Questions tagged [riemann-surfaces]

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

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Algebraic Curves and Riemann Surfaces: self study [closed]

For my bachelorthesis as a third year student in my bachelorsdegree i have to study algebraic geometry with the book Algebraic Curves and Riemann surfaces by Rick Miranda. Does anybody have tips or ...
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There exist holomorphic maps from a torus $T$ to the sphere $S$ of all degree $\geq 2$

While reading through "Restrictions on harmonic maps of surfaces" by J. Eells and J. C. Wood I came across the lines: We know that there are holomorphic maps from a torus $T$ to the sphere $...
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Why does a holomorphic degree $1$ map imply that the Jacobian is strictly positive?

I am currently reading "Lectures on Harmonic Maps" by R. Schoen and S. T. Yau and have problems understanding one step in the proof of a Corollary on p. 13. Corollary: Suppose $N = S^2$, ...
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Clarification of the proof of the form $\partial_{1,0}$ in Restrictions on harmonic maps of surfaces by J. Eells and J. C. Wood

I am currently reading "Restrictions on harmonic maps of surfaces" by J. Eells and J. C. Wood and have trouble understanding the proof of the lemma on p. 265. The paper states: Let $X$ and $...
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The existence of a larger compact set containing all minimising geodesics of a compact subset

On a hyperbolic surface, is a compact set $K$ necessarily contained in a larger compact set $K’$ so that any two points of $K$ can be joined by a minimising geodesic within $K’$? The question came out ...
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How to prove that the following composition of mappings from Riemann sphere to Riemann sphere is a möbius transformation?

I’m reading O.Foster’s «Lectures on Riemann surfaces» and trying to solve the edited version of exercise 1.3. O. Foster’s «Lectures on Riemann surfaces» Instead of proving that the mapping is ...
salmonella 's user avatar
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Riemann Surface of $z^3$

I'm new to the topic of Riemann surfaces and complex analysis, and I tried to compute the Riemann surface of $z^3$ on Wolfram Alpha but couldn't because its not a multivalued function. How can I ...
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Seeking References for Visual Verification of Dirichlet Regions in Triangle Groups

I'm self-studying the impact of base point placement on the shape of Dirichlet regions within triangle groups. Theoretical findings suggest the region is a quadrilateral when the base point lies on a ...
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Holomorphic 1-forms as a subspace of the first de Rham cohomology group of a surface

If we fix a complex structure $c$ on a closed oriented surface $S$, then the space of holomorphic 1-forms $\Omega^1(S,c)$ has complex dimension equal to the genus $g$ of $S$ and - since they are all ...
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What is the branch cut of composite of multivalued complex function

I have the following function where I want to identify the Riemann surface. $$ f(z)=\log\left(\sqrt{z^2+1}\right). \quad\quad\quad (1) $$ The square root function has a Riemann surface $R_{SR}$ with ...
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Considerations for branches of the square root on the complex plane: a down-to-earth approach

Disclaimer: I'm not a native English speaker and I'm not familiar with the English terms for mathematical objects. I will gladly edit my post based on suggestions concerning grammar mistakes and the ...
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Detail in the proof of page 300, proposition IV.2.1 of Hartshorne's Algebraic Geometry

I have seen that a question on this specific problem has been already asked and answered here A map is injective if it is nonzero at the generic point? Just to give a context we have a finite ...
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Understand the geodesic, $\gamma$, induced by an abelian form $\omega$ and their integral

in my phd thesis, informaly I use this result: Let $(A_i,\omega_i)$, $i=1,2$, where $A_i$ is an annulus and $\omega_i$ is an abelian differential form with a translation structure on $A_i$ that ...
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Volume of strata of abelian differentials

I am reading this paper https://arxiv.org/pdf/math/0006171.pdf on the volume of strata in the moduli space of abelian differentials, and just have a small clarificatory question. On page 4, the volume ...
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Non-compact Riemann surfaces with zero Euler characteristics

The question arises because of a statement in John H.Hubbard's book "Teichmuller theory and Applications to Geometry, Topology and Dynamics, vol 1" Page 130, Chapter 4.4.It says that: The ...
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Definition of the energy functional for harmonic maps between compact Riemann surfaces

I am currently reading "Restrictions on harmonic maps of surfaces" by J. Eells and J. C. Wood and don't understand how they get to the definition of the energy functional. The first ...
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Riemann Roch Canonical Divisor - show the divisor of a differential form does not depend on choice of local coordinates

I'm studying algebraic geometry and we've just been introduced to differential forms. The book (Algebraic Geometry by Garrity .et al) gives us this exercise: Exercise 3.6.33: Suppose $x$ and $y$ are ...
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Dimension of the space of holomorphic quadratic differentials

The Riemann-Roch theorem for a holomorphic line bundle $L$ over a Riemann surface $X$ states that $$h^{0}(X,L)-h^{0}(X,L^{-1}\otimes K)=\deg(L)+1-g.$$ For $L=K^2$, the bundle of holomorphic quadratic ...
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Riemann surface of the hypergeometric function

The hypergeometric function $_2F_1(a,b,c,z)$ has a branch cut extending from $z=1$ to $z=\infty$. Does this define an infinite-sheeted Riemann surface (like that for $\log{z}$) or one with a finite ...
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Help: Exercises about Types of Mobius transformations

I'm working on the following exercise of Jones, Gareth A., and David Singerman. Complex functions: an algebraic and geometric viewpoint. Cambridge university press, 1987. Exercise: 2D Let $ S $ be a ...
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Transformation for isothermal coordinates of energy of harmonic maps between Riemann surfaces

I am currently reading section "3.6 Maps Between Surfaces. The Energy Integral. Definition and Simple Properties of Harmonic Maps" in "Compact Riemann surfaces" by Jürgen Jost and ...
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Exercise about beahviour of Mobius transformation

I'm working on the following exercise 2A of ...
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Is non-constant holomorphic map $f:\mathbb P^n\to \mathbb P^n$ surjective?

Let $\mathbb P^n$ denote the $n$-dimensional complex projective space. From Forster's book Lectures on Riemann surfaces p.11, Theorem 2.7, we know any non-constant holomorphic map $f:\mathbb P^1\to \...
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trace map of jacobians of Riemann surface

I'am reading 'A first course in modular form'. On page 221 Given two compact riemann surface and holomophic map $h$ : $X \to Y$. Suppose $h$ is a surjection of finite degree $d$, that $h$ is locally $...
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Quotient by sheaf of ideals associated to effective divisor

Let $X$ be a Riemann surface, and $D$ a divisor (we define a divisor to be a map $D:X \to \mathbb Z$ with closed and discrete support). Define the sheaf of $\mathcal O_X$-modules $\mathcal O_X(D)$ ...
runyoucleverboy's user avatar
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Euler characteristic of not necessarily proper curve

Given a field $k$ of characteristic $0$ and a (not necessarily proper) smooth curve $C$ over $k$, i.e. a geometrically connected, geometrically reduced, smooth, separated scheme of finite type over $k$...
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second betti number of surface groups

Let $\Gamma$ be a surface group. Denote $b^i(\Gamma):=\operatorname{dim}_{\Bbb Z}H^i(\Gamma, \Bbb Z) $. Using group cohomology for the trivial action of $\Gamma$ on $\Bbb Z$, we can see that $H^0(\...
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$H_1(X, \mathbb{Z}) \to \Omega(X)^*$ is a lattice

Let $X$ be a compact connected Riemann surface. Letting $\Omega^1(X)$ be the set of holomorphic $1$-forms on $X$, we have a natural map $\Omega^1(X) \to H^1(X, \mathbb{C}) =H_1(X, \mathbb{C})^*$ given ...
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$\mathcal O_X^n$ is a coherent sheaf for each $n$

I am reading a proof on these notes about $\mathcal O_X$ modules but there are some details I cannot understand. Proposition: The sheaf $\mathcal O_X^n$ is coherent Obviously, it is of finite type. ...
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2D tiling with regular pentagons (and generalizations)

We know that the only regular polygons that can tile the 2D plane are triangles, squares, and hexagons. One way of seeing this is that, if we try to place regular pentagons (for instance) around a ...
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Show that $\frac{1}{2\pi i}\int_{\gamma_p}z\frac{h'(z)}{h(z)}dz$ is in $\mathbb{Z}+\mathbb{Z}\tau$

This is the problem IV.3.F I found at page 127 of the book "Algebraic Curves and Riemann Surfaces" of Rick Miranda. Let $\tau \in \mathbb{C}$ such that $Im(\tau)>0$ and define the lattice ...
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Is a module over a (c-)soft sheaf (c-)soft?

In the context of Riemann Surfaces, or $\mathbb C$ manifolds (or even more generally) I wonder if a module over a c-soft sheaf $\mathcal S$ is itself c-soft which I believe implies soft if the space ...
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Vertex locations of Dessin w/h Circle packing in Discrete Dessin d'Enfant

I'm reading the following paper. Bowers, Philip L., and Kenneth Stephenson. Uniformizing Dessins and BelyiMaps via Circle Packing. Vol. 170. No. 805. American Mathematical Soc., 2004. In the paper, ...
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0 cohomology of Riemann sphere

Consider the Riemann sphere $S^2$, together with the sheaf $\Omega_{S^2}$ which is defined to be $\Omega_{S^2}:=\operatorname{Ker}\overline{\partial}$ with $\overline{\partial}\colon\mathcal A^{1,0}\...
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Why is this construction of an affine curve not uniformization?

I'm learning the Shimura curve. When I reading the note of Pete L. Clark (SC2-Fuchsian.pdf (uga.edu)), I was stuck on a thinking question. First, there is a theorem (Uniformization Theorem) about ...
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Description of Riemann surface of polynomial inverse

My question is about page 4 of the pdf of the following paper , one does $\textbf{not}$ need to read pages 1-3 of the paper to understand my question ( the $\textbf{only}$ part that needs to be read ...
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Möbius transformation taking line segment between two complex numbers to the segment $[-1,1]$

I am reading Kodaira's Complex Analysis, and trying to understand his proof of the uniformization theorem for simply connected Riemann surfaces. Due to what I assume is translation errors, this book (...
John Cavanaugh's user avatar
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1 answer
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Evaluate function at $\infty$ in $\mathbb{C}^\infty$

What is the definition of $f(\infty)$ in $\mathbb{C}^\infty$? Supposing that I have a function like $f(z) = \frac{az+b}{cz+d}$, how I can evaluate the function at $z=\infty$? Because $f(\infty) = \...
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Smooth part of an Affine Plane Curve defined by irreducible polynomial is a Riemann Surface

Miranda states the following. Given an irreducible polynomial in $\mathbb{C}[x,y]$, the singular points of its locus of roots $X$ forms a finite set. If we delete these points of $X$ the the ...
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About Galois branched covers of Riemann Surfaces

I am reading Giraud's notes (sorry they are in French) on Riemann surfaces and I have some questions about the end of the fifth section. To begin with, a branched cover $f: X \to Y$ of degree $n$ (i.e....
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Are rational functions on the Riemann sphere invariantly characterized? If so, what about general Riemann surfaces?

Consider the Riemann sphere $\hat{\mathbb{C}} := \mathbb{C} \cup \{ \infty \}$. One often considers polynomials on the Riemann sphere, i.e. maps \begin{equation} P: \hat{\mathbb{C}} \rightarrow \hat{\...
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If the degree of a divisor on a Riemann surface is $\deg(D) \geq 2g$, then $L(D-(p)) \subsetneq L(D)$ for any point $p$

Let $S$ be a compact connected Riemann surface, $D$ a divisor on $S$, and $p\in S$ a point. I want to show that if $\deg(D) \geq 2g$ (where $g$ is the genus of $S$), then we have a strict inclusion ...
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Checking that a differential form is holomorphic on an algebraic curve and that it extends to a holomorphic form on its compactification

Consider the following algebraic curve: $$C=\left\{ (x,y) \in \mathbb{C}^2 | x^3+y^3+3\lambda xy + 1=0 \right\} $$ …where $\lambda^3 \neq -1$. I think this is called the Hesse pencil, and it’s a ...
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$\bar{S}=\left\{[z,w,t]\in \mathbb{C}P^2 : w^2t^{k-2}=(z-a_1t)\ldots (z-a_kt)\right\}$ is a Riemann surface iff $k \leq 3$

I know that if $P \in \mathbb{C}[z]$ is a monic polynomial with distinct roots, $$P(z)=(z-a_1) \ldots (z-a_k),$$ then the set $$S= \left\{ (z,w)\in \mathbb{C}^2 : w^2=P(z) \right\}$$ is a connected ...
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Formula for gaussian curvature of a holomorphic curve in complex 2-space

Define h : ℂ → ℂ2 via     h(z) = (f(z), g(z)), where f, g : ℂ → ℂ are analytic functions such that the complex velocity h'(z) = (f'(z), g'(z)) ∈ ℂ2 is never equal to (0,0). Suppose further that h is ...
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Curl of normal unit vector of a smooth and closed surface?

Let's say we have a curvilinear coordinate system $(\rho,\theta,\zeta)$. Also, let's say we have a smooth and closed surface $\Gamma$ parameterized as $\Gamma: \mathbf{S}(\rho(\theta,\zeta),\theta,\...
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Prove Stereographic projection is a homeomorphism from Sphere except North Pole to Complex Plane

I'm self-studying exercises of Jones in Reference and I'm stuck at 1D which is the following; 1D My question The (standard) proof that I know (for instance Q4 of this note) does not seem to involve ...
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Why is conformal mapping important for Riemann Sphere

The mapping of the complex plane to the Riemann sphere is conformal. As a result, many sources contend that it's this property of conformal that makes the concept of Riemann Sphere non-trivial. ...
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Calculating the difference between $f(x+iy)$ and $f(x-iy)$ when $y\to0+$

I have asked the question about the limit of two functions as they're approaching the real axis before. Today I considered the function $f(z)=\sqrt{1-z^2}$, trying to calculate the difference in ...
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Why is the set of points in $\mathbb{C}P^2$ where a non-degenerated quadratic form vanishes biholomorphic to $\mathbb{C}P^1$?

Let $Q: \mathbb{C}^3 \rightarrow \mathbb{C}$ a non-degenerated quadratic form and let $S=\{[z_1,z_2,z_3] \in \mathbb{C}P^2 : Q(z_1, z_2, z_3) \}$. My lecture notes on Riemann surfaces mention that $S$ ...
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