Questions tagged [riemann-surfaces]

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

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Limits of abstract smooth surfaces

For $r >0$, let $K_r \subseteq \mathbb{C}$ be the closed subset, $K_r = \mathbb{C} \setminus D(0,r)$. Define $S_r$ to be the quotient of $K_r$ under the identification: $$ z \sim -z, \hspace{1cm} ...
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Why use orientation-preserving diffeomorphism (instead of all diffeo's) in the construction of the moduli space of a Riemannian manifold

The question is basically in the title, but I want to make it more precise: Given an oriented Riemannian 2-manifold $\Sigma$ one can take a quotient of the set $$ \mathcal{M}_+(\Sigma)=\{~c~\mid (\...
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How to compute the ramification indices in the degree-genus formula

I've been study the proof of the genus formula and I am a little confused about the ramification index of one of the maps involved. Let $C$ be a projective algebraic curve in $\mathbb{C}P^2$ such that ...
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Special definition of “algebraic curve” for Riemann surfaces?

On the book Algebraic curves and Riemann Surface by Rick Miranda, page 169, I see the following definition: (Last part of definition 1.1) A complex Riemann surface $X$ is an algebraic curve if the ...
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Why are certain contour integrals of doubly periodic meromorphic functions zero?

Consider the following theorem: suppose $f:\mathbb C \to \mathbb C_{\infty}$ is a non-constant doubly periodic meromorphic function. (So, $f(z) = f(z+ \omega_1) = f(z+\omega_2)$ for all $z \in \mathbb ...
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does the definition of a $C^k$ differential form rely on a smooth structure?

I have learnt some basic content of smooth manifolds from Tu's Introduction to Manifolds, in which a smooth 1-form on a smooth manifold $M$ is defined as a smooth section of the cotangent bundle $T^*M$...
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Proper map of Riemann surfaces

Consider a proper holomorphic map $f:X\to Y$ between two (connected, but not necessarily compact) Riemann surfaces. Is it true that $f$ is surjective whenever it is non-constant? In a lecture about ...
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linearly independent coordinate functions implies nondegenerate image

I am currently working through chapter 5 Rick Miranda book on Riemann surfaces. On page 157 he makes the comment that if $f_0, \dots, f_n $ are meromorphic functions and $ \psi: X \rightarrow \mathbb{...
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Is there a software with which you can visualise a complex function?

I just discovered that complex functions exist, and that you need four dimensions to represent them (i.e. three space dimensions and one represented by colours => Riemman’s surfaces). I was ...
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The Monodromy Representation of a Riemann Riemann Surface with a Singular Projective Structure

Let $R$ be a Riemann Surface with a Regular Projective Structure $\nu$. Questions: What is the Monodromy Representation $\mu: \pi_1(R) \longrightarrow PSL(2,\mathbb{C})$, how to define it? If the ...
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If E is non-singular then it has non-singular weierstrass equation

Let $E$ be a non-singular projective curve of genus one. There exist regular functions $x,y$ on $E$ satisfying a Weiestrass equation $$ y^2 = x^3 + ax + b$$ Is this equation necessarily non-singular? ...
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Equivalent Statements for the Fibre Bundle

Suppose $S$ is a Riemann Surface and $M$ is given as a $\mathbb{C} \mathbb{P} (1)-$bundle over $S$ where $ \pi:M \longrightarrow S$ is the corresponding projection. Is not the previous statement (...
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Question about how a divisor ensure zero in $l(D)$

Let $D$ be a divisor on a Riemann surface, prove if $d(D)<0$, then $l(D)=\{0\}$. Here $d(D)$ is the degree of $D$, $l(D):=\{f\quad meromorphic|(f)+D\geq0 \}$.
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Unramified holomorphic map is isomorphism

In my course about Riemann surfaces, the professor briefly mentioned the following as a fact that we shall just accept: If $X$ is a connected, compact Riemann surface and $f:X\to\mathbb C_\infty$ is ...
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Integration of complex $(p,q)$-form

In complex geometry, we have $(p,q)$form $\in$ $\wedge^{p,q}T^*X$, I wonder how to define their integration on submanifold, or top-form on all manifold. For instance, in the Riemann surface book I ...
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Genus 1 Riemann surface admits meromorphic $f$ and $g$ satisfying $g^2=f^3+a_1f+a_2$

I am working on the following problem: Let $M$ be a genus 1 Riemann surface. By studying the spaces $L(-kp)$ for a point $p\in M$ and $k=1,\dots,6,$ show that $M$ admits meromorphic functions $f,g$ ...
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Biholomorphism to Riemann sphere

Supose $X$ is a compact Riemann surface of genus 0 and suppose we have a meromorphic function $f:X\to \bar{\mathbb{C}}$. How can we see that $X$ is biholomorphic to the Riemann sphere?? I think it ...
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The proper holomorphic map removed set of remification points needs not to be proper [duplicate]

Let $f:X\to Y$ be proper holomorphic map of the connected Riemann surface.Let $R$ be the set of ramification points. Prove $f:X\setminus R \to Y$ needs not to be proper,however the map $f:X\setminus (...
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multiplicity formula for the meromorphic 1-form on compact Riemann surface

Let $S$ be a compact oriented surface, given a smooth real 1-form $\alpha$ which vanishing at disrete set $\Delta$.under local coordiante $$\alpha = \alpha_1 dx+ \alpha_2 dy$$ Define the multiplicity ...
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Construct meromorphic function that has single simple pole on $\Bbb{CP}^1$

I was trying to prove there exist a meromorphic function on the complex projective line $\Bbb{CP^1}$ which has single simple pole.(therefore it biholomorphic to the Riemann sphere). In order to let it ...
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Compact Riemann surface is sequentially compact.

Now, I try to prove that; M:a compact Riemann surface. $\forall \{P_j\}_{j\in N}\subset M$ (sequence of points) $\exists\{P_{j_k}\} _{k\in N}$ (subsequence of $\{P_j\}$) s.t. the subsequence converge....
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Why consider $dx/x$ on a complex curve?

In a paper I'm reading, the author considers a compact Riemann surface -- or smooth algebraic curve, you pick -- $X$ given by the equation $y^d=x^n-1$ for some natural numbers $n,d$. My understanding ...
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Algebraicity of compact Riemann surfaces

I am taking a course in Riemann surfaces, in which the classical result about algebraicity of compact riemann Surfaces has been proven. However, I think there are some dubious points in the proof. ...
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Integration of forms on the Riemann sphere

Imagine you want to integrate a specific differential form around the equator of the Riemann sphere. This form is such that it is holomorphic at all points above the equator but there is a pole ...
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Orientation of the Riemann surface corresponding to $w=\sqrt{\lambda-z^2}$

I am trying to describe the Riemann surface $F_{\lambda}$ corresponding to the function $w=\sqrt{\lambda-z^2}$ where $\lambda\in \mathbb{C}, \lambda\ne 0$ over $|z|\le 2$. What I have so far are the ...
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There does not exist any non-zero holomorphic quadratic differentials on the Riemann sphere S^2. [duplicate]

I don't know how to prove there does not existt any non-zero holomorphic quadratic differentials on the Riemann sphere S^2. I need your help. A Conception: For any holomorphic chart (U,z), a ...
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Extending an automorphism of the upper-half plane to the whole Riemann sphere

We know the automorphism group of the upper-half plane $\mathbb{H} \subset \mathbb{C}$ is given by $\Big\{ \frac{az+b}{cz+d} \quad \big|\quad a,b,c,d \in \mathbb{R} \quad \text{and } ad-bc>0 \Big\}$...
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The least poles a meromorphic functon can attain on a Riemann Surface of genus g

I am recently reading some materials about Riemann Surface. When I met Riemann Roch theorem, I came up with a question as follows: If f is a non constant meromorphic function on a surface of genus g, ...
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The exponential map is generally not holomorphic.

Prove that $\exp_0:\mathbb{C}\to \mathbb{D}$ is not holomorphic, where $\mathbb{D}$ is the unit disk with hyperbolic metric $ds^2=\frac{4dzd\bar{z}}{(1-|z|^2)^2}$. I am reading Jost's Compact Riemann ...
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Finitely many Speiser graphs for a given entire holomorphic map of finite type?

Recently, I read definition of Speiser graph or, also called, line complex (see, for example there ). There is a certain ambiguity in its definition and I am going to formulate my question about it. ...
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Proving that the group of holomorphic automorphisms of the Riemann Sphere $\mathbb{C}_\infty$ are the Möbius Transformations

Is the following proof correct? Proof: Let $F\in\text{Aut}(\mathbb{C}_\infty)$. Let $L:\mathbb{C}_\infty\to\mathbb{C}_\infty$ be a Möbius transformation that maps $F^{-1}(\infty)$ to $\infty$. For ...
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$h^0(S^nF(nD_1+D_2))=O(n^2)$ for a rank 2 vector bundle $F$ on a smooth curve

The following proposition and proof are given in Lemma 2.5 of https://mathscinet.ams.org/mathscinet-getitem?mr=1272710, and I have some questions about it. Proposition. Let $F$ be a rank 2 vector ...
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Does any symmetric surface look like a surface of revolution? [closed]

Let $(M^2,g,\boldsymbol{\xi})$ be a two-dimensional Riemannian manifold endowed with a Riemannian metric $g$ (strictly positive signature) and a Killing field $\boldsymbol{\xi}$. Will there be an ...
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analytically continuing a function into a two-leaf function

My function is defined as $$f(z) = \int_0^\infty \frac{\rho(x) dx}{x-z } , $$ where $\rho(x)$ is a real function. It is easy to see that $f$ has a branch cut along the positive real axis. Now let us ...
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How to understand the induced complex structure of a quotient space? [duplicate]

I saw a proposition in page 60-61 in https://math.berkeley.edu/~teleman/math/Riemann.pdf Proposition: $\mathfrak{H} / S L(2, \mathbb{Z}) \cong \mathbb{C}$, and the bijection is implemented by the (...
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The quotient of homogeneous polynomial on a smooth projective curve is meromorphic if the denominator is not identically zero.

Suppose $X \subset \mathbb{CP}^n$ is a smooth projective curve. Why is $G/H$ a meromorphic function on $X$ if $G$ and $H$ are homogeneous polynomials of degree $d$ and $H$ does not vanish identically ...
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Components of Riemann Tensor on sphere

Since the surface of a sphere is a 2D manifold, the Riemann Tensor should have only one component. But if I calculate according to $$R^i_{~rkj}= \frac{\partial\Gamma^i_{~jr}}{\partial u^k} -\frac{\...
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Tiling of Klein quartic with 56 equilateral triangles

I am interested in the regular hyperbolic 14-gon associated with the Klein quartic (Klein's famous "Hauptfigur"). This 14-gon, or more precisely the genus-3 surface obtained by identifying ...
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1 vote
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Help understanding a construction of the lift of an analytic map between complex tori

I'm trying understand the construction of the lift of an analytic map $$f:\mathbb{C}/\Lambda_1\to \mathbb{C}/\Lambda_2$$ (with $\Lambda_i$ lattices in the complex plane) to a map $$F:\mathbb{C}\to \...
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Constructing meromorphic differential over compact Riemann surface

On a compact Riemann surface, one can construct meromorphic differential having only simple poles by using dipole Green function by means of Perron method. (the construction is here and having such ...
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Analytic continuation of $y=\sqrt{x^4-x^2+1}$

Question. Consider the function $y=\sqrt{x^4-x^2+1}$. Consider a sufficiently large circle $C$ of $0$ in $\mathbb{C}$, and let $x_0$ lies in this circle (suppose $x$ is not a branch point of this ...
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Extending holomorphic function between compact riemann surfaces

I apologize if this is not the right way to go about asking such a question. I am interested in understanding the proof in this question about this theorem. Let $M$,$N$ be compact connected Riemann ...
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Sections of pullback bundle

Let $X$ be a genus 3 curve canonically embedded in $\mathbb{CP}^2$. Why is it that the line bundle $L$ obtained by pulling back the hyperplane bundle $\mathcal{O}_{\mathbb{P}^2}(1)$ has 3 independent ...
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Embedding of complex torus into $\mathbb{P}^3$

I'm dealing with Riemann Surfaces and I saw how a basis for the space $L(D)$ (where $D\in Div(X)$ is a divisor for the complex torus $T=\mathbb{C}/\Lambda$) of D-bounded meromorphic functions give ...
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What is a non-effective holomorphic line bundle $L$ of degree 0 such that $L \otimes L$ is effective?

I am trying to solve an exercise that asks for a non-effective holomorphic line bundle $L$ of degree 0 such that $L \otimes L$ is effective. I think I have mostly solved it but since I am a bit shaky ...
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Meromorphic functions on a Riemann surface as quotient of polynomials

Consider a compact Riemann surface $C$ embedded in a projective plane with coordinates $[x:y:z]$. Is it possible to write any meromorphic function $f\in M(C)$ as a fraction $f=\frac{P(x,y,z)}{Q(x,y,z)}...
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Isomorphism of meromorphic function fields implies Riemann surfaces are isomorphic

This is about from Forster, Lectures on Riemann Surfaces, Exercise 8.1. The exercise is the following: ($\mathcal{M}(X)$ represents the field of meromorphic functions on $X$, and likewise for $Y$.) ...
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How to compute the degree of this branched covering between two compact Riemann surfaces?

Let $S_g$ be the compact Riemann surface of genus $g$. For $g>1,\, $we can construct a holomorphic map from $S_g$ to $S_1$ as below: $S_g \rightarrow \mathbb{C}^g/\mathbb{Z}^{2g} \rightarrow S_1=\...
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Possible degrees of nonconstant map $f:C\rightarrow \mathbb{P}^1$ for a plane curve $C$

I am looking for the possible degrees of nonconstant map $f:C\rightarrow \mathbb{P}^1$ for a plane curve $C$. By combining the Brill-Noether theorem with the equality $g={d-1\choose 2}$ for a plane ...
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Confusion about the change of variable $z \to \frac{1}{z}$ for a multivalued function

I'm currently struggling with something that came up in my studies. I'm trying to integrate a multivalued function like the square root on a given path, specifically a function with two branch points, ...
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