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# Questions tagged [riemann-surfaces]

Riemann surfaces(Riemannian surfaces) is one dimensional complex manifold. For questions about classical examples in complex analysis, complex geometry, surface topology.

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### What finite simple groups appear as factors of surface fundamental groups?

Let $\Sigma_g$ be the a closed orientable surface of genus $g$. My somewhat naive question: what is known about simple finite factors of $\pi_1(\Sigma_g)?$ In particular, I know that the composition ...
55 views

### Disk with punctures and convex geodesical hull of the punctures isomorphic?

Consider a unit disk with marked points $z_i$, $i=1, \dots , n$ on its boundary. Let us call this surface $X$. As it is well known, the disk can be equipped with an hyperbolic metric and is then ...
120 views

### On a map between Riemann surfaces of genus $1$

Let $C$ be a compact Riemann surface of genus $1$, and $p\in C$, and $w$ be a local holomorphic coordinate on $C$ near $p$ with $w=0$ at $p$. As usual, for a divisor $D$ denote by $L(D)$ the vector ...
52 views

Let $M$ be an open Riemann surface and $f$ a multivalued holomorphic function from $M$ to $\mathbb{H}$, where $\mathbb{H}$ is the upper half plane. Suppose that the monodromy of $f$ lies in $A=\left\{\... 0answers 116 views ### Algebra of meromorphic functions on a Riemann surface Let$C$be compact Riemann surface of high genus. Let$p$be a general point on$C$and$z$a local coordinate around$p$. Given a meromorphic function on$C$, regular outside$p$, we can look at its ... 0answers 230 views ### Is there an algorithm to compute a Belyi map for the Riemann surface? Let$y^2=x^5-x-1$be an affine model of a projective complex curve, is there an algorithm to compute the Belyi map (preferably of small degree), i.e., map to the projective line ramified only at$\{0,...
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### Existence of a connection $A$ on a holomorphic line bundle $L$, s.t $F(A)=(\deg L)\omega$

I'm reading this paper and at page 67, he states that for any line bundle $L$ over a Rieman surface there is a connection $A$ whose curvature is $$F(A)=(\deg L)\omega,$$ where $\omega$ is a positive ...
239 views

### On a corollary of a paper by Colin and Honda

The question is about the last sentence of the last corollary of Stabilizing the monodromy of an open book decomposition by Vicent Colin and Ko Honda. This question is also related to this other ...
124 views

### Space of biholomorphic maps into a Riemann surface

Let $F$ be a Riemann surface and $Q\in F$. Consider $U:=(\mathbb{C}\cup\infty)\setminus [-1,1]^2$. I am interested in the space X:=\{f:U\to F;\,\text{$f:U\to f(U)$ biholomorphic and $f(\infty)=Q$}\},...
121 views

### Ramification concept in complex analysis and algebraic number theory

I have a question about the connection between the concept of ramification/branching out for Riemann surfaces and algebraic number theory: In the theory of Riemann surfaces we have following ...
271 views

I know that this is supposed to be standard, but I don't know how to search for it... hence the question: Let $J$ be an almost complex structure on $M:=\mathbb R^2$, i.e., a $C^\infty$ section of $\... 0answers 49 views ### Criteria for a limit to be a proper function This question is obviously broad; turning this broadness into something sharp is part of the problem. Given a sequence of functions defined on a Riemann Surface$R$, valued in$\Bbb C^2$, what ... 0answers 130 views ### Stabilizing an open book with Anosov piece It was proven by Colin and Honda in Stabilizing the monodromy of an open book decomposition that any diffeomorphism can be made pseudo-Anosov and right-veering after a series of positive ... 1answer 158 views ### Genus of non-reduced curves Let$X$be a smooth projective variety of dimension$3$, and$L$an ample line bundle with$h^0(X,L)\geq 2$. Let$s$and$t$be two generic linearly independent sections of$L$, and$C$the curve that ... 1answer 155 views ### Notational question about quadratic differentials in Strebel's book “Quadratic differentials” In Kurt Strebel's book "Quadratic Differentials", in Chapter 2,$\S4$, he begins by saying: "Every analytic function$\varphi$is a domain$G$of the$z\$-plane defines, in a natural way, a field of ...

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