All Questions
Tagged with complex-analysis riemann-surfaces
799
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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 ...
- 85
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36
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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 ...
- 1,249
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28
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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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43
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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 ...
- 94
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34
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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 ...
- 424
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36
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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 ...
- 715
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34
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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 \...
- 1,642
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67
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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 $...
- 7,499
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64
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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 ...
- 7,499
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13
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$\operatorname{det}(Z_0T_0 + Z_1T_1 + Z_2T_2) = 0$ 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 $\operatorname{det}(Z_0T_0 + Z_1T_1 + Z_2T_2) = 0$ ...
- 37
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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
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31
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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
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22
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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
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16
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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 ...
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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 ...
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