sphere with local coordinate $z$ in one chart and $w=1/z$ in ...
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59 views

Why is this sequence exact?

Let $X$ be a smooth complex projective curve. Let $D$ be any divisor and let $p$ be a point. Rick Miranda claims this sequence is exact on page 285 of his book on Riemann surfaces: $$ 0 \to \mathcal ...
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1answer
38 views

Base point free linear system

Let $X$ be a (compact) Riemann surface. Let $D$ be a divisor. In Rick Miranda's book on Riemann surfaces, on page 160, there is a bijection between Base-point-free linear systems of dimension $n$ on ...
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1answer
58 views

De Rham interpretation of $H^1(R,p,\mathbb{C})$

Let $R$ be a Riemann surface of genus $g\ge 2$ and $p\in R$ a point. This is my question: Is there a way to interpret the relative cohomology group $H^1(R,p,\mathbb{C})$ as a De Rham cohomology group ...
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1answer
35 views

Ratio of holomorphic forms on a Riemann surface

Let $R$ be a Riemann surface of genus $g>1$ and $\omega$, $\sigma$ two holomorphic 1-forms on $R$. This means that locally we can write $\omega=fdz$ and $\sigma=gdz$ with $f$ and $g$ holomorphic. ...
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1answer
69 views

First cohomology group on a Riemann surface with all wedge products equal to zero

Sorry for the strong edit, but I realized my question had a easier formulation: Can there be a Riemann surface $X$ with the property $\sigma\wedge \tau=0$ for every $\sigma,\tau\in H^1(X,\mathbb{C})$?...
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1answer
27 views

What is a primitive element in a Fuchsian group?

I am reading some introductory texts about Selberg's trace formula for hyperbolic surfaces and I have encountered the concept of "primitive element" $\gamma$ in a hyperbolic Fuchsian group $\Gamma \...
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2answers
33 views

Books on Riemann Surfaces

I am starting a scholarship on geometry and the subject of research is going to be Riemann surfaces (we will focus on compact Riemann surfaces). I am finishing my undergraduate studies so my knowledge ...
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1answer
67 views

Why Differential Forms on Riemann surfaces?

I am working with Rick Miranda's "Algebraic Curves and Riemann Surfaces". Right now I am in chapter four "Integration on Riemann Surfaces" and struggle with it a lot!:( It starts with the definition ...
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1answer
24 views

Complement of compact subspace of surface

Let $X$ be a smooth 2-manifold, $K$ be a compact subset of $X$, such that only one component of $X\backslash K$ does not have compact closure, call this component $U$ (there may be other components). ...
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46 views

Exciting applications of the Riemann-Roch-theorem for Riemann-surfaces

This semester I took a lecture on Riemann surfaces. The professor proved the Riemann-Roch-theorem (stated below). As an application of it, he proved elementary results, we did earlier in the course ...
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41 views

Is there only one complex structure on complex plane $\mathbb{C}$? [duplicate]

There is a trivial complex structure on $\mathbb{C}$. Do we have other complex structures on complex plane $\mathbb{C}$? If not, how to prove it?
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38 views

Ext group of bundles on moduli space of curves

Let $\mathcal{M}_{g}$ be the moduli space of curves of genus $g$. Let's suppose $g \geq 2$. Let $T$ be the tangent bundle of $\mathcal{M}_{g}$. Is the Ext group $\text{Ext}^1(\bigwedge^2T, T)$ trivial?...
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16 views

Local representation of antiholomorphic map

Let $f:\mathbb{C} \to \mathbb{C}$ be an antiholomorphic map, $f(0)=0$. How can I show that there exists a holomorphic function $z(w)$, $z(0)=0$, defined in a neighborhood of $0$, such that $f(z(w))=z(\...
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1answer
33 views

How to determine all the complex structures on torus $T^2$?

I have known that the lattice given by the pair $(\tau_1,\tau_2)$ can determine a complex structures on torus $T^2$. But how to prove that all the complex structures of torus can be obtained in this ...
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1answer
39 views

More than 3 branch point Dessign d' enfant

I wanted read about Dessign d' enfants most of the reference define it as (X,D) where X is compact orientable surface and D is the bipartite graph with some properties that is there is a bijection ...
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36 views

dimension of $\Omega^1 \left( X \right)$ the space of holomorphic $1$-forms.

I'm reading $1$-forms on "Rick Miranda, Algebraic Curves and Riemann surfaces". According to the book's notation: Let $X$ be a compact Riemann surface of genus $g$ and $\Omega^1 \left( X \right)$ be ...
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36 views

Sheeted-covering space degree 2 of Riemann Surfaces

In Milne - Elliptic curves, one finds the following on page 92: Branched-covering maps are not local isomorphisms at the ramified points; so could somebody explain to me what Milne means by 'a ...
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9 views

Showing that $(x,y)\mapsto(x^2,xy)$ is unramified

Let $S_{1},S_{2}$ be the hyperelliptic curves given by \begin{align*}S_{1}&: y^{2} = x^{8} - 1,\\S_{2}&: y^{2} = x^{5} - x,\end{align*} respectively. Let $f: S_{1} \rightarrow S_{2}$ be the ...
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1answer
47 views

conformal structure of a disc

I wonder if the conformal structure of the unit disc $D^2=\{(x,y):x^2+y^2\leq 1\}$ is unique. More precisely, given a Riemannian metric $g$ on $D^2$, is it always true that $g=e^{2u}g_0$, where $g_0$...
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121 views

The Teichmüller space $T_g$ of a closed riemann surface $S_g$ of genus $g \geq 2$ can't be parametrized by $6g-6$ geodesic length functions

Setting: It is well known that the Teichmüller space $T_{g,b}$ of an oriented Riemann surface $S_{g,b}$ of genus $g \geq 2$ with $b \geq 1$ boundary components (satisfying $2g + b \geq 3$) can be ...
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local representation of “logarithmic connection”

Let X be a Riemann compact surface, $D\subset X$ be a finite subset, and (E,$\nabla$) be a logarithmic connection. And let $z$ be a local coordinate at $p\in D$, why $\nabla $ can be written by: $\...
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62 views

Roots of canonical line bundles that are not necessarily square roots

I understand that holomorphic square roots of the canonical line bundle of a compact Riemann surface always exist, and that there are $2^{2g}$ choices of such a root. But what about further roots? ...
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66 views

Riemann-Roch and quartic

I know very little in algebraic geometry, but I want to learn!! So I know the Riemann-Roch theorem as follow: let $$L(D)=\{\text{ meromorphic functions, s.t. }\operatorname{div}(f)\geq D \}$$ and $$...
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1answer
18 views

Show that $\Omega^1(X) \to \operatorname{Rh}^1(X)$ is injective.

Problem: Let $X$ be a compact Riemann surface. Show that $$\Omega^1(X) \to \operatorname{Rh}^1(X) = \frac{\ker (d : \mathcal{E}^{(1)}(X) \to \mathcal{E}^{(2)}(X))}{\operatorname{im}( d: \mathcal{E}(X) ...
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24 views

Help with corollary 4.6 in Griffiths

Corollary 4.6 (P.72) in Griffith's 'Introduction to Algebraic Curves' proves that $\mathcal{O}=\mathbb{C}\{x,y\}=$set of all holomorphic functions in $x,y$ is a UFD, using the Weierstrass preparation ...
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98 views

What does it mean when a differential form “stays the same”?

For example, consider the differential one-form $$\frac{\mathrm dw}{1-w^2}$$ If we make the change of coordinates $w=1/z$ then we see that $$\frac{\mathrm dw}{1-w^2} \longrightarrow \frac{\mathrm dz}{...
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The multivalued behaviour of complex exponential $z^\lambda$

On Gustav Doetch's Introduction to the Theory and Application of the Laplace Transform, it says: The power series $\sum_{n=0}^\infty a_nz^n$ converges on a circular disc. Replacing the integers $n$...
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1answer
30 views

Hodge decomposition on Riemann surface

On a compact Riemannian manifold $M$ the Hodge decomposition takes the form $$\Omega^k(M)=d\Omega^{k-1}(M)\oplus\mathcal{H}(M)\oplus d^*\Omega^{k+1}(M)$$ Where $d^*$ is the adjoint of $d$ w.r.t. the ...
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94 views

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
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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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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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1answer
103 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
53 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 $U_0=Z(...
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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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13 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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29 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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81 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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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 ...