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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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Help with gluing together surfaces of infinite genus

Let $S=\bigwedge_{m\in\mathbb{Z}}\mathbb{S}^{1}$ denote the wedge sum of countably infinitely many circles—that is, the topological space obtained by removing countably infinitely many distinct points ...
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Checking Holomorphicity/Meromorphicity of a Differential 1-Form On A Curve

Let $f\left(x,y\right)$ be an elementary function of $x$ and $y$. Examples: $$4x^{3}-ax-b-y^{2}$$ $$a^{x}-y^{2}+x^{3}y-1$$ $$x^{3}-3xy+y^{3}$$ $$\frac{\sin^{2}x}{x^{2}+\sin^{2}y}-\frac{1}{3}$$ Let ...
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Finding a condition on a polynomial $P(z,w)$ for an affine curve to extend to a smooth algebraic curve in $(\mathbb{CP}^1)^2$

I have the following problem: Let $\alpha:\mathbb{CP}^1\to\mathbb{CP}^1$ be the map $z\mapsto z^{-1}$. Then the group $G=\mathbb{Z}/2\times\mathbb{Z}/2$ acts on $\mathbb{CP}^1\times\mathbb{CP}^...
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Computing global monodromy of branched covering of Riemann sphere

Given an irreducible polynomial $F(X,Y)\in\mathbb{C}[X,Y]$, there is an associated Riemann surface $S$ obtained from the zero set of $F$. Projection onto the second coordinate gives a map $\pi:S\to \...
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Rigidity of isometries of finite covers of Riemann surfaces

Let $\Theta$ and $\Sigma$ be compact Riemann surfaces with $\widetilde\Theta=\widetilde\Sigma=\mathbb{H}^2$ (so $\theta$ and $\Sigma$ are quotients of $\mathbb{H}^2$ by Fuchsian groups) such that $\...
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How could I draw (or at least imagine) a Riemann surface obtained by quotient?

In the study of Riemann surfaces, we can get lots of cool examples by identifying parallel sides of polygons. For example, by identifying the parallel sides of a square we obtain a torus. This is easy ...
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Global sections of square root line bundle

Let $C$ be a smooth curve in $\mathbb{P}^2$ over field $\mathbb{C}$. Suppose that I have a very ample line bundle $L$ on $C$ of even degree. Then $L$ has $2^{2g}$ square roots in $Pic\ C$. These are ...
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About branch points of a holomorphic map

Let $F:X \to Y$ be a holomorphic map between Riemann surfaces. $q \in Y$ is a branch point if it is the image of a ramification point. How to prove that the set of branch points is a discrete subset ...
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$L/K$ is finite separable field extension with $O_v$ valuation ring of $K$ w.r.t valuation $v$. Integral closure of $O_v$ in $L$ is DVR?

Let $L/K$ be a finite separable field extension with $O_v$ valuation ring of $K$. Here I will not assume $O_v$ complete and similarly for $K$ as well.(i.e. There is a completion $\hat{O}_v$ of $O_v$ ...
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Integrate a top form over a surface without partition of unity

Suppose we are given a compact Riemann surface $M$, an open cover $\mathscr{U}=\{U_1,U_2,\dots\}$ of $M$, charts $\{(U_1,\phi_1),(U_2,\phi_2),\dots\}$, holomorphic coordinates, $\phi_m:p\in U_m\mapsto ...
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Reference request: introductory level book for Riemann surfaces

I have been told by different people that probably no subject can unify algebra, analysis and geometry better than Riemann surfaces. Regardless of how true it is, I'm looking for a textbook that ...
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Is there a term for Gaussian curvature based on geodesic curvature?

Gaussian curvature is usually defined as the product of maximal and minimal normal curvatures of curves through a point on a surface. What if we use geodesic curvatures of the ambient space instead? ...
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Which smoothness properties are preserved under ramified covering maps?

Setting. Let $M$ be a Riemann surface and $\Gamma$ a discrete group that acts properly discontinuously on $M$ by holomorphic maps. It is well known that each $x \in M$ has a finite stabilizer, that ...
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Is the genus of biholomorphic Riemann surfaces the same?

Is the statement above true for $X \cong_{bihol} Y$? I would say yes, since I can transform any holomorphic function on a open set in $X$ to one in $Y$ and vice versa.
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Weak solutions for divisors

I have a question on the following definition in the Forster: I don't get the part where it says "Clearly a weak wolution $f$ is a proper, i.e., meromorphic function, solution precisely if $f$ is ...
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Branch points on Riemann surfaces

I have a question on the exact definition of branch points (on Riemann surfaces) I have the following definition:: Let $f:X \rightarrow Y$ be a holomorphic function between Riemann surfaces. $x \in X$...
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Every line bundle on a complex algebraic curve has a meromorphic section

Every line bundle $L$ on a complex algebraic curve $X$ is of the form $\mathcal{O}(D)$, where $D$ is some divisor on $X$. This means $L$ has at least one nonzero meromorphic global section, i.e. $$H^...
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Exact sequence of sheafs

I am not quite sure whether my solution for this exercise is correct, more precisely the part $d: \mathcal{O} \rightarrow \Omega$. My solution: each holomorphic 1-form $\omega$ is locally exact (...
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Holomorphic coordinates on Riemann surfaces

I have a big problem understanding the meaning of holomorphic coordinates on Riemann surfaces, especially in relation to 1-forms. Holomorphic coordinates on a Riemann surface $X$ is an open set $U \...
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Question on divisors of meromorphic functions on Riemann surfaces

I have a question on the following definition I don't quite get why the definition says " if f is identically zero in a neighborhood of a", I mean there's nothing wrong with that I just don't get why ...
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Surface covered by hyperbolic plane admits a hyperbolic metric

Let $S$ be a surface. Is it true that if $S$ is covered by the hyperbolic plane (or a subset thereof) then it admits a Riemannian metric of constant negative curvature? How does the metric (or ...
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Quotient topology clarification, what happens if we glue together a point on a boundary with a point in the middle

When I first learned about the quotient topology $X/\sim$ on a topological space $X$ the quotient space was defined to be $X$ with all the points identified by $\sim$ glued together. So if $X = [0,2]$ ...
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Miranda - Classification of Curves of Genus 4

I'm trying to understand pag. 207 of Miranda's book "Algebraic Curves and Riemann Surfaces" where it is explained which homogenous polynomial equations define a Riemann Surface of Genus Four embedded ...
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What does the Color and height of a Riemann surface represent

The title says it, but I think Color may represent angle and in part because magnitude, as height, is unbounded. Its frustrating that I haven't been able to find this from searching the web. Any ...
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Analytic functions on the cylinder

It is known that the cylinder $X:=S^1\times \mathbb R$ is a complex manifold. Does $X$ have analytic functions. i.e, if $f:X\to \mathbb C$ is analytic then is $f$ necessarily constant?
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What's going on when we compute $d(\gamma(z)) = \frac{1}{|cz+d|^2}dz$, where $\gamma \in \operatorname{SL}_2(\mathbb Z)$

Let $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb Z)$. Consider the space $\Omega^1(\mathbb H)$ of smooth complex $1$-forms on $\mathbb H$. These ...
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Ordinary r-fold point on the dual curve

Let $C$ a projective curve in $\mathbb{P}^2(\mathbb{C})$. We say a line $L \subset \mathbb{P}^2$ is mulltiple tangent of $C$ if there are $P_1, \dots P_k$ points on $C$ such that $L$ is the tangent of ...
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Meromorphic Functions on Riemann Surfaces

My question refers to a step in the proof of Prop. 3.3.5 Szamuely and Tamásin's "Galois groups and fundamental groups": Here the statement and Thm 3.3.3 & lemma 3.3.6: The main ingredients for ...
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Genus of a smooth projective curve

I was trying to prove that the genus of a smooth projective complex curve $F=0$ of degree $d$ is $(d-1)(d-2)/2$. My attempt was to take the standard projection $$\pi:\mathbb{P}^2 \to \mathbb{P}^1$$ $$...
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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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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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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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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}$...