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

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Question on Forster's proof of the residue theorem

I have some questions about the proof of the residue theorem in Lectures on Riemann Surfaces by Otto Forster. The Residue Theorem. Suppose $X$ is a compact Riemann surface and $a_1,\cdots,a_n$ are ...
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64 views

Complex structures on Riemann surfaces

Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq H^0(M;K^2)$. Considering $\alpha$ as a map $T^{0,1} M \to T^{1,0} M$, the bundle $$ \{v + \alpha(v) \mid v \in T^{0,1} M\} ...
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82 views

When functional equations determine a unique global analytic function?

I'm learning about analytic continuations and global analytic functions which were seen to be connected components of the sheaf of analytic germs. Sometimes we get problem sets in which we are ...
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28 views

Mobius transformation of Algebraic curve

I am working on the uniformization of algebraic curve problem. Currently, my adviser gave me a question about build a Mobius transformation between algebraic curves, and then lift it to the Rimeann ...
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1answer
35 views

How does Ahlfors define derivative on a Riemann Surface?

I'm reading a passage in Ahlfors (3rd Edition page 298) and he says the following: He has previously defined $G_0(f)$ to be the connected component of any germ generated by $f$. Then he wants to ...
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1answer
23 views

Covering of Projective line

This is an exercise given during the course in Riemann Surfaces that I attended this year. Let $X$ be a compact Riemann Surface that is a degree $3$ cover of $\mathbb{P}^1(\mathbb{C})$ given by ...
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1answer
69 views

Holomorphic line bundle with degree zero

I'm studying algebraic geometry and I need some help to understand the Riemann-Roch theorem. Let us consider a holomorphic line bundle $\xi$ over a Riemann surface $X$. The unique invariant of a ...
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36 views

The number of holomorphic coverings (with given degree) of the punctured sphere is finite.

I'm looking for a proof of the following theorem: Fix a finite set $B=\{y_1,\ldots,y_k\}\subseteq \mathbb P^1(\mathbb C)$, then there is only a finite number of isomoprhism classes of ...
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Any no-zero homomorphism of holomorphic vector bundles over a compact Riemann surface factors through a maximal rank homomorphism.

I was reading the paper "Stable and Unitary vector bundles on a compact surface" by Narashiman and Seshadri. I quote from the paper- Can someone please explain how does any non-zero homomorphism ...
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51 views

How can we compute the order of 1-form on Riemann surfaces

Let X be a hyperellictic curve defined by $y^2=h(x)$. Let $\pi:X\rightarrow\mathbb{P}^1$ be the double covering map seding $(x,y)$ to $x$. Let $\omega=\pi^*(dx/h(x))$. How can we compute the orders of ...
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1-form on Riemann Surface

Good evening, I can not prove the following result: Let $\omega $ be a meromorphic 1-form on $ \mathbb {C} _ {\infty} = \mathbb {C} \cup \infty $ such that $ \omega_{|\mathbb{C}} = f (z) dz $. Show ...
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Computing geodesic distances from structural data

I am attempting to compute geodesic distances on manifolds where structural data have been sparsely sampled. First, off I am not well versed in the mathematics of differential geometry but I do have ...
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30 views

Simple Branched covering over sphere.

A simple branched covering is a branched covering with branching points of degree at most 2, in some context, it is also required to have at most one branching point in each fiber. My question is ...
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56 views

de Rham cohomology group of $\mathbb{CP}^1$

Prove the de Rham cohomology group of the projective space $H^0(\mathbb{CP}^1, \mathbb{C})$ is isomorphic to $\mathbb{C}$ and $H^1(\mathbb{CP}^1, \mathbb{C})=0$.
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universal covering of punctured plane and Poincaré metric

I want to prove the following result: Let $\Omega$ be the domain $\mathbb{C}\backslash \{a_1,a_2,a_3,a_4\}$. Its universal covering is the unit disk, and the standard Poincaré metric is pulled back to ...
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1answer
22 views

Real part of integral over holomorphic 1-form is zero implies the one-form is zero

Suppose we are working on a Riemann Surface $X$, assume of genus g $\geq$ 1. Let $\omega$ be a holomorphic 1-form on $X$, with $$ \textrm{Re} \int_\gamma \omega = 0$$ for every closed contour ...
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1answer
31 views

Base-point-free linear systems (elementary?) property

I'm having troubles solving exercise K on page 167 of the book "Algebraic curves and Riemann surfaces" of Miranda. The question is the following one : Let Q be a base-point-free linear system, let ...
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37 views

Is the image of the Riemann sphere a Riemann surface?

I am looking for conditions on $f$ such that $f:S^2 \to \mathbb C$ is such that the image, $f(S^2)$ is a Riemann surface. Must $f$ be analytic, or something stronger? Or are there no simple conditions ...
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1answer
32 views

Hyperbolic (and related) structures on open unit disk

I am facing some confusion about different structures on the open unit disk $D:=\{ z \in \mathbb{C}, |z|<1 \}$. By Riemann Mapping Theorem we know there is just one complex structure on $D$, up to ...
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27 views

Function fields for genus 2 curves

Let's say you were given a genus one algebraic curve by the equation $y^2 = (x-a)(x-b)(x-c)$ and you wanted to parametrize it. We could go ahead and convert it to Weierstrass form: $y^2 = 4t^3 - g_2t ...
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96 views

Metric Tensors and its Taylor Expansion in Normal Coordinates

With metric tensors of the unit sphere in normal coordinates, their Taylor series for $p\in S$ near the north pole $N$ can be written as follows. $$g_{rr}(p) \equiv 1; g_{r\theta}(p) = g_{\theta ...
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57 views

Divisors and holomorphic map between a compact Riemann surface and a torus

Let $X$ be a compact Riemann surface (or more generally a compact manifold?) and let $\mathbb{C}^a/\Lambda$ be a complex torus. Suppose we have a holomorphic map $$g: X\to\mathbb{C}^a/\Lambda.$$ By ...
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60 views

Orientation on Riemann surfaces

$\mathcal{X}$ is a Riemann surface and $\mathcal{E}^{(2)}(\mathcal{X})$ is the $\mathbb{C}$-Vector space of all differentiable $2$-forms on $\mathcal{X}$. I want to define the orientation of ...
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affine curve. analysis

Can anyone explain what is written in book? $\Gamma(w,E)\equiv w^{2}-E^{n}+\sum_{i>0,j\ge 0, ni+2j<2n} g_{ij}w^{i}E^{j}=0, $ where $g_{1,0}$ doesn't equal to zero. "At infinity it compactified ...
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some intuition about the degree of a map

Consider a map $$ f: \Sigma \to X/\sigma,$$ where $\Sigma=\Sigma_g/\Omega$ is a quotient of a Riemann surface by an antiholomorphic involution, $\sigma:X\to X$ is an antiholomorphic involution of some ...
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1answer
61 views

References for a standard result about coverings of Riemann surfaces

I my thesis I have to cite the following standard result: Let $Y$ be a compact Riemann surface and let $B\subseteq Y$ be a finite subset. Given a natural number $d$, there are only finitely many ...
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83 views

Why is the projection map proper?

It might be a silly question. I got stuck there. For the context, see S.K.Donaldson's Riemann Surfaces, chapter 4, section 4.2.3. Suppose $P(z,w)=a_0(z)+a_1(z)w+\dotsb+a_n(z)w^n\in\mathbb C[z,w]$ is ...
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83 views

Explicit description of flat connections under pullback on principal bundles over Riemann surfaces

I'm trying to find a proof/reference for a statement that I've seen quoted in some way or the other, but without reference. The setting: let $P\longrightarrow M$ be a flat principal $G$-bundle over ...
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9 views

Short-time representation of variations of metrics on principal bundles via exp?

Let us consider a principal $G$-bundle $P\longrightarrow M$ together with an $H$-reduction $s$, where $H$ is a maximally compact Lie subgroup. As an $H$-reduction, $s\in\Gamma(M,P/H)$, hence we can ...
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Automorphisms of rational curves

Let $X$ be a non-empty open subscheme of $\mathbb P^1_{\mathbb C}$. What is the automorphism group of $X$ in terms of PGL$_n(\mathbb C)$ and the points on the boundary?
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Riemann surface associated with complete analytic function of $(z^2-1)^{1/3}$

I'm trying to define an analytic function on '$\mathbb{C}$' of the form $f(z)=(z^2-1)^{1/3}$, i.e. I first remove two semi-infinite rays $l_1$ and $l_2$, one going from $1$ to $\infty$ along the ...
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20 views

Automorphism group of torus fixing origin

I've got a short question: Suppose that you have some lattice $\Lambda$, say $\Lambda=\mathbb{Z}+\mathbb{Z}i$, and let $T$ be the torus $\mathbb{C}/\Lambda$, coming with the quotient map ...
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1answer
49 views

“Bundle of metrics” on a principal bundle?

I've come across the term "bundle of metrics" on a principal bundle. In particular, my setting is that for $N\longrightarrow M$ a universal cover of a compact Riemann surface, $P\longrightarrow M$ a ...
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25 views

Equivalence of atlases on Riemann surface

I've got a very vague question about Riemann surfaces, more like a meta-question: One of the first things one defines is, obviously, an atlas on a Riemann surface $R$ ($R$ is a connected Hausdorff ...
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1answer
69 views

Galois extension and morphism of curves

Let $\phi: C \rightarrow \mathbb P^1$ a morphism (over a field of characteristic 0) from a rational curve $C$ to $\mathbb P^1$ of degree 3. By the Riemann-Hurwitz formula the degree of the ...
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62 views

Constructing a meromorphic function

I need help with the following problem. "Let $C : y^2 = x^3 − 5x^2 + 6x$ be a cubic curve with the standard group law. Find a meromorphic function on $C$ having the pole of order two at ...
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64 views

Universal property of the Abel map

In the book Algebraic Geometry I edited by Safarevich, the following universal property of the Jacobian variety of an algebraic curve is given page 158 (with no more details): The Abel mapping $a: ...
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33 views

Constructing maps of degree $k$

One of the common constructions one finds when first learning about the (topological) degree of a map is the construction of maps $f_k:S^n\rightarrow S^n$ of degree $k\in\mathbb{Z}$ (i.e. ...
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Fermat Quartic Tiling

I have been reading about the Fermat quartic $Q \subset \mathbb{P}^{2}$, defined in homogeneous coordinates as $X^{4}+Y^{4}+Z^{4}=0$. This is the second most symmetric non-hyperelliptic surface of ...
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2answers
54 views

Topological space underlying this curve

I have to solve this exercise but I have really no clue even how to start with it: Identify the topological space underlying the cubic $Y^2Z=X^2(X-Z)$ in $\mathbb{PR}^2$. How does it fit with the ...
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Triple Cover of the Riemann Sphere

I have the triple branched covering $X$ of $\mathbb{P}^{1}$ defined by $y^{3}=x^{6}-1$. I want to show the following: (i) The canonical embedding $\phi: X \rightarrow \mathbb{P}^{3}$ can be given in ...
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1answer
98 views

Proving the Existence of an Automorphism on $\mathbb{P}^{1}$

I recently came across the following problem while reading: Suppose that a compact Riemann surface $X$ has genus $g>1$. Let $\phi_{i}:X \rightarrow \mathbb{P}^{1}$ for $i=1,2$ be a pair of ...
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Holomorphic functions on algebraic curves

I have been asked to solve the following problem, but I really need some help... How are the holomorphic functions $f:C\to D$, where $C,D$ are nonsingular algebraic curves of genus 1? I know that I ...
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Diffeomorphism between a regular surface and the plane

Do Carmo states that (example 2, page 74) if $\mathbf x: U\subset\mathbb R^2\rightarrow S$ is a parameterization, then $\mathbf x^{-1}: \mathbf x(U)\rightarrow \mathbb R^2$ is differentiable. Why is ...
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1answer
55 views

The relation between principal curvature and curvature tensor?

To me, there are two systems of curvature of a surface, one is consist of 'principal curvature, mean curvature, Guass curvature, normal curvature' while the other is consist of 'curvature tensor'. I ...
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75 views

Fixed Point Involutions

In recent reading on Riemann surfaces and complex manifolds (primary Miranda with a few random finds online), I encountered the notion of involutions, in particular fixed point involutions. We recall ...
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51 views

Parametrization of this elliptic curve

What's the simplest way to parametrize the curve given by the equation $$y^2 = (x^2-a^2)^2 - b^2,$$ namely simple functions (polynomials?) $x(z)$, $y(z)$, that would satisfy the above relation. This ...
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38 views

Analytic Continuation of a Function Containing a Square Root to a Second Riemann Sheet

Consider the function $f(z) = g_1(z) + \sqrt{z} \, g_2(z)$, where $g_1(z)$ and $g_2(z)$ are entire functions, and we take the principal branch of the square root. $f$ is analytic on $\mathbb{C} / \{z ...
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Mittag-Leffler Problem

We have: $X$ a compact Riemann surface defined by $y^{2}=1-x^{6}$ and $P=(0,1) \in X$ a point given in local coordinates $(x,y)$. Furthermore, we have a meromorphic function $f(x,y)=y/x$ such that $f ...
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1answer
58 views

Existence of $g_{d}^{r}$ implies existence of which $g_{d'}^{r'}$'s?

Suppose I have a Riemann surface with a $g_{d}^{r}$. I am wondering what $g_{d'}^{r'}$'s exist for sure. For instance, since $h^{0}\left(L\left(-p \right)\right)$ is either $h^{0}\left(L\right)$ in ...