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Questions tagged [riemann-surfaces]

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

Riemann surfaces are one-dimensional complex manifolds that are deformations of $\mathbb{C}.$

Riemann surfaces are orientable as a real manifold, and every simply connected Riemann surface is conformally equivalent to one of the following:

Elliptic (Positive curvature): The Riemann sphere (the complex plane with an extension to a point at infinity)

Parabolic (zero curvature): The complex plane

Hyperbolic (negative curvature): The open disk or upper half-plane

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one 1-form on the Riemann surface of an algebraic function

Given the many-valued algebraic function $w =\sqrt{(z-1)(z-2)(z-3)}$, we can get a Riemann surface $S$ that is topologically equivalent to a torus. I am wondering whether the $1$-form $ \displaystyle\frac{dz}{w}$ is a holomorphic form on the torus…
Jacob.Lee
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Differential Form on Riemann Surface

Can one explain what is differential form on a Riemann Surface? What is motivation to define it?
user11049
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How many Riemann surfaces homeomorphic to the sphere are there?

In a previous question, I learned that there exist infinitely many non biholomorphic Riemann surfaces homeomorphic to the torus. Is it also true for the sphere?
Seirios
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lifting of automorphisms of Riemann surface

If $X$ is a compact Riemann surface, $Y$ is a universal cover of $X$, and $f\colon X\rightarrow X$ is a biholomorphic map, then, can $f$ be lifted to a $biholomorphic$ map on $Y$? (I mean, topologically, it can be lifted to a homeomorphism from $Y$…
user8186
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meromorphic functions on the Riemann sphere

I am studying the following exercise: Show that the meromorphic functions on the Riemann sphere have the form $p(z)/q(z)$, where $p$, $q$ are coprime polynomials. Is an exercise in Donaldson Riemann Surface. I thought the following way to solve:…
Manoel
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Global sections of holomorphic line bundles $\mathcal{O}(n)$ over $\mathbb{P}^1(\mathbb{C})$

What are global sections of holomorphic line bundles $\mathcal{O}(n)$ over Riemann sphere? We define $\mathcal{O}(n)=\mathcal{O}(-1)^{\otimes (-n)}$ for $n<0$ and $\mathcal{O}(n)=(\mathcal{O}(-1)^{\otimes n})^*$ for $n>0$,…
ralleee
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Putting a Riemann surface structure on a set of equivalence classes in a torus

I'm looking at the torus given by $X = \mathbb{C}/\Lambda$ where $\Lambda$ is the lattice spanned by $1$ and $\omega$ where $\omega$ is a primitive cube root of unity. I've shown that $\sigma(z) = \omega z$ is a well-defined map on the torus and now…
Wooster
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Does there exist a complex analytic map between compact Riemann surfaces with certain conditions?

I am wondering if there exists a complex analytic map $\pi:X\rightarrow Y$ with $g(Y)=g(X)-1.$ By Riemann-Hurwitz formula, I think the simplest of such maps is a complex analytic map between compact Riemann surfaces of degree 1 with one branch point…
awllower
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the meromorphic 1-form on the riemann surface

This is the question from Lectures on Riemann surfaces (Otto forster), exercise 10.3: Suppose $X$ is a Riemann surface and $\omega$ is a meromorphic 1-form on $X$ which has residue zero at every pole. Show that there is a covering $p: Y \to X$ and…
Antoine
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Multiplicity of a holomorphic map at every point is one

Let $F$ be a non constant holomorphic map between two compact Riemann surface $X,Y$. Assume for each point $p\in X , mult_p(F)=1$. Then ,what can we say about the degree of that map (deg(F))? Locally $F$ looks like a coordinate chart function. Can…
Infinity
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Why doesn't the order of permutations matter when I build Riemann surfaces à la Hurwitz?

I'm currently trying to construct simple Riemann Surfaces in the way of Hurwitz (see e.g. here): Given the complex plane $E$, which originates at $O$ and $w$ non-identical points $a_k$. Cut the plane from $O$ to each $a_k$, which shall be denoted…
draks ...
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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 of $Y$. This is from Rick Miranda's Algebraic…
yuan
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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: $$ S_1S_2S_3\cdots S_w=1, \tag{1} $$ which ensures that there is…
draks ...
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Action of Automorphism Group on Homology Group

How the group of automorphisms of a compact Riemann surface acts on the first homology group of the Riemann surface? [Here, by an automorphism of a Riemann surface, we mean conformal self homeomorphism of the Riemann surface. In the paper "Action of…
user8186
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Riemann Surface Comprehension

I have a question about an example in following script: https://math.berkeley.edu/~teleman/math/Riemann.pdf Here the excerpt: Why there don't exist a continuous choice of $w$ near $\pm1$ and $\pm k$? Why it does lead to the opposite choice upon…
user267839
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