# Questions tagged [riemann-surfaces]

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

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### Definition of a (holomorphic) differential form in the context of translation surfaces

I am a beginning graduate student with an interest in geometry, in particular, in translation surfaces. I am trying to learn from the recent text by Athreya & Masur. My biggest point of confusion ...
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### Compute the genus of a double cover of $\mathbb{P}^1(\mathbb{C})$ branched at 12 points

I am studying Riemann Surfaces and the lecturer gave the following as an exercise: Compute the genus of a double cover of of $\mathbb{P}^1(\mathbb{C})$ branched at 12 points I would appreciate some ...
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### Order 2 branch cut and different sheet structures on Riemann surfaces

I am trying to understand some simple branch structures of Riemann surfaces with order 2 ramification / branch points. Let's talk about surfaces in $\Sigma \subset \mathbb{C}^2$ cut out by a ...
1 vote
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### Holomorphic map with no singular values is a covering map?

I am working on Problem 8-c from Milnor's Dynamics in One Complex Variable, which describes a necessary and sufficient condition for a holomorphic map $f : S \to S'$ between Riemann surfaces to be a ...
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### Local behaviour of the action of a finite subgroup of the automorphism group of a Riemann Surface fixing a point

Consider $(\Sigma,j)$ a closed Riemann surface, and $G \subset Aut(\Sigma,j)$ a finite subgroup that fixes a point $z_0 \in \Sigma$. How can one construct a $G$-invariant neighbourhood $U \ni z_0$ ...
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### Induced homomorphism of Abel-Jacobi map is surjective for genus 1

I'm trying to prove that every Riemann surface of genus 1 is biholomorphic to a torus, via a sequence of exercises in Richard Hain's lecture 'Moduli of Riemann surfaces, transcendental aspects'. He ...
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### the symbol defining a cocycle ? Or rather its square?

Let S be a Riemann surface. A quasi-meromorphic function on S is a function holomorphic everywhere except finitely many points and such that locally it can be written as $re^{\phi}$ where both r and φ ...
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### Potentials and orientation-preserving isometries

I'm trying to prove the following Lemma: If $U,V \subseteq \mathbb{C}$ are open, and are equipped with conformal metrics $g_U=\lambda^2dzd\bar{z}$ and $g_V=\mu^2dzd\bar{z}$. If $f:U \rightarrow V$ is ...
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### Classification of complex structures of $\mathbb{C}^{*}$

Riemann's theorem states that simply connected Riemann surfaces are biholomorphic to $\mathbb{C}, \mathbb{P}^1(\mathbb{C})$ or $H$ the upper-half complex plane. It is also easy to check that the cover ...
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### What do the axes represent on this Riemann surface of the complex logarithm?

My question seems to a special case of the answer to this question: What does the Color and height of a Riemann surface represent, but that post seems to make use of more advanced techniques than I ...
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### Covering properties of non-constant holomorphic function $f: X \rightarrow \mathbb{C}$

I'm working through a proof that Riemann surfaces are second countable, and one of the main steps is showing that if $X$ is a connected Riemann surface such that there is a non-constant holomorphic ...
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Let $\Gamma \subset PSL(2,\mathbb{R})$ be discrete, and consider the Riemann surface $\mathbb{H} / \Gamma$ with the unique complex structure for which the quotient map $\pi : \mathbb{H} \rightarrow \... • 1,642 1 vote 0 answers 41 views ### Covering Map Associated to Map from Homology Let$X_n$be a connected$2$-dimensional manifold with nonempty boundary and set$X_n^*:=X_n-\{x_0\}$. Assume that$H^1(X^*_n;\mathbb{R})\cong \mathbb{R}$. Then since $$H^1(X_n^*;\mathbb{R}) = \text{... • 2,085 2 votes 0 answers 67 views ### Universal property of Abelian-Jacobi Map/Jacobi variety for Riemann Surfaces I have a question about universal property of Abel Jacobi Map and the Jacobi variety in the (classical) context of Riemann surfaces / complex smooth proper curves. Let C be such RS/complex sm curve ... • 7,561 0 votes 0 answers 64 views ### Few Questions about Properties of Exponential Map \text{exp}: \text{Lie}(G) \to G of Compact Complex Lie Group Let G be compact Riemann surface with the structure of a complex commutative Lie group, ie the multipliciation map m:G \times G \to G is holomorphic (+certain usual diagrams satisfy axiomatic ... • 7,561 0 votes 0 answers 35 views ### Compact Riemann Surfactes with Group Structure Let X be compact Riemann surface with the structure of a complex Lie group, ie the multipliciation map m:G \times G \to G is holomorphic (+certain usual diagrams satisfy axiomatic group law ... • 7,561 0 votes 0 answers 13 views ### \operatorname{det}(Z_0T_0 + Z_1T_1 + Z_2T_2) = 0 defines a complex curve C with kernel as holomorphic line bundle This is Exercise 12.4. Riemann Surfaces (Simon Donaldson). Let T_0, T_1, T_2 be generic n \times n complex matrices. Show that the equation \operatorname{det}(Z_0T_0 + Z_1T_1 + Z_2T_2) = 0 ... 0 votes 0 answers 25 views ### Five Islands/Five Points Theorems (Reference Request) Does anyone have a modern reference for the Five Islands Theorem of Ahlfors and/or the Five Points Theorem of Lappan? I know that there are proofs of the Five Islands that don't involve the Ahlfors-... • 233 1 vote 0 answers 31 views ### For which c \in \mathbb{C} does multi-valued function \sqrt{e^z -c} define 2 single-valued functions? For which c \in \mathbb{C} does multi-valued function \sqrt{e^z -c} on \mathbb{C} define 2 single-valued functions? I guessed that \sqrt{e^z -c} is single-valued iff c = 0, for if c = 0 ... • 11 0 votes 0 answers 25 views ### Checking for understanding about integrating 1-forms and 2-forms on a complex torus Let S= {(z,w):w^2=4z^3-z}$$a=\frac{(\Gamma(1/4))^2}{2\sqrt{\pi}}t\in[0,a]\gamma_1(t)=\frac{d}{dt}\wp(t; 1,0)\gamma_2(t)=\frac{d}{dt}\wp(it; 1,0)$$Let \alpha_1, \alpha_2 be ... • 887 1 vote 1 answer 55 views ### Examples of Riemann surfaces such that a given point on the surface does not necessarily determine a unique value of w If you Google search The basic idea of Riemann surface theory is to replace the domain of a multi-valued function with a Riemann surface to make it single-valued \tag1 then you will find many, many ... • 887 2 votes 2 answers 97 views ### Is the Riemann surface ℂ of the form f(z,w)=0? If so, what are the possible options for an everywhere-analytic f? What are all of the possible options for an analytic (over some region) - expressible as a polynomial OR locally as a convergent power series in both z and w over that region - bivariate function ... • 887 1 vote 0 answers 60 views ### Uniqueness of Poincaré metric on disk It is well-known that the unit disk \mathbb{D} admits a complete conformal metric of constant curvature -1, called the Poincaré metric. Is the Poincaré metric the unique metric on \mathbb{D} ... • 2,625 0 votes 0 answers 22 views ### Confusion in Lemma 21.3 in Forster's Riemann Surfaces Lemma. Suppose X is a compact Riemann Surfaces of genus g. Then there are g distinct points a_1,\dots,a_g \in X with the following property: Every holomorphic 1-form \omega \in \Omega(X) ... • 986 0 votes 0 answers 16 views ### Constructing a meromorphic function on a Riemann surface Suppose that X is a compact and connected Riemann surface nad that p \in X. I want to construct a meromorphic function f on X such that {\rm old}_p(f) = 1434. Note that there are no ... 0 votes 1 answer 33 views ### Question about proof regarding holomorphic maps between Riemann surfaces So in our lecture notes, we have this proof; My question is, why the discussion on accumulation points? It seems to me like no part of this proof required any notion of accumulation points. Even the ... • 146 1 vote 0 answers 38 views ### Describe the monodromy of a holomorphic map f:\mathbb{P}^1\to \mathbb{P}^1 I am currently studying monodromy from Rick Miranda's book "Algebraic Curves and Riemann Surfaces" and I have the following problem: Let f:\mathbb{P}^1\to \mathbb{P}^1 be defined via the ... • 301 6 votes 3 answers 246 views ### How did Jacobi find his connection between theta functions and q? I was 'reading' (I can't actually read german, but I can read math!) Jacobi's derivation of the ODE for y(q) = \sum_{n=-\infty}^{\infty} q^{n^2} . On page 2 of the paper Jacobi states the following ... • 17.5k 5 votes 1 answer 74 views ### Is the curve y^7=x^2(x-1)^2 hyperelliptic? When playing around with genus 3 curve X which has an automorphism \sigma of order 7. Using the Riemann-Hurwitz formula, one find the map X\to Y:=X/\langle\sigma\rangle is a degree 7 map ... • 786 1 vote 0 answers 49 views ### Proving that \{([w_0 : w_1], [z_0 : z_1 : z_2]) \in \Bbb {CP}^1 \times \Bbb {CP}^2| w_0z_1 = w_1z_0\} is a smooth complex hypersurface Let F be the subset of \Bbb {CP}^1 \times \Bbb {CP}^2 defined by F = \{([w_0 : w_1], [z_0 : z_1 : z_2]) \in \Bbb {CP}^1 \times \Bbb {CP}^2| w_0z_1 = w_1z_0\} I want to prove that F is a smooth ... • 3,162 0 votes 0 answers 22 views ### Understanding the Harmonic Energy for Maps Between Riemann Surfaces？ I am trying to understanding this functional, the thing I got confused about is the meaning of term u_z\overline{u}_{\overline{z}}+ \overline{u}_{z}u_{\overline{z}} ? Because harmonic map is almost ... 1 vote 0 answers 28 views ### Verifying that M has curvature -1 iff f''=f when g=dr^2+f(r)^2d\theta^2,M=[0,\infty)_r\times S_\theta^1 I am currently at a difficult position, because I have to check some definitions/examples regarding hyperbolic surfaces, but I have not taken a proper course on Riemannian manifolds or surfaces in the ... • 1,030 1 vote 1 answer 47 views ### Confusion in proof of H^1(X,\mathcal{O})=0 where X is an open disk In Otto Forster's Lecture on Riemann Surfaces, I have faced a small confusion in which I am certainly overlooking something in the proof of Theorem 13.4 that for X:= \{z\in \mathbb{C}: |z|<R\}, 0&... • 986 3 votes 2 answers 152 views ### Open sets on a surface with locally connected boundary Let \Sigma be a surface and \Omega be an open subset of \Sigma. Suppose that \Omega is homeomorphic to the open unit disk \mathbb{D} and is relatively compact in \Sigma. I'm interested in ... • 1,107 1 vote 0 answers 29 views ### Characterizing the complex structure on a non-compact Riemann surface (This is a crosspost from MathOverflow) A consequence of Torelli's theorem is that a closed Riemann surface X is determined by its period matrix. More precisely, fix (\alpha_i) a basis of H_1(X, \... 3 votes 1 answer 56 views ### Hyperbolic surfaces with only one short geodesic \textbf{Question}: Let R>0. Does there exist a compact hyperbolic surface S which has one and \underline{only\ one} primitive geodesic of length \le R? I am aware of the fact that the ... 2 votes 1 answer 44 views ### Correspondence between basepoint free linear systems and Holomorphic maps to \mathbb{P}^n I am currently reading's Miranda's book on Riemann Surfaces. I have a question about codimension 1 subspaces of L(D). I am stuck on the proof of Propoisiton 4.15 in chapter 5. Questions regarding ... 0 votes 0 answers 69 views ### Obstruction for the existence of almost complex structures on open surfaces? Let \Sigma be a smooth open connected orientable real surface. Are there obstruction results for the existence of an almost complex structure J on \Sigma? (Since almost complex structures are ... • 1,174 1 vote 1 answer 37 views ### Understanding the Absence of Odd-sided Shapes in Dirichlet Domains: Theorems and Discussion I am studying the shape of dirichlet domains. I understood that the following Theorem 10.5.1 of Beardon provides the bounds on the number of sides. And the Umemoto's paper (See References) also ... • 740 1 vote 0 answers 30 views ### Homology groups of genus g Riemann surface with n punctures. I want to know whether one can compute H_k(\Sigma_{g,n}) with elementary methods like singular chain complex? By \Sigma_{g,n} I mean a genus g surface with n circular holes. If not, I'm not ... 5 votes 1 answer 111 views ### What is the meaning of the notation {\cal O}_{\mathbb{CP}^n}(k)? I am studying some papers in which the notation {\cal O}_{\mathbb{CP}^1}(-1), {\cal O}_{\mathbb{CP}^1}(-2) and {\cal O}_{\mathbb{CP}^1}(1) appear. I am not familiar with that notation and while ... • 26.9k 0 votes 0 answers 32 views ### plot3d hyperelliptic singular curve I am trying to draw a singular hyperelliptic curve of genus two in Sage. My goal is to obtain something that looks like (including the oriented one-cycles): I think that the equation of such a ... • 3,270 1 vote 0 answers 53 views ### Metric choice in the tangent space of a Riemannian manifold obtained through the Log map After using the Log map, as defined in this paper Riemannian approaches in Brain-Computer Interfaces: a review (Section III. A page 2&3), to project points from the manifold onto the tangent space ... 0 votes 0 answers 47 views ### Do surjective morphism exists between the Riemann surfaces \Bbb P^1 and \Bbb C ? Can they be extended to a morphism \Bbb P^1 \to \Bbb P^1? How can I prove the existence or not of a surjective morphisms of Riemann surfaces \Bbb P^1 \to \Bbb C ? and \Bbb C \to \Bbb P^1? In case that it exists, how can I prove if there's any that can ... • 3,162 2 votes 0 answers 63 views ### Compact Riemann surfaces as algebraic curves I'm trying to grasp proofs of the fact that compact Riemann surfaces are algebraic curves. It seems as though no textbook fully treats this, and I wanted to ask about what was going on. On one hand, ... • 133 3 votes 0 answers 69 views ### zero locus of irreducible polynomial I am learning Riemann surfaces and one of the books I use is Algebraic curves and Riemann surfaces by Rick Miranda. He mentions if f(z,w)\in \mathbb{C}[z,w] is irreducible the zero locus is ... • 51 2 votes 0 answers 42 views ### Associated equivariant holomorphic map from a surface group representation Given a (smooth but not holomorphic) surface bundle over surface S_g\to M\to S_h (where g,h>1, there is an induced surface group representation$$\pi_1(S_h)\to \text{SP}(2g,\mathbb R).$$I was ... • 1,907 0 votes 1 answer 51 views ### Do surjective morphisms$\Bbb C^\times \to \Bbb C$and viceversa between Riemann surfaces exist and can they be extended to$\Bbb P^1\to\Bbb P^1$? How can I prove the existence or not of a surjective morphism of Riemann surfaces$\Bbb C^\times \to \Bbb C$? and viceversa? In case that it exists, how can I prove if there's any that can be ... • 3,162 1 vote 1 answer 77 views ### Is it possible to give a complex structure on two dimensional cone? Let$G$be group of$n^{th}$roots of unity. Assume that$G=<\alpha>$, where$\alpha ^n=1$. Now we define a group action,$G$act on$\mathbb{C}$, by$(x,z)\mapsto xz$. The Quotient space$\...
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Let $R$ be a Riemann surface, and define two quasiconformal maps $f_i : R \rightarrow R_i, i=1,2$ to be equivalent if there exists a conformal map $c : R_0 \rightarrow R_1$ and a homotopy \$g_t : R \...