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Questions tagged [riemann-surfaces]

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

Riemann surfaces are one-dimensional complex manifolds that are deformations of $\mathbb{C}.$

Riemann surfaces are orientable as a real manifold, and every simply connected Riemann surface is conformally equivalent to one of the following:

Elliptic (Positive curvature): The Riemann sphere (the complex plane with an extension to a point at infinity)

Parabolic (zero curvature): The complex plane

Hyperbolic (negative curvature): The open disk or upper half-plane

2132 questions
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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 prefer an explanation that avoids talking about…
Edward Hughes
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Differential forms

I failed to understand the definition of holomorphic $1$-form on Riemann surfaces. Can one explain it here? I saw two definitions in the books of Miranda and Farkas-Kra. Definition 1.: Suppose that $w_1=f(z)dz$ is a holomorphic $1$-form in the…
user8186
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Is there any obstruction other than Riemann-Hurwitz to the existence of covers of Riemann surfaces?

Suppose $X$ is a genus $g$ Riemann surface, and $h,d,e_i$ are positive integers such that $2-2g = d(2-2h) + \sum (e_i-1)$. Is there necessarily a Riemann surface $Y$ with a map $f: Y \rightarrow X$ such that $f$ has degree $d$, $Y$ has genus $h$,…
Tony
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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 holomorphic map implies that the local angles are…
Myshkin
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Proof of equivalence of conformal and complex structures on a Riemann surface.

I am trying to understand the fundamentals of Riemann surface theory and so far I have the following: --Definition 1. A conformal structure on a Riemann surface $\Sigma$ is an equivalence class of metrics $$ [g]=\{e^{2u}g \colon u\in…
Aerinmund Fagelson
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Intuitive Explanation of Riemann Surfaces and what they are used for?

I took my first class on Complex Analysis last semester and I wanted to continue learning more about it this semester doing independent readings. I was advised to read up on Riemann Surfaces but I'm having a hard time grasping what the motivation…
Sam
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Miranda Pag. 66 Plugging Holes

Let $X$ be a Riemann surface. A hole chart on $X$ is a complex chart $\phi: U \mapsto V$ on $X$ such that $V$ contains an open punctured disc $D_0=\{z: 0 < ||z-z_0 || < \epsilon \}$ with the closure in $X$ of $\phi^{-1}(D_0)$ inside $U$, and this…
TheWanderer
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de Rham cohomology on moduli of curves

I do not know the constructions of Deligne-Mumford; so let us suppose that the moduli space $\mathcal{M}_g$ of Riemann surfaces of genus $g$, with $g>1$, is constructed using the moduli of abelian varieties. Now given a point $x \in \mathcal{M}_g$,…
Espresso
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Flat embedded surfaces in arbitrary 3-manifolds.

It is generically known that any surface (at least embedded in $\mathbb{R}^3$ - and see edit below) can be deformed so that it is flat - i.e. has a metric of zero curvature - as long as one adds some conical singularities (for reference, here's an…
levitopher
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Any line in $\mathbb{P^2}$ is isomorphic to $\mathbb{P^1}$.

Any line in $\mathbb{P}^2$ is isomorphic to $\mathbb{P}^1$. I have a doubt we have to show isomorphism topologically or vector space or anything else? I have found a map which is the following: Let $X=\{[x:y:z]:F(x,y,z)=ax+by+cz=0\}$ ,where $a,b,c$…
jay sri krishna
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Proper map of Riemann surfaces

Consider a proper holomorphic map $f:X\to Y$ between two (connected, but not necessarily compact) Riemann surfaces. Is it true that $f$ is surjective whenever it is non-constant? In a lecture about Riemann surfaces, we proved the following…
Zuy
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Meromorphic functions on a Riemann surface

Let $X$ be a non-hyperelliptic Riemann surface of genus $g = 4$. Let $p,q,r$ be three points on $X$, not necessarily distinct. Is there any obstruction to the existence of a meromorphic function $f: X\to \mathbb{C}$, such that $f$ has simple poles…
Raziel
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compact Riemann surface and branched cover

Show that a compact Riemann surface admits a branched cover of the sphere with only simple branch points. I have this problem and it seems to me that the way is to use Riemann Roch theorem, but I do not know how to exactly apply! Any help is…
Manoel
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Meromorphic Function in a Compact Riemann Surface

Let $x_{1}, . . . , x_{n}$ be distinct points in a compact Riemann surface $\Sigma$ and let $w_{1}, . . . w_{n}$ be distinct points in $\mathbb{C}$. Show that there is a meromorphic function on which maps $x_{i}$ to $w_{i}$ for $i = 1, . . . , n$.
Manoel
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2 Definitions of Holomorphic functions on Riemann surfaces

In a lecture that I currently attend we defined Riemann surfaces and holomorphic mappings on it somewhat different than in another lecture that I attended a year ago. My question is: Are these definitions of Riemann surfaces and holomorphic mappings…
Kimmel
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