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Questions tagged [riemann-surfaces]

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

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Verify Herwitz's formula for $z^3/(1-z^2)$

This is an exercise from Miranda's book "Algebraic curves and Riemann surfaces". Consider $f(z)=z^3/(1-z^2)$ as a holomorphic map from the Riemann sphere $\mathbb{C}_\infty$ to itself. Verify ...
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Ricci Tensor in an Einstein Manifold

I must prove that an hypersurface $M$ on $\mathbb{R}^{n+1}$ that is Einstein and compact can be only the $n-$dimensional sphere when $n>2$ The Einstein condition we permits to say that scalar ...
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Unique complex structure on the modular curve $\mathbb{H}/\operatorname{PSL}(2,\mathbb{Z})$

Is the complex structure on the modular curve coming from the quotient $\mathbb{H}/\operatorname{PSL}(2,\mathbb{Z})$ unique? (Here $\mathbb{H}$ is the upper half plane in $\mathbb{C}$) According to ...
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1answer
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Torus and period integrals

I'm following a course in Riemann surfaces, and I'd like to solve the exercise below. Let $L$ a lattice in $\mathbb{C}$, and let $T:= \mathbb{C}/L$ the corresponding torus. i) Prove that $dx$ and $...
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Ricci Tensor and Einstein Manifolds

What can we say about an hypersurface Einstein manifolds on $\mathbb{R}^{n+1}$ when $n\geq 3$ ? The manifolds is Einstein so Ricci tensor is a multiply of the metric $g$ on manifold: $Ric=\lambda g$ ...
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Show that a space of holomorphic sections of a line bundle is isomorphic to the space of meromorphic functions on a Riemann surface.

Let $\Sigma$ be a Riemann surface, $x \in \Sigma$ and let $z: U \rightarrow D \subset \mathbb{C}$ be a local coordinate system centered at $x$. For every $k \in \mathbb{Z}$, a holomorphic line bundle $...
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Algebraic dependence of meromorphic functions on a compact Riemann surface

I am trying to show given two meromorphic functions $f,g$ on a compact Riemann surface $X$ that there exists a non-zero polynomial $P \in \mathbb{C}[X,Y]$ such that $P(f,g) = 0$. I have seen this ...
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Word length vs hyperbolic length of curves on a hyperbolic surface

Suppose S is a surface which admits a hyperbolic metric - by this I mean a complete Riemannian metric of constant negative curvature, with totally geodesic (possibly empty) boundary. Fix some ...
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3D-plots of complex functions

Commonly it's believed that one cannot fully visualize a complex function $f:\mathbb{C}\rightarrow \mathbb{C}$ because the full plot would have to be 4-dimensional. But is this true? And why would the ...
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Riemann bilinear relations and meromorphic abelian differentials

I am getting quite confused with Riemann bilinear relations. Let $\Sigma$ be a compact Riemann surface of genus $g$, with a canonical homology basis $a_1,b_1,\dots,a_g,b_g$, with associated ...
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1answer
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Space of univalent mappings $f: \mathbb{D} \to \mathbb{C}$ has no nesting

Let $S$ be the space of univalent (i.e. injective) mappings from the disk $\mathbb{D}$ to the plane $\mathbb{C}$ normalized so that $f(0) =0$ and $f'(0)=1$. So $$f(z) = z+a_2z^2+a_3z^3+\cdots.$$...
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A homotopy where all intermediate maps have holomorphic antiderivatives

I will denote by $\mathbb{C}^*$ the punctured complex plane, $\mathbb{C} \setminus \{0\}$. Let's say I have a holomorphic map on the punctured plane, $w: \mathbb{C}^* \to \mathbb{C}$, such that the ...
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Question about total branching number

Given a map $f:\mathbb{CP^1} \leftarrow \mathbb{CP^1}$ by $f(z)=\frac{4z^2(z-1)^2}{(2z-1)^2}$ Find all branching points and their degrees. If my calculation is correct I got 4 branching points and in ...
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Jordan curve and Conformal maps

Let $\mathbb D$ be the unitary open disk, $D$ a bounded domain(open and connected) with boundary a jordan curve and $f$ a conformal map from $\mathbb D$ to $D$, is it true that we can extend $f$ as an ...
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Constructing a homotopy of nonzero holomorphic functions using local homotopies

I'll denote by $\mathbb{C}^*$ the punctured complex plane $\mathbb{C} \setminus \{0\}$. Say that I've got some open cover $\{V_j\}_{j \in J}$ of the closed unit interval $[0,1]$, and continuous ...
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conformal coordinates on a Riemann surface

Let $\Sigma$ be a Riemann surface with complex structure $j$ and a volume form $dvol_\Sigma$. I read somewhere that one can take the so-called 'conformal coordinates' $z=s+it$ so that $j\partial_s = \...
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Applications of Riemann surfaces in engineering or physics

I know that the maximal analytic continuation of a holomorphic function is an example of Riemann surfaces but don't know what it is used for. What can we do with this surface?
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Showing algebraic dependence of meromorphic functions on a compact Riemann surface

I have been given the following question to do: Let $f,g$ be meromorphic functions on a compact Riemann Surface $R$. Show that there is some polynomial such that $P(f,g) = 0$ (i.e. show that any two ...
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1answer
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Computation of $L(p+q+r)$ on a smooth projective curve

Let $X$ be a smooth projective curve in $\mathbb{P}^2(\mathbb{C})$ of degree $4$ and $p,q,r \in X$. What's $L(p+q+r)$? With a standard computation, the genus of $X$ is $3$, so applying Riemann-Roch ...
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Showing that an elliptic function has no poles

Let $\Lambda = \{m \omega_1+n\omega_2; m,n \in \mathbb{Z}\}$ with $\omega_i \in \mathbb{C}$ with $\omega_2/\omega_1 \notin \mathbb{R}$ be a lattice. Define the Weierstrass $\mathscr{P}$ function on ...
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Weierstrass $\wp$-function defines a map from the torus to an elliptic curve. Why is it injective?

For $L$ a lattice in $\mathbb C$, the Weierstrass $\wp$-function is the meromorphic function $$\wp(z) = \frac{1}{z^2} + \sum\limits_{0 \neq \lambda \in L}\frac{1}{(z-\lambda)^2} - \frac{1}{\lambda^2}$...
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1answer
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Symplectomorphisms of a Riemann surface

Let $X$ be a compact Riemann surface (one dimensional complex manifold) of genus $g > 1$, fix a non-degenerate two form $\omega$ on $X$, it is automatically closed by dimension reasons, so it is a ...
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Real world applications of Riemann surfaces of holomorphic functions [closed]

The maximal analytic continuation of a holomorphic function is an example of Riemann surfaces. What is it used for? Please edit the question to limit it to a specific problem with enough detail ...
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Hurwitz' Construction of Riemann Surfaces, but $S_1S_2\cdots S_w\neq 1$

Hurwitz' construction of Riemann surfaces (see e.g. here), asks for all permutations $S_k$ of the copies of the cutted complex planes $E^*$ at the branching points $a_k$, to fulfill: $$ S_1S_2S_3\...
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Smooth curves of odd genus

Let $C$ be a smooth curve of genus $g$ and $J_C$ its intermediate Jacobian. Recall that $J_C$ is a ppav of dimension $g$. Fixing a point $p\in C$, one can define the Abel-Jacobi map $$a\colon C\...
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Integration of a (0,1)-form on the boundary of a Riemann surface

In Simon Donaldson's book, he says that for any (0,1)-form $\theta$ on a compact connected Riemann surface $X$, the integral of $\partial\theta$ over $X$ is zero by Stokes' theorem - but that seems ...
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1answer
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Globally Generated Vector Bundle on a Riemann surface

This is a very vague question, in fact not really a question at all more of a search. I am studying some vector bundle theory on Riemann surfaces and would just like some non-trivial example of ...
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Cutting out a Riemann surface inside its Jacobian variety

After choosing a base point $P_{0}$ in a compact Riemann surface $X$ of genus $g$, the Abel-Jacobi map gives an embedding of $X$ into its Jacobian variety $Jac(X).$ This map can also be extended to ...
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Any two dimensional orientable manifold is complex [duplicate]

Let $M$ be a $2$-dimensional orientable manifold. Is $M$ a Riemann surface? If it is true, how can I show it? Thank you very much.
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Riemann mapping theorem with pathological boundary

From wikipedia: In complex analysis, the Riemann mapping theorem states that if $U$ is a non-empty simply connected open subset of the complex number plane $\mathbb{C}$ which is not all of $\mathbb{C}$...
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Number of branch points for a projection to $\Bbb{CP}^1$

Based on a helpful response to my previous post which offered advice on how to split the image of a map into affine and non-affine components, I've come up with a solution to the following problem. ...
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1answer
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Branch points of a projection to $[X:Y] = \Bbb{CP}^1$ using homogeneous coordinates

I have a question that I've been stuck on for some time now, and although I'm able to understand similar questions I'm stuck on this particular type of projection to the complex projective line. ...
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1answer
132 views

A guide to Algebraic Geometry

I have completed one semester course on Commutative Algebra and Riemann Surfaces, and currently I am trying to read Algebraic Geometry. While reading from different books I feel that I must need a ...
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Rational expressions in the function $f$

By definition $\mathbb{Q}(\sqrt2):=\{a+b\sqrt2\,\, |\,\, a, b \in \mathbb{Q}\}$. Ok. No problem with that! I read that for $f$ a nonconstant meromorphic function on a Riemann surface $X$, $\mathbb{...
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Homology of Fermat curve

Let $C(n):X^n+Y^n=Z^n$ be the plane projective Fermat curve of degree $n$ over $\mathbb{C}$. Shorter version of the question: How can I describe explicit representatives for a basis for the singular ...
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If two minimal surfaces equal to neighborhood then they are equal

Does anyone know if there is any result that says "if two minimal surfaces equal to neighborhood then they are equal"? My teacher said that you think the result is true, but I do not find this result ...
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Type I pencils on Riemann surfaces (complex curves)

I was reading the paper https://www.researchgate.net/publication/265872523_On_the_number_of_pencils_of_minimal_degree_on_curves_with_small_gonality and found the statement in the second paragraph to ...
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Hyperelliptic curves express by branch points

In a book of algebraic geometry, the author says that equation $$ y^2=x(x^4-1) $$ defined a curve $S$ of genus $2$. My problem is this: Because the curve has genus 2, it can be expressed as follows $...
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Degree of a canonical divisor on a compact Riemann surface

I'm reading Jürgen Jost's "Compact Riemann Surfaces" Springer textbook 3rd ed (a very good read!). Jost defines the divisor of a meromorphic differential $\eta$ on a compact Riemann surface by \...
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1answer
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A basis for the holomorphic differentials of a hyper-elliptic Riemann surface

Bumped in to this problem while trying to understand hyper-elliptic Riemann surfaces, and being a bit new to the subject of Riemann surfaces, was not that much confident on a couple of things. ...
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Non-trivial Extensions of Sheaves of $\mathcal{O}$-Modules

I am working on Problem XI.4. E of Miranda’s book “Algebraic Curves and Riemann Surfaces”. Let $X=\mathbb{P}^1, p = \infty, \mathcal{O}[n]=\mathcal{O}[n\cdot p]$. Write down a nontrivial extension ...
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Functions on hyperelliptic curves

I'm working with the hyperelliptic curve of genus $2$ $$ \mathcal{C} \colon Y^2-X^5-1=0$$ over the complex numbers, so the function field is $$F:= \frac{\mathbb{C}(X,Y)}{(Y^2-X^5-1)}.$$ I ...
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Is this exact: $0\to\mathcal{O}^{hol}\to\mathcal{O}(p)\to \Bbb{C}_p,$

Let $X$ be a Riemann surface and $p\in M$ some point. Let $\mathcal{O}(p)=\mathcal{O}((-p))(U)=\{f\in \mathcal{O}^{hol}(U)\mid f\text{ has a zero of order atleast 1 at p}\}$ I.e. we have the divisor ...
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Uniqueness of Algebraic Belyi Pairs from Dessin

From a Dessin d'Enfant on a surfaces $X$ we get a meromorphic function $f:X\rightarrow\mathbb{CP}^{1}$ such that the only critical values are $\{0,1,\infty\}$ i.e a Belyi pair $(X,f)$. Belyi's ...
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$\Bbb C/(1+ib)\Bbb Z$ is an elliptic curve

I have to prove that $\Bbb C/(1+ib)\Bbb Z$ is an elliptic curve, where $b\in\{1,2,3\}$. The problem is that the teacher was a bit broad giving the definition of elliptic curve. He presented them as ...
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Gap Numbers for the Canonical Linear Systems

I am working on an exercise in Miranda’s book “Algebraic Curves and Riemann Surfaces” [Chapter VII.4.S]. Let $X$ be a nonhyperelliptic curves of genus $g\ge 3$. Denote by $G_p(|K|)=\{n_1<\dots<...
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138 views

Are concrete Riemann surfaces Riemann domains over $\mathbb C$?

While reading about Riemann surfaces I stumbled upon the following two definitions. A Riemann surface of a complete analytic function is an example of both definitions while Abstract Riemann Surface ...
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What is the genus of $y=x^n$?

I would like to say that the Riemann surface S defined by $y=x^n$ has genus 0 but I don't know if I make a mistake anywhere, could you check if I'm right? First option: $Y: S \to \mathbb{P}^1$, $(x, ...
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1answer
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Is there a function with prescribed zeros and poles on an elliptic curve?

Let $T$ be the complex tore from the lattice $(1, \tau)$ where $im(\tau)>0$. How to prove the existence of a meromorphic function on $T$, with divisor $(0) + (\frac{1}{2}) - 2 (\frac{\tau}{2})$ ? (...
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The function field of a modular curve

I write $X(N)$ for (the compactification of) the quotient $$\mathbb{H}\ / \ \text{PSL}(2,N).$$ For instance, $X(1)=\mathbb{P}^1$ is the compactification of $\mathbb{H}\ / \text{SL}(2,\mathbb{Z})$ and ...