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

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53 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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16 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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33 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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37 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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31 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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11 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(\...
1
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1answer
24 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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32 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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33 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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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
45 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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28 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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16 views

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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54 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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64 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
16 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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2answers
96 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
28 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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47 views

Riemann surface defined by polynomial

Let $$P(z,w)=w^3-(z-\alpha_1)^2(z-\alpha_2)\cdots(z-\alpha_k)$$ with $3\mid k+1$, $\alpha_1,\dots,\alpha_k\in\mathbb C$ distinct, and let $X$ be the compact Riemann surface defined by this polynomial. ...
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2answers
93 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 ...
2
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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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18 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 ...
2
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1answer
89 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
50 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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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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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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28 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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80 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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26 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
32 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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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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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 $X(\mathbb C)$...
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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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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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4answers
133 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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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
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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
80 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
53 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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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
64 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
41 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. ...