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 ...
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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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Border of riemann surface given by quotient of fuchsian group
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 \...
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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{...
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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 $...
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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 ...
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