Questions tagged [riemann-surfaces]
For questions about Riemann surfaces, that is complex manifolds of (complex) dimension 1, and related topics.
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Geometric interpretation of an elliptic point on a Riemann surface / Hyperbolic surface
Let $\Gamma$ be a Fuchsian group of signature $[g;m_1,\dots,m_r;s]$. When we quotient $\mathbb{H}^2$ by $\Gamma$ we obtain a genus $g$ surface with $s$ cusps or 'cones' and $r$ elliptic points with ...
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0answers
53 views
Direct image of a line bundle is a vector bundle
Let
$$f:X\to Y$$
be a holomorphic map between compact connected Riemann surfaces and let
$$L\to X$$
be a holomorphic line bundle on $X$. I'm having some trouble understanding the fact that the direct ...
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1answer
28 views
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 ...
1
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0answers
30 views
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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0answers
26 views
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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0answers
13 views
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 ...
1
vote
0answers
39 views
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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0answers
19 views
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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2answers
55 views
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 ...
2
votes
1answer
64 views
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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1answer
23 views
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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0answers
29 views
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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0answers
29 views
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 ...
1
vote
1answer
48 views
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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1answer
51 views
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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0answers
58 views
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 ...
0
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1answer
49 views
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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0answers
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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 ...
1
vote
1answer
69 views
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
$$H^1(X,\{P_1,\dotsc,P_n\};\mathbb{Z}...
0
votes
0answers
27 views
Identifying the branch points of a covering
Consider the compact Riemann surface $\overline{X}$, which is the compactification of $$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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1answer
59 views
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}$. ...
2
votes
0answers
41 views
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 ...
0
votes
1answer
30 views
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 ...
1
vote
0answers
51 views
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.
...
1
vote
1answer
63 views
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$...
2
votes
1answer
35 views
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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0answers
30 views
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 ...
1
vote
0answers
29 views
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 ...
0
votes
1answer
44 views
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 ...
1
vote
0answers
13 views
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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0answers
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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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votes
0answers
43 views
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 \...
1
vote
1answer
47 views
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 $\...
2
votes
1answer
31 views
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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vote
0answers
41 views
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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vote
0answers
22 views
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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0answers
27 views
$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$ ...
2
votes
1answer
51 views
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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votes
1answer
89 views
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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votes
0answers
20 views
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?
...
0
votes
1answer
36 views
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 ...
0
votes
1answer
47 views
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.
0
votes
0answers
39 views
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 ...
0
votes
1answer
47 views
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$...
1
vote
0answers
54 views
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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0answers
36 views
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 (...
0
votes
0answers
63 views
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 \...
0
votes
0answers
18 views
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 ...
0
votes
1answer
25 views
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 ...
