# Questions tagged [riemann-surfaces]

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

113 questions
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### Quotient Space of Hausdorff space

Is it true that quotient space of a Hausdorff space is necessarily Hausdorff? In the book "Algebraic Curves and Riemann Surfaces", by Miranda, the author writes: "$\mathbb{}P^2$ can be viewed as the ...
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### How do different definitions of “degree” coincide?

I've recently read about a number of different notions of "degree." Reading over Javier Álvarez' excellent answer for the thousandth time finally prompted me to ask this question: How exactly do ...
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### What is the Riemann surface of $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$?

The following appears as the second-to-last problem of Stewart's Complex Analysis: Describe the Riemann surface of the function $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$. This problem ...
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### Why the emphasis on Projective Space in Algebraic Geometry?

I have no doubt this is a basic question. However, I am working through Miranda's book on Riemann surfaces and algebraic curves, and it has yet to be addressed. Why does Miranda (and from what little ...
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### Toward “integrals of rational functions along an algebraic curve”

In a talk by V.I. Arnold, this is said: When I was a first-year student at the Faculty of Mechanics and Mathematics of the Moscow State University, the lectures on calculus were read by the set-...
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### $\mathbb{C}\mathbb{P}^1$ is homeomorphic to $S^2$

I just read on wikipedia that the the complex projective line is homeomorphic to the riemann sphere. How do I prove this? But, before that I have an extremely silly doubt that has been eating me. In ...
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### On the Usual Orientation of Cubic Graphs in Random Construction of Riemann Surfaces

In "Random Construction of Riemann Surfaces", Robert Brooks and Eran Makover say : Definition 2.1 A left-hand turn path on $(\Gamma, \mathcal O)$ is a closed path on [the cubic graph] $\Gamma$ such ...
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### Applications of Belyi's theorem

Belyi's theorem (1979) is stated as follows: A smooth projective curve over $\mathbb C$ is defined over a number field if and only if there exists a finite morphism (of varieties over $\mathbb C$...
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### Canonical divisor on algebraic curve

Can someone help me with this problem? Let $D$ be a divisor on an algebraic curve $X$ of genus $g$, such that $\deg D = 2g-2$ and $\dim L(D) = g$. Then $D$ must be a canonical divisor. By Riemann-...
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### Space of Germs of Holomorphic Function

A bit of a general question, but here goes. Morally, what is the space of germs of a holomorphic function? I know that a germ is simply an equivalence class of function elements, where we regard two ...
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### Books on complex analysis

Is there any book on $1$-dimensional complex analysis, where all is written in the language of sheaf theory? It's clear, that a lot of constructions can be formulated in simplier way using it. There ...
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I'm stuck with p.238 of Diamond's book on modular forms. If $\{f_j\}_{j=1}^g$ is the eigenform basis of $S_2(\Gamma_0(N))$, fixing $p_0 \in X_0(N)$ we have an holomorphic map $$\phi : X_0(N) \to J(X_0(... 1answer 202 views ### Extending isomorphism of punctured Riemann surfaces In Schlag's book on Riemann surfaces, we have Problem 4.15, which is: Let M, N be compact Riemann surfaces and suppose f: M\setminus\mathcal{S} \to N\setminus\mathcal{S}' is an isomorphism, ... 2answers 566 views ### Hyperbolic metric geodesically complete Consider the upper half plane model of the hyperbolic space (\mathbb{H} with the riemannian metric g=\frac{dx^2+dy^2}{y^2}). It is known that (\mathbb{H},g) is geodesically complete, which means ... 0answers 137 views ### Relation between n-tuple points on an algebaric curve and its pre-image in the normalizing Riemann surface This question is sort of an extension to this previous question of mine, Hyperellipticity (or not!) of a Riemann surface and the singularities of the curve If one knows the multiplicity of a ... 1answer 221 views ### Show that \int_{\partial P}z\frac {f'(z)} {f(z)} dz  is on the lattice \Lambda Problem: Let f(z) be a meromorphic function on the complex torus \mathbb C/\Lambda that as a function on \mathbb C has no zeros and no poles on \partial P, the boundary of the fundamental ... 2answers 2k views ### A more general definition of branched covering. If f:X\longrightarrow Y is a holomorphic map between two compact Riemann surfaces, then f is called also a branched covering map. This is because the branched points of f form a finite set S\... 0answers 193 views ### Why is every one dimensional Complex Manifold paracompact? I read on the page Why are smooth manifolds defined to be paracompact? in one of the answers that every one dimensional complex manifold is automatically paracompact, i.e. there is no complex analogue ... 0answers 165 views ### Is this complex structure unique? Let G be a subgroup of PSL(2,\mathbb{C}), so that G acts on \mathbb{C}\cup\{\infty\} by linear fractional transformations. We say that G acts properly discontinuously at a point z\in \... 3answers 209 views ### Show that a function from a Riemann Surface g:Y\to\mathbb{C} is holomorphic iff its composition with a proper holomorphic map is holomorphic. I'm trying to show the following: Let f:X\to Y be a proper holomorphic map between connected, non-empty Riemann Surfaces. Show that a map g:Y\to\mathbb{C} is holomorphic if and only if its ... 4answers 4k views ### Perspectives on Riemann Surfaces So, I have come to a somewhat impasse concerning my class selection for next term, and I have exhausted all the 'biased' sources. So, I was wondering if anyone in this fantastic mathematical community ... 1answer 3k views ### Understanding Ramification Points I really don't understand how to calculate ramification points for a general map between Riemann Surfaces. If anyone has a good explanation of this, would they be prepared to share it? Disclaimer: I'd ... 1answer 1k views ### The importance of modular forms I'm studying modular forms and my professor started the course talking about elliptic functions. These particular functions form a field (once that the lattice \Lambda is fixed) called E(\Lambda)=\... 1answer 2k views ### Describing Riemann Surfaces We've just learned about Riemann surfaces in my complex analysis class and to get a better understanding, I've been trying to find similar problems. I came across this: Describe the Riemann surfaces ... 2answers 3k views ### Complex analysis book with a view toward Riemann surfaces? I am considering complex analysis as my next area of study. There are already a few threads asking about complex analysis texts (see Complex Analysis Book and What is a good complex analysis textbook?... 2answers 1k views ### What can Lefschetz fixed point theorem tell us about the group of automorphisms of a compact Riemann surface? Let X be a compact Riemann surface of genus g>1, f\in Aut(X), a biholomorphism of X onto itself, x\in X a fixed point of f. Since tangent map of a holomorphic map (on the real tangent ... 1answer 384 views ### Constant Curvature Metric and Biholomorphic Equivalence This is probably a dumb question, but let's try it anyway. I know two versions of the uniformization theorem, and I don't understand their equivalence. The first says that every Riemann surface has a ... 2answers 3k views ### Good book for Riemann Surfaces I am considering reading one of 'Algebraic curves and Riemann Surfaces' by Rick Miranda or 'Lectures on Riemann Surfaces' by Otto Forster. Which one of these is more advanced and comprehensive ? What ... 2answers 335 views ### Can there be a point on a Riemann surface such that every rational function is ramified at this point? Let X be a compact connected Riemann surface, and let S\subset X be a finite subset. Does there exist a morphism f:X\to \mathbf{P}^1(\mathbf{C}) which is unramified at the points of S? I'm ... 1answer 1k views ### Computing the monodromy for a cover of the Riemann sphere (and Puiseux expansions) Given an irreducible polynomial P(X,Y) \in \mathbb{C}[X,Y], one obtains by analytic continuation a Riemann surface M with a branched covering f \colon M \to \mathbb{P}^1_{/\mathbb{C}}, ... 1answer 338 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 ... 3answers 2k views ### orientability of riemann surface Could any one tell me about the orientability of riemann surfaces? well, Holomorphic maps between two open sets of complex plane preserves orientation of the plane,I mean conformal property of ... 1answer 1k views ### Riemann surface with punctures corresponds to a hyperbolic surface with cusps I am reading a paper on Riemann surfaces and the author used the fact that \{Riemann surfaces with genus g and n punctures\} is in one-to-one correspondence with \{ hyperbolic surfaces ... 1answer 1k views ### Can a meromorphic function be written as ratio of holomorphic function? Well, I want to know whether a meromorphic function can be written as ratio of two holomorphic function on \mathbb{C} or on a Riemann surface. Thank you for help. 1answer 1k views ### Projective closure of an algebraic curve as a compactification of Riemann surface Assume f \in \mathbb{C}[x,y] a polynomial such that the affine algebraic curve X=V(f) has no singular points. Then there is a natural structure of non-compact Riemann surface on X, which can be ... 1answer 286 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 ... 1answer 244 views ### Divisor on curve of genus 2 I suffer from lack of concrete examples in Algebraic Geometry, so I will appreciate it if somebody can help me in understanding a bit better this one: Let \mathcal{C} be a genus 2 curve (... 1answer 810 views ### Application of Riemann-Roch The Riemann-Roch theorem states l(D)-l(K-D)=1 + deg(D) - g where D is a divisor on a compact Riemann surface X and K is the divisor of meromorphic 1-form. In the case g=2 and D=K this is ... 1answer 2k views ### Connected sums and their homology Edit: I already received a good answer to my second question. I'd be interested in a hint about the first one, as well. Thanks in advance! I'm interested in compact Riemann surfaces and their ... 2answers 529 views ### Why is a holomorphic map between compact connected Riemann surfaces a branched covering? I have seen it claimed that a non-constant holomorphic map f:X \rightarrow Y between compact connected Riemann surfaces is a branched covering i.e. surjective and there is a finite set \Sigma \... 3answers 3k views ### Can \mathbb RP^2 cover \mathbb S^2? I know \mathbb{S}^2 is the universal cover of \mathbb{R}P^2, but can \mathbb{R}P^2 be a covering space (at all) of \mathbb{S}^2? Attempt at solution It's clear that for a umramified covering ... 1answer 475 views ### Computing Riemann surfaces of a given algebraic function I've never seen written in a book a way or an algorithm for computing Riemann surfaces of a given algebraic function. I would like to know how to construct such Riemann surface using intuitive cutting ... 1answer 2k views ### Euler characteristic of a connected sum of surfaces. How do I see that the Euler characteristic of a connected sum of surfaces S_1 and S_2 is given by$$\chi(S_1 \# S_2) = \chi(S_1) + \chi(S_2) - 2?$$2answers 900 views ### Riemann surface arising as a quotient of the upper half-plane. Let H be the upper half-plane \{z \in \mathbb C \mid \Im(z) > 0\}. For a fixed real \lambda > 0, let be the automorphism$$d_\lambda : H \to H, z \mapsto \lambda z . Denote $\Gamma$ the ...
Let $S$ denote the Riemann Sphere. Recall that a Mobius transformation is a function $f:S \to S$ defines as $z \to \frac {az+b}{cz+d}$ where $a,b,c,d \in \mathbb C$ with $ad-bc=1$. What is the ...
Recall the hairy ball theorem : any continuous vector field on a sphere $S^n$ of even dimension must vanish at least once. Now consider the Riemann sphere $\mathbb{C} \cup \infty \simeq S^2$. Let \$X(...