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

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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 ...
4
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0answers
34 views

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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1answer
37 views

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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67 views
+50

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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20 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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54 views

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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3answers
129 views

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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1answer
48 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
53 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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28 views

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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39 views
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$\{(x,y)\in \mathbb C^2|y^2=\sin x\}$ 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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42 views

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
57 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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35 views

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
74 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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1answer
57 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
70 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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45 views

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 ...
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14 views

Finding degree and branching index of projection map

I'm looking at the Riemann surface $R = \{(z,w) : f(z,w) = w^3 - z^3 + z = 0 \}$. I'm looking at the projection map $f: (z,w) \to z$ and I'm trying to find the degree and the branching index. I can ...
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102 views

Not taking holomorphic bundles for granted

When we say something to the effect of, "Consider a holomorphic bundle $V$ on a complex manifold $X$...", we are saying that the transition functions of $V$ are holomorphic functions with respect to ...
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22 views

Sum of preimages of holomorphic function is holomorphic

In Milne's notes on Modular Forms, he sketches the proof of Prop 1.16 (p. 20): "Let $f$ be a nonconstant meromorphic function with valence $n$ on a compact Riemann surface $X$. Then every meromorphic ...
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1answer
46 views

First Chern Class of divisors on compact Riemann surfaces

let $X$ be a compact Riemann surface and $D$ a divisor on $X$. I'm looking for a argument for the statement $c_1(\mathcal{O}_X(D)) = \deg(D)$, where $\mathcal{O}_X(D)$ is the associated line bundle to ...
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2answers
46 views

Analytic continuation of holomorphic function along clockwise/counterclockwise path

"Write down (say, as a power series) a holomorphic function $f(z)$ on $D(1, 1)$ which satisfies $f(z)^5 = z$ and $f(1) = 1$. What is the result of analytically continuing $f$ along a path which ...
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0answers
59 views

On a method to compute dimension of moduli space of Riemann surfaces

Consider genus $g$ Riemann surfaces, and thier moduli space $\mathcal M_g$. To determine dimension of $T\mathcal M_g$, start with a complex structure, which in some coordinates can be written ...
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1answer
46 views

Meromorphic function with bounded order of zeros and poles

The following problem has been bothering me for a long time; Let $X$ be a compact Riemann surface of genus $g$. Is there a non-zero meromorphic function on $X$ with all of its poles and zeros have ...
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34 views

Are generalised configuration spaces related to holomorphic maps?

A branched cover of the Riemann sphere is a non-constant holomorphic map $\phi: \Sigma \to \mathbb{C}P^1$ where $\Sigma$ is a compact Riemann surface. The Hurwitz space of branched coverings of the ...
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1answer
124 views

Describe the Riemann surface for $w=z^2-1$.

Question: Describe the Riemann surface for $w=z^2-1$. My thoughts so far: the Riemann surface needs two cuts emanating from the origin across the real line since it maps every input to one point ...
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1answer
45 views

Riemann Mapping Theorem, the concept of a Riemann mapping

If I construct a composition of mappings that map the upper half of the unit disk conformally to the entire unit disk, then this mapping is a Riemann mapping, by the Riemann Mapping Theorem, since ...
3
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1answer
58 views

A question regarding sheaf cohomology

I am trying to understand a statement from http://arxiv.org/abs/1312.1562, saying "... $H^1(\Sigma^x,T\Sigma) = 0$ since $H^1(\Sigma^x,\mathcal{O}) = 0$ and the Mittag-Leffler problem is solvable on ...
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35 views

Two generating meromorphic functions seperate points on a compact Riemann surface?

Problem Suppose $z,f$ are two meromorphic functions on a compact Riemann surface $M$, whose meromorphic function field is $\mathbb C(M)=\mathbb C(z,f)$, where $\mathbb C(M)$ is a finite extension of ...
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27 views

Euler Characteristic $\mathbb{C} \backslash\{0,1\}$ and non-compact surfaces

I am looking at various proofs of the Little and Big Picard Theorems. I am interested in the following question: Without the Uniformization Theorem, can one calculate the Euler characteristic of ...
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38 views

Misplaced complex analysis intuition on Riemann Surfaces

Next week I will be giving a lecture, based on Chapter 2.6 from Jost's book Compact Riemann Surfaces. He states the following theorem: Theorem 1 (Jost Theorem 2.6.2) Let $S$ and $\Sigma$ be Riemann ...
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1answer
43 views

Show that something is a subsystem of a complete linear system

I have a simple and basic question concerning degree of projective curves and I'm referring to something I've read on Miranda's book, Algebraic curves and Riemann Surfaces, Chapter VII, 3. The Degree ...
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1answer
12 views

Sum over the branches of a composition of an entire function with the branches of an algebraic function is entire.

Let $l(t)$ be the solution of the polynomial equation $g(l,t)=\det(lE-(A-tB))=0$, where E is the identity and A, and B are $n\times n$ matrices. The natural domain of definition is a Riemann surface ...
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1answer
58 views

Divisor of meromorphic section of point bundle over a Riemann surface

Let $X$ be a compact connected Riemann surface (not $\mathbb{P}^1$), $p\in X$ be a point on it. Let $L$ be the holomorphic line bundle associated to the divisor $D=p$. By construction $L$ comes with a ...
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43 views

fundamental group and covering space

I have a question about the fondamental group of the following covering space $$ p : Y \rightarrow X ; \; Y \owns (x,y) \mapsto x \in X $$ where $X = {\mathbb P}^1$ and $$ Y := \{(x,y) \in {\mathbb ...
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Approximation of holomorphic functions and topological properties

So, in the last couple of lectures of my complex analysis class we've proved some approximation theorems of holomorphic functions. Eventually, we showed the following propositions: Theorem 1. Let ...
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53 views

Proof of Riemann-Roch using Mittag-Leffler

In the introduction for Rick Miranda's book "algebraic curves and Riemann surfaces", it says that they will prove Riemann-Roch "in an algebraic manner, via an adaptation of the adelic proof, expressed ...
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0answers
35 views

Why is this function on the Riemann surface holomorphic?

Forster defines analytic continuation of a germ of a holomorphic function at a point on a Riemann surface as follows. Suppose $ X$ is a Riemann surface, $a\in X$ and $\phi\in\mathcal{O}_a$ is a ...
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Investigate the covering of the sphereby the sphere associated with the rational function

Here there is an example of Singerman Complex Functions, i think i understand it, the point is that having $f$ degree 3, do i have three copies of $\mathbb{C}$? if so, how can i glue them? What i ...
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3answers
106 views

How to define the complex square root $ \sqrt{z} $?

We need to define the complex square root $ \sqrt{z} $ on a small open $ U \subset \mathbb{C} $, for example a disc. Let put : $ \mathcal{F} (U) = \{\ f: U \to \mathbb {C} \ / \ f \ \text{is ...
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0answers
27 views

Creating a holomorphic map from a three holed torus to a two holed torus.

I'm trying to make a non-constant holomorphic map, $f$, between a 3 holed torus and a 2 holed torus, with no branch points. Now I can see that $deg(f) = 2$ from the Riemann Hurwitz formula. So ...
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1answer
44 views

Discrete faithful representation in $PSL(2,\mathbb R)$ and horocycles in hyperbolic space

Let $S$ be a closed oriented surface of genus $g>1$. Is the following true ? Let $\alpha,\beta\in \pi_1(S)\backslash \{1\}$ and $\rho:\pi_1(S)\rightarrow PSL(2,\mathbb R)$ be a discrete ...
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2answers
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Comparing genus of domain and image for maps between Riemann surfaces.

I've been asked to show that if $R$ and $S$ are compact, connected Riemann surfaces, and $f: R \to S$ is holomorphic then $g(R) \ge g(S)$ (g is the genus). Now surely this fact follows from the fact ...
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2answers
76 views

Maps between Riemann surfaces are open and continuous

I'm having some trouble with a couple of concepts in Riemman surfaces that I would really appreciate some help clarifying! Firstly, is it true that a holomorphic map between two Riemann surfaces $f:R ...
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2answers
70 views

computation on hyper surface $z=x^2+y^2$

I have problem with following exercise Consider the hypersurface $M$ parametrized by $z=x^2+y^2$. Endow this with the Riemannian metric induced from the $\mathbb{R}^3$. Compute the sectional ...
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1answer
52 views

Intuitively understanding Riemann surfaces

I'm looking at the Riemann surface of $f(z) = z^{1/2}$ so the set $\{(z,w) \in \mathbb{C}^2 : w^2 = z \}$. I understand that the point of the riemann surface is to understand this multi-valued ...
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How does one define the universal curve over the moduli space of stable curves

In relation to the Prym class. How does one define the universal curve $\pi:\bar{\mathcal{C}_g}\longrightarrow\bar{\mathcal{M}_g}$ for genus $g\geq 2$ and their dualizing sheaf $\omega_g$.
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118 views

Two surfaces are not isometries of each other, but have the same Gaussian Curvature

How can you show that two surfaces are not isometries of each other, but have the same Gaussian Curvature. For example, I see that: the helicoid given by X = (ucosv, usinv, v) & the ...
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23 views

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 ...