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

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Question on Fulton's coverage of Riemann surfaces

Riemann surfaces beginners question: (I am learning about normalization of algebraic curves for the first time using Fulton's Algebraic topology and was doing fine until i hit a this snag) SHORT ...
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1answer
30 views

Surface constructed using curves

Suppose that $E$ and $F$ are two complex compact Riemann surfaces with genus greater or equal than $2$. Set $$S=E \times F$$ the surface composed by the cartesian product of thees curves. What can i ...
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Torus-like Riemann surface for a genus 1 Cassini oval

Among the Cassini ovals there is the lemniscate of Bernoulli. This latter curve has genus $0$ and can be mapped to a standard Riemann sphere: $$ (R(t+t^3), R(t-t^3), (1+t^4)/\sqrt{2}) \qquad t\in ...
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1answer
30 views

Distance on riemann sphere [duplicate]

Let we have $C$ the set of complex numbers and $z_1 , z_2 \in C $ we have $Z_1 , Z_2 \in S$ correspond on riemann sphere and we will define : $$ d(Z_1,Z_2)=\frac{2|z_1-z_2|}{\sqrt{1+|z_1|^2} ...
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a question about finding umbilical points in an elipsoid.

Determine the umbilical points of the elipsoid $${x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2}=1.$$ My thoughts: let $x=asin(\theta)cos(\phi),y=bsin(\theta)cos(\phi),$and $z=cos(\theta)$. Thus, I ...
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1answer
39 views

Extending the metric of a hyperbolic surface with boundary to its double

Let $M$ be a hyperbolic surface with totally geodesic boundary. Taking the double $DM$ of $M$, it is easy to see using Euler characteristic that $DM$ is itself a hyperbolic surface (without boundary). ...
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27 views

Rational section of the canonical line bundle of a smooth curve

Let $C$ a complex Riemann surface with genus $g>0$, $L$ a theta characteristic on $C$ i.e $L \in Pic(C)$ such that $L^2 \equiv \omega_C$ where $\omega_C$ is the caninical line bundle on C and ...
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19 views

Constructing Riemann surfaces and holomorphic functions

How do I construct a Riemann surface $S$ and holomorphic functions $f:\mathbb{C}-[\pm1] \rightarrow S$, $g:S \rightarrow \mathbb{C}-[\pm1]$ such that f is conformal, g surjective (every point having ...
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1answer
93 views

First derivative of the Weierstrass $\wp$ function as a function on $\mathbb{C}/\Lambda$

I am currently trying to prove various facts about $\wp'$, considered as a meromorphic map from $\mathbb{C}/\Lambda\to\mathbb{C}$, where $$\wp'(z) = -2\sum_{w\in\Lambda}\frac{1}{(z-w)^3}.$$ In ...
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Question about a notation of line bundle

If $C$ is a complex Riemann Surface with positive genus, $D$ a divisor on $C$, $L$ a line bundle of $C$, with the term $L(D)$ what do we mean? I have this idea: $L(D)$ set of all sections of $L$ ...
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Question to understand how sections work

Suppose that $C$ is a compact Riemann surface o positive genus $g$. Let $L$ a linear bundle on $C$ . Chosen a divisor $D$ on $C$ we consider the set $L(D)$ that is the tensor product between $L$ and ...
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2answers
62 views

Is the Riemann surface for the square root simply connected?

I am looking for universal covering spaces and I am now wondering if the Riemann surface for the square root $z^{1/2}$ (or even more general for $z^{1/n}$) is simply-connected and therefore a ...
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1answer
38 views

Are all 2D tensors in a specified flat metric equal to that same metric conformally scaled?

I have a tensor $T_{mn}$ where its indices coorespond to a flat metric $g_{mn}$. I want $T_{mn}$ to be a new metric $\tilde{g}_{mn}$, such that $T_{mn}(g_{rs}) = \tilde{g}_{mn}$. A theorem says that ...
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34 views

Line bundle of degree 1 on a genus 2 surface with 2 independent global holomorphic sections

By Riemann-Roch, for a degree 1 line bundle on a genus 2 Riemann surface the space of global holomorphic sections has dimension between $0$ and $2$. Is there an explicit example of a degree 1 line ...
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34 views

Degree and Ramification points of an holomorphic map between Riemann Surfaces

The question is the following: we have an holomorphic map from $\Bbb P^1$ to $\Bbb P^1$, defined by $f(z)=z^3-3z$. I need to find the degree and the ramification points and their orders, then verify ...
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1answer
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Spectral representation of an analytic function

I have a question about the spectral representation of an analytic function $G$ on a Riemann surface (specifically, the complex plane with a finite amount of cuts), i.e. the representation of the form ...
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38 views

Line bundle of degree 1 on a genus 2 surface without global holomorphic sections

By Riemann-Roch, for a degree 1 line bundle on a genus 2 Riemann surface the space of global holomorphic sections has dimension between $0$ and $2$. Can someone show an explicit example of a degree 1 ...
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39 views

Graph Jacobian (Sandpile group) usages

Let $\Gamma$ be a graph (say, finite) and $S_\Gamma$ be it's Jacobian (also known as the sandpile group or Picard group). I'm wondering about what fundamental things one can learn about $\Gamma$ from ...
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1answer
37 views

dimension of space of modular functions using the Riemann-Roch theorem?

Let $H$ be the upper half-plane, and $M_k$ be the space of modular forms of weight $k$ on $H$ under the action of $SL(2,\mathbb{Z})$. I have read (Koblitz, Introduction to Elliptic Curves and Modular ...
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32 views

Coordinate on the boundary of Riemann surface

Let $\Sigma$ be a Riemann surface with boundary. Question: Is there canonical way to parameterise the boundary components up to shift? By shift I mean change of coordinate $\phi$ to $\phi + c$. ...
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1answer
30 views

Existence of Gluing of Riemann surfaces

Consider two copies of holomorphic disks $\{ z \in \mathbb{C} \ | \ |z| \leq 1 \}$. Denote them by $\Delta_1$ and $\Delta_2$. Let $f$ be a diffeomorphism from boundary of first disk to boundary of ...
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27 views

Union of holomorphic atlases is holomorphic atlas.

Let $S$ be a surface with open subsets $V$ and $W$ such that $s = V \cup W$. Suppose that $V$ and $W$ have holomorphic atlases $\Phi$ and $\Psi$ such that the holomorphic atlases $\Phi|_{V \cap W}$ ...
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Topological structure of the Riemann sheets?

In complex analysis, for me, the most intriguing thing is the branch cut of a function. The multi-valuedness makes life very difficult. Presumably, for some functions, like $$ f(z) ...
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1answer
19 views

Canonical map from fundamental group to Fuchsian group?

Suppose we have a Riemann Surface $S$ of constant negative curvature $-1$. What is the canonical map from the fundamental group $\pi_1(S)$ to the discrete subgroup $\Delta \subset PSL_2(\mathbb{R})$ ...
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35 views

Alebgraic curve and Riemann surfaces

How do we prove that any smooth complex algebraic curve $C\subset\mathbb{P}^2$ is a Riemann surface? Does there exist a complex version of the implicit function theorem?
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Maximum principle of harmonic function on compact manifold

Thm . (Maximum Principle) Let h be a harmonic function on a domain D in C . (a) If h attains a local maximum in D then h is constant. (b) Suppose that D is bounded and h extends continuously to the ...
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“Pulling back a function from a neighborhood of 0 in $\mathbb{C}^2$ to $\mathbb{U}$ is analogous to computing the derivative of that function.”

I'm reading on Riemann Surfaces and after defining $$\mathbb{U} \equiv \{(z,\ell)\in\mathbb{C}^2\times\mathbb{P}_1;z\in\ell\},$$ and $B\ell_0:\mathbb{U}\to\mathbb{C}^2$, the author mentions that, ...
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Given a smooth 2-form $\varphi$ on a punctured Riemann surface, is there $\nu$ of type (1,0) such that $\varphi=d\nu$?

Let $X$ be a compact Riemann surface, $p\in X$, and $\varphi$ be a smooth 2-form on a $X-\{p\}$, and hence exact. I'm wondering if it is possible to find a form of type (1,0) whose differential is ...
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tiling of Riemann surface of genus 2 by 12 heptagons

Let S be a Riemann Surface of genus 2. Is there a picture in the litterature for a tiling of S by 12 heptagons (I presume such a tiling exists; I am also aware of the beautiful pictures of the famous ...
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Compactness of the moduli space of bundles with fixed determinant

The moduli space of semistable holomorphic vector bundles of fixed rank and fixed determinant line bundle on a compact Riemann surface is known to be compact itself. (In particular, when the rank is ...
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How to write a divisor: Exercise n 20 pag 285 from Geometry of algebraic curves by Arbarello, Cornalba, Griffiths, Harris

Let $C$ a complex Riemann surface (compact) and $\alpha:C^{'} \rightarrow C$ an unramified double cover of $C$. Define the application $\alpha_{*} :Div(C^{'}) \rightarrow Div(C)$ as follow $$\forall ...
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1answer
58 views

Dehn twist as isometries on hyperbolic surface

[I am editing the question to a most correct and precise one thanks to comments of Lor and studiosus] Let (S,g) be a compact hyperbolic surface. On a simple closed geodesic $\gamma $ I can realized a ...
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44 views

Tangential derivative vs covariant derivative

My question is basically the same as this, but the answer in that page was not clear to me. Let me restate the question here: let $\Omega\subset\mathbb{R}^3$ be a domain with boundary $\Gamma$, and ...
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Obtaining a single-valued branch of $\ln \left( \frac{z-a}{z-b} \right)$ with a branch cut

It is rather easy to see that the function $$f(z) = \ln \left( \frac{z-a}{z-b} \right)$$ has branch points at $z=a$ and $z=b$, My question is why considering a branch cut "connecting" $a$ and $b$ ...
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Completely self-contained (and as elementary as possible) introduction to Teichmuller Theory

Can you recommend a completely self-contained and elementary (as much as it can be) introduction to Teichmuller Theory?
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153 views

Exercise from Geometry of algebraic curves by Arbarello, Cornalba, Griffiths, Harris

Let $\pi:C^{'} \rightarrow C$ an unramified double cover of a complex Riemann surface $C$ of genus $g$. With the symbol $Nm_{\pi}$ we mean the norm application that takes a meromorphic function on ...
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1answer
34 views

Complex cohomology $\cong$ sheaf cohomology of constant sheaf on Riemann surface?

I am currently reading in a rather down to earth book on Riemann surfaces. They define the first complex cohomology group $H^1(X, \mathbb{C})$ associated to a Riemann surface $X$ via $H^1(X, ...
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Is this quotient of meromorphic functions of finite order?

Let $f$ and $g$ be two non-zero meromorphic functions of finite order, in the sense that two numbers $\rho_f$ and $\rho_g$ exist such that $$f(z)=\mathcal{O}(e^{|z|^{\rho_f}})$$ ...
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Fact check: global geometry / topology of moduli space of curves

Question: Is the moduli space of smooth complex curves of genus $g\geq2$ isomorphic to the affine space $\mathbb A_{\mathbb C}^{3g-3}$? (Note: I am not asking about the compactification of this ...
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60 views

constant-curvature Riemannian metric for Bring's surface

There is a well-known and very symmetric space that is called either "Bring's curve" or "Bring's surface", depending upon the context. (Bring was a Swedish mathematician in the 18th century.) Let's ...
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1answer
61 views

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) = ...
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Ramification: Riemann surfaces vs Number fields

I am trying to understand the connection between Riemann surfaces and number fields. I am wondering if there an inconsistency in the definition of ramification in terms of Riemann surfaces vs number ...
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100 views

$\{(x,y)\in \mathbb C^2|y^2=\sin x\}$ as interior of compact Riemann Surface with Boundary

A takehome exam problem for my Riemann Surfaces class, which used Griffith's Introduction to Algebraic Curves, was the following: Show that $S=\{(x,y)\in \mathbb C^2|y^2=\sin x\}$ is not interior ...
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An alternative description of an holomorphic map associated to a complete linear system

I need an help with an exercise in Miranda's book "Algebraic curves and Riemann surfaces". More precisely is the exercise in Problems V.4 I. Given a Riemann surface $X$ and a divisor $D$ on $X$ ...
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1answer
91 views

Is the Riemann sphere conformal equivalent to the 2-sphere?

Today I stumbled across the calculation (mentioned in this post) of the transition maps of the stereographic projections from the 2-sphere to the plane. And I wondered about the result that the last ...
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Non-isomorphism of topological line bundles on a Riemann surface, from first principles only

Although this question is in the same vein as my previous query, Isomorphisms (and non-isomorphisms) of holomorphic degree $1$ line bundles on $\mathbb{CP}^1$ and elliptic curves, it is nonetheless ...
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1answer
85 views

Complex structure on the Jacobian of a Riemann surface

Let $X$ be a fixed smooth, connected, compact Riemann surface of genus $g$. The Jacobian variety $\mbox{Jac}(X)$, which parametrises isomorphism classes of holomorphic degree $0$ line bundles on $X$, ...
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61 views

Prove the holomorphic line bundle $\lambda(p+q)$ is the dual of the natural projective bundle

Let $M=\mathbb{C}P^1$ be the complex projective space, $U_0=\{[z_0,z_1]:z_0\ne 0\}$, $U_1=\{[z_0,z_1]:z_1\ne 0\}$ be the coordinate charts and define ...
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1answer
86 views

Isomorphisms (and non-isomorphisms) of holomorphic degree $1$ line bundles on $\mathbb{CP}^1$ and elliptic curves

I have two highly-coupled questions concerning holomorphic line bundles, and so I will go ahead and ask them together. The first concerns line bundles on $\mathbb{CP}^1$ and the other concerns line ...
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Is there a natural ring structure on $\operatorname{Pic}(\mathbb{CP}^1)$?

The set of isomorphism classes of holomorphic line bundles on a complex manifold $X$ is a group under tensor product. This group is called the Picard group and is denoted $\operatorname{Pic}(X)$. We ...