Tagged Questions

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Holomorphic and meromorphic functions on Riemann surfaces

On any domain $\Omega\subset \mathbb{C}$, the set of all holomorphic functions form an integral domain. Its field of quotient is the set of all meromorphic functions on $\Omega$. However this is not ...
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Are there any linkage or transformation between some period transcendental numbers and algebraic irrational numbers?

Are there any linkage or transformation by combination of integral and algebraic function like in the definition of period number between some period transcendental numbers and algebraic irrational ...
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What is the Riemann surface of $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$?

The following appears as the second-to-last problem of Stewart's Complex Analysis: Describe the Riemann surface of the function $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$. This problem ...
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Some questions about complex curves in $\mathbb CP^2$

I would like to ask for some clarifications in the following questions about complex curves. My first question is if I correctly understand what the complex curve in $\mathbb CP^2$ is. Is it only a ...
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Is it possible to realize a general compact Riemann surface in $\mathbb CP^2$?

Let $X$ be a compact Riemann surface with smooth boundary $\partial X$. Is it always possible to realize $X$ as a complex submanifold of $\mathbb CP^2$? In other words, is it true that there exists a ...
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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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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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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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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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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 PoincarĂ© metric is pulled back to ...
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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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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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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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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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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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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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Constructing a meromorphic function

I need help with the following problem. "Let $C : y^2 = x^3 âˆ’ 5x^2 + 6x$ 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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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 ...