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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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A basic question on mutually orthogonal coordinate systems

I am reading the first chapter of Information Geometry and its applications by Amari. I am struggling to grasp a basic concept about mutually orthogonal coordinate systems. Since the book is not ...
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Complex structure of branched cover over Riemann surface

Suppose $X$ is a Riemann surface, $Y$ is a Hausdorff topological space and $p: Y\to X$ is a local homeomorphism. Then there is a unique complex structure on $Y$ such that $p$ is holomorphic. Now if $\...
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Proof of existence of maximal analytic continuation of a holomorphic germ

The following is from Lectures on Riemann Surfaces by O. Forster: 7.8. Theorem. Suppose $X$ is a Riemann surface, $a\in X$ and $\varphi\in\mathcal{O}_a$ is a holomorphic function germ at the point $...
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Winding number and ramification index

It is known that if $f:X\rightarrow Y$ is a non-constant holomorphic map of Riemann surfaces, then there is an integer $k$ such that locally around each $x\in X$, we can choose coordinate functions ...
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Galois cover and solvability

This is a very general question: let $f:X\to \mathbb P^1$ be a branched cover of compact Riemann surfaces, where $X\subset \mathbb P^n\times \mathbb P^1$ and $f$ is the natural projection. For any ...
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Meromorphic Function on a Riemann Surface

I have a question about an argument used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (or look up at pages 73/74): In the proof we construct locally in $U$ a polynomial $...
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Existence of non-constant holomorphic map between two given compact Riemann surfaces

Given two compact Riemann surfaces $X,Y$, can we always find a non-constant holomorphic map from $X$ to $Y$? In particular, when $Y$ is a elliptic curve, does that map exist? Michael Albanese has ...
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Confusing step in a proof of Weyl's Lemma

I am reading Kra's and Farkas' book on Riemann surfaces, and Theorem II.2.1 is Weyl's Lemma: Let $\varphi$ be a measurable square integrable function on the unit disk $D$. The function $\varphi$ is ...
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Intuition behind $\mathcal{M}_g\cong\mathcal{T}(S)/\text{Mod}(S)$.

Let $S$ be a compact Riemann surface of genus $g$. The mapping class group $\text{Mod}(S)$, constitued by all homotopy classes of orientation-preserving diffeomorphisms, acts on $\mathcal{T}(S)$ (...
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Reference request: Fuchsian Model of compact Riemann surface

Let $S$ be a compact Riemann surface with genus $\geq$ 2. Then by the Uniformization theorem it has universal covering space the upper half-plane $\mathbb{H}$(up to conformal equivalence). Now I read ...
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Zeros of meromorphic function on Riemann surface and paracompactness

I am looking at an exercise at the end of this video on YouTube about Riemann surfaces. We are asked to show that for a nonzero complex-valued meromorphic function $f$ defined on a connected open ...
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Why does a holomorphic differential has $2g-2$ zeros?

If $X$ is a compact Riemann surface, then any holomorphic differential on $X$ has $2g-2$ zeros. I would like to know how to prove this. If possible, without some "heavy machinery" like divisors and ...
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Reference for Uniformization Theorem

I would appreciate if someone could point me to some introductory literature/resources where I can learn about Poincare's Uniformization Theorem at a basic level. Any good powerpoint notes, short ...
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The genus of a smooth algebraic curve in $(\mathbb{CP}^1)^2$

I came across this exercise in Riemann Surfaces by S. Donaldson: Let $Z$ be a smooth algebraic curve in $\mathbb{CP}^1\times\mathbb{CP}^1$. Let $d_1,d_2$ be the degrees of the projection maps ...
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The sum of orders of zeros of a holomorphic function in a genus $g$ Riemann surface is equal to $2g-2$

Let $X$ be a compact Riemann surface and $f:X\to\mathbb{C}$ be a holomorphic function. By the local normal form, for each point $p\in X$ there exists a chart $\varphi:U\to V$ in $X$ centered at $p$, ...
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Explicit construction and computations on higher genus Riemann surfaces

I'm learning about the higher genus ($g>1$) Riemann surfaces and I find it hard due to the lack of explicit examples. Specifically I'm interested in the basis of holomorphic forms, Abel map, prime ...
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How should I interpret $H^1(X,\{P_1,\dotsc,P_n\};\mathbb{Z}\oplus i\mathbb{Z})$?

I am reading A. Zorich's article "Square Tiled Surfaces and Teichmuller Volumes of the Moduli Spaces of Abelian Differentials" and there a main object is the space $$H^1(X,\{P_1,\dotsc,P_n\};\mathbb{Z}...
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Identifying the branch points of a covering

Consider the compact Riemann surface $\overline{X}$, which is the compactification of $$X=\{(z,w)\in\mathbb{C}^2\mid z^{2a}-2w^bz^a+1=0\}.$$ Here, $a,b>0$ are integers. We let $p:\overline{X}\to\...
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Is $H^1(X,\mathbb{C})$ the cohomology group or the vector space of holomorphic differentials in a Riemann surface $X$?

I am an undergraduate studying Riemann surfaces so I don't have a huge background in algebra. As far as I know, $H^1(X,\mathbb{C})$ is the cohomology group of $X$ with coefficients in $\mathbb{C}$. ...
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Uniqueness of Riemann surface

Note: I'm learning complex analysis from Gamelin and Riemann surfaces appear there in the first chapter as prerequisites. I have no background in Topology/Complex Analysis so please don't use that ...
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two lattices are equal if there is a matrix such that …

I am trying to prove the previous lemma and I have thought to do the following: To go from right to left, I do not know how to proceed, but to go from left to right, it occurs to me to do the ...
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Contour integral of square root on its Riemann surface

Consider a branch of the square root function $f(z)=z^{1/2}, z\in\mathbb{C}$ with $Im \thinspace {f(z)}>=0$, i.e.                        ...
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1answer
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Line bundle from $\mathcal{O}(p)\cong \mathcal{O}(q)$

Let $X$ be a compact Riemann surface. If $\mathcal{O}(p)\cong \mathcal{O}(q)$. How to see there exists a line bundle $L$ and $s_1,s_2$ two sections of $L$ such that $s_1$ vanishes only at $p$ and $s_2$...
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Are the only Riemann surfaces which are quotients of $\mathbb{C}$the cylinder and the toruses? Why?

Consider the Riemann surfaces $\mathbb{C}^\times=\mathbb{C}\setminus\{0\}$ and $\mathbb{C}/\Lambda$, where $\Lambda$ is a lattice in the complex plane (i.e. a discrete additive subgroup of $\mathbb{C}$...
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The holomorphic map from a compact smooth curve $C$ to $\mathbb{C}P^1$ when $H^0(C,\mathcal{O}(p))=2$

Let $C$ be a compact smooth complex curve with $H^0(C,\mathcal{O}(p))=2$. I feel confused with the following words: Denote by $a$ and $b$ two non-collinear sections in $H^0(C,\mathcal{O}(p))$. Then ...
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Composite of certain Finite Subextensions $L|K$ Galois

I have a question about a step used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (see page 120): The point of my interest is a statement in Proposition 4.6.1: Fix a ...
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1answer
50 views

Residue of $\frac{dz}{w^3}$ on the riemann surface

Let compact riemannian surface X satisfy the equation $w^3=z(z-1)(z-2)$. Let $p \in X$ be the point where $w=0, z=1$. Compute $Res_p(\frac{dz}{w^3}) $ By implicit function theorem, I see, that ...
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Universal covering of $n$ punctured 2-sphere

Actually, I just want to understand first sentence of Here. It says that Let $X$ be the $\mathbb{CP}_{1}$ with $n$ points deleted. Let $n \geq 3$. If I understand correctly, the universal covering ...
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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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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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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1answer
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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 ...