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

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Complex vector identity

Let $f=(f_1,f_2,f_3)$ be a complex vector. Can we see that $$G:=\frac{2\Im(f_2\bar{f_3})+i2\Im(f_3\bar{f_1})}{|f|^2-2\Im(f_1\bar{f_2})}=\frac{f_3}{f_1-if_2}$$ I tried using $f_j=x_{j,u}-ix_{j,v}$ ...
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1answer
22 views

Norm of a complex cross product

Let $c=(c_1,c_2,c_3)$ be a complex vector. How can we see that $\|c\|^2=\|c\times \bar{c}\|$? Here the bar means component wise complex conjugation, the norm is the Hermitian norm, and the cross ...
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1answer
20 views

Finding the Riemann surface of $w = z^{1/2}$

I'm trying to find the Euler characteristic of $R = \{(z,w) : f(w,z) = w^2 - z = 0\}$. To do this I'm using the Riemann-Hurwitz theorem with the projection $\Pi: R \to \mathbb{C}P^1$. Now in local ...
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21 views

Inverse of a constant function on an open set

I was working on holomorphic functions and Riemann surfaces, and I was wondering about the inverse of a constant function: Let $f:U\rightarrow V$ be a holomorphic function between two Riemann ...
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54 views

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 ...
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18 views

Equation of the curve corresponding to a principal polarization

Let $\mathbb{C}^2/\Lambda$ be a principally polarized abelian surface. I think it is well-known how to write down the equation of the divisor (Riemann surface) corresponding to the polarization, in ...
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18 views

Proving Riemann-Hurwitz formula for riemann sphere

Given a rational map $f:\hat{\mathbb{C}} \to \hat{\mathbb{C}}$, where $\hat{\mathbb{C}}$ is the Riemann sphere, I need to prove that $2\deg(f) - 2 = \sum (v_f(p)-1)$, i.e. prove the Riemann-Hurwitz ...
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41 views

Link between Riemann surfaces and Galois theory

In my notes for a Geometry of Surfaces course that I'm studying, there is the following quote: (For those of you who like algebra and Galois theory) Studying compact connected Riemann surfaces is ...
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14 views

relation between degree and residues

Let $C$ a compact riemann surface of positive genus and $\omega_C$ the canonical divisor over $C$ with standard degree $2g-2$. Take on $C$ a divisor of positive degree $d$ and set ...
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2answers
89 views

$\operatorname{Gal}(\overline{K}/K)=\mathbb{Z}$ possible?

Is it possible to have $\operatorname{Gal}(\overline{K}/K)=\mathbb{Z}$? My question comes from the link beetween covering and field extensions. For covering the simplest example is ...
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1answer
45 views

Finding holomorphic map on Riemann surface from a map between two Riemann surfaces

I have a non-constant degree two map between Riemann surfaces $R$ and $S$, $f: R \to S$. I'm trying to find a holomorphic homeomorphism $\tau: R \to R$ such that $f(\tau) = f$ and $\tau^2$ is the ...
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1answer
62 views

Sheaf, étalé space with Riemann surfaces.

Let $f:X\rightarrow Y$ be an holomorphic map betwen two Riemann surfaces and let: $\Gamma:=${ $(x,y)\in X\times Y|y=f(x)$ } $\subset X\times Y$ be the graph of $f$. I have to show that ...
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1answer
70 views

Riemann Roch Meromorphic section on a line bundle.

Let $g:\mathbb{C}\times \mathbb{C^*}\rightarrow \mathbb{C}\times\mathbb{C^*}$ defined by $g(z,w)=(w^n z,\alpha w)$ where $0<|\alpha|<1$. Let $G$ be the cyclic group spanned by $g$ and $A$ the ...
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1answer
30 views

How to prove that the flat torus is indeed flat?

The $n$-dimensional torus can be obtained as a quotient: $T^n=\mathbb{R}^n/\mathbb{Z}^n$. As pointed out here, the standard metric on $\mathbb{R}^n$ is invariant under translation by the elements of ...
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1answer
77 views

Riemann surfaces with Riemann Roch theorem, linear fiber over an elliptic curve

Let $g:\mathbb{C}\times \mathbb{C^*}\rightarrow \mathbb{C}\times\mathbb{C^*}$ defined by $g(z,w)=(w^n z,\alpha z)$ where $0<|\alpha|<1$. Let $G$ be the cyclic group spanned by $g$ and $A$ the ...
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23 views

Automorphisms of simple covers of Riemann surfaces

Can anybody give me a simple proof that simple covers of a Riemann surface have no covering automorphisms?
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18 views

Linearity of map between holomorphic tangent spaces

The proposition I am trying to prove is as follows: Let $f:X\to Y$ be a holomorphic map between two Riemann surfaces. For each $x\in U$ the map $$ Df(x):T_{X,x}^{1,0}\ni [h]_x\mapsto [f\circ ...
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13 views

Local conformal coordinates on a surface

Let $\mathcal{M}\subset\mathbb{R}^3$ be a smooth enough regular surface. We want to show that around a point $p\in\mathcal{M}$ there is a neighborhood about $p$ in $\mathcal{M}$ which is parametrized ...
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1answer
67 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 ...
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39 views

Mapping a curve into projective space

Let $\mathcal{C}$ be a (smooth, complex, projective) genus 2 curve. Take two different points $p,q\in\mathcal{C}$ and let $K$ be the canonical divisor class. I know (by means of Riemann-Roch) that the ...
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1answer
39 views

$j$-invariants of isogenous elliptic curves

Suppose that $E,E'$ are isogenous smooth complex elliptic curves - is there some relation between their $j$-invariants?
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57 views

Difference between Euler characteristics of a Riemann surfaces

Let $X$ be a compact connected Riemann surface of genus $g$. Let $U$ be the complement of $r$ points in $X$. The Euler characteristic of $X$ = $2-2g$. That I understand. But I'm confused about the ...
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Question on Fulton's coverage of Riemann surfaces

Riemann surfaces beginners question: (I am learning about normalization of algebraic curves for the first time using Fulton's Algebraic topology and was doing fine until i hit a this snag) SHORT ...
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31 views

Surface constructed using curves

Suppose that $E$ and $F$ are two complex compact Riemann surfaces with genus greater or equal than $2$. Set $$S=E \times F$$ the surface composed by the cartesian product of thees curves. What can i ...
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7 views

Torus-like Riemann surface for a genus 1 Cassini oval

Among the Cassini ovals there is the lemniscate of Bernoulli. This latter curve has genus $0$ and can be mapped to a standard Riemann sphere: $$ (R(t+t^3), R(t-t^3), (1+t^4)/\sqrt{2}) \qquad t\in ...
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37 views

Distance on riemann sphere [duplicate]

Let we have $C$ the set of complex numbers and $z_1 , z_2 \in C $ we have $Z_1 , Z_2 \in S$ correspond on riemann sphere and we will define : $$ d(Z_1,Z_2)=\frac{2|z_1-z_2|}{\sqrt{1+|z_1|^2} ...
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28 views

a question about finding umbilical points in an elipsoid.

Determine the umbilical points of the elipsoid $${x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2}=1.$$ My thoughts: let $x=asin(\theta)cos(\phi),y=bsin(\theta)cos(\phi),$and $z=cos(\theta)$. Thus, I ...
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1answer
46 views

Extending the metric of a hyperbolic surface with boundary to its double

Let $M$ be a hyperbolic surface with totally geodesic boundary. Taking the double $DM$ of $M$, it is easy to see using Euler characteristic that $DM$ is itself a hyperbolic surface (without boundary). ...
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32 views

Rational section of the canonical line bundle of a smooth curve

Let $C$ a complex Riemann surface with genus $g>0$, $L$ a theta characteristic on $C$ i.e $L \in Pic(C)$ such that $L^2 \equiv \omega_C$ where $\omega_C$ is the caninical line bundle on C and ...
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23 views

Constructing Riemann surfaces and holomorphic functions

How do I construct a Riemann surface $S$ and holomorphic functions $f:\mathbb{C}-[\pm1] \rightarrow S$, $g:S \rightarrow \mathbb{C}-[\pm1]$ such that f is conformal, g surjective (every point having ...
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111 views

First derivative of the Weierstrass $\wp$ function as a function on $\mathbb{C}/\Lambda$

I am currently trying to prove various facts about $\wp'$, considered as a meromorphic map from $\mathbb{C}/\Lambda\to\mathbb{C}$, where $$\wp'(z) = -2\sum_{w\in\Lambda}\frac{1}{(z-w)^3}.$$ In ...
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27 views

Question about a notation of line bundle

If $C$ is a complex Riemann Surface with positive genus, $D$ a divisor on $C$, $L$ a line bundle of $C$, with the term $L(D)$ what do we mean? I have this idea: $L(D)$ set of all sections of $L$ ...
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58 views

Question to understand how sections work

Suppose that $C$ is a compact Riemann surface o positive genus $g$. Let $L$ a linear bundle on $C$ . Chosen a divisor $D$ on $C$ we consider the set $L(D)$ that is the tensor product between $L$ and ...
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2answers
70 views

Is the Riemann surface for the square root simply connected?

I am looking for universal covering spaces and I am now wondering if the Riemann surface for the square root $z^{1/2}$ (or even more general for $z^{1/n}$) is simply-connected and therefore a ...
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1answer
45 views

Are all 2D tensors in a specified flat metric equal to that same metric conformally scaled?

I have a tensor $T_{mn}$ where its indices coorespond to a flat metric $g_{mn}$. I want $T_{mn}$ to be a new metric $\tilde{g}_{mn}$, such that $T_{mn}(g_{rs}) = \tilde{g}_{mn}$. A theorem says that ...
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37 views

Line bundle of degree 1 on a genus 2 surface with 2 independent global holomorphic sections

By Riemann-Roch, for a degree 1 line bundle on a genus 2 Riemann surface the space of global holomorphic sections has dimension between $0$ and $2$. Is there an explicit example of a degree 1 line ...
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39 views

Degree and Ramification points of an holomorphic map between Riemann Surfaces

The question is the following: we have an holomorphic map from $\Bbb P^1$ to $\Bbb P^1$, defined by $f(z)=z^3-3z$. I need to find the degree and the ramification points and their orders, then verify ...
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41 views

Spectral representation of an analytic function

I have a question about the spectral representation of an analytic function $G$ on a Riemann surface (specifically, the complex plane with a finite amount of cuts), i.e. the representation of the form ...
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38 views

Line bundle of degree 1 on a genus 2 surface without global holomorphic sections

By Riemann-Roch, for a degree 1 line bundle on a genus 2 Riemann surface the space of global holomorphic sections has dimension between $0$ and $2$. Can someone show an explicit example of a degree 1 ...
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44 views

Graph Jacobian (Sandpile group) usages

Let $\Gamma$ be a graph (say, finite) and $S_\Gamma$ be it's Jacobian (also known as the sandpile group or Picard group). I'm wondering about what fundamental things one can learn about $\Gamma$ from ...
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1answer
41 views

dimension of space of modular functions using the Riemann-Roch theorem?

Let $H$ be the upper half-plane, and $M_k$ be the space of modular forms of weight $k$ on $H$ under the action of $SL(2,\mathbb{Z})$. I have read (Koblitz, Introduction to Elliptic Curves and Modular ...
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1answer
35 views

Coordinate on the boundary of Riemann surface

Let $\Sigma$ be a Riemann surface with boundary. Question: Is there canonical way to parameterise the boundary components up to shift? By shift I mean change of coordinate $\phi$ to $\phi + c$. ...
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33 views

Existence of Gluing of Riemann surfaces

Consider two copies of holomorphic disks $\{ z \in \mathbb{C} \ | \ |z| \leq 1 \}$. Denote them by $\Delta_1$ and $\Delta_2$. Let $f$ be a diffeomorphism from boundary of first disk to boundary of ...
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33 views

Union of holomorphic atlases is holomorphic atlas.

Let $S$ be a surface with open subsets $V$ and $W$ such that $s = V \cup W$. Suppose that $V$ and $W$ have holomorphic atlases $\Phi$ and $\Psi$ such that the holomorphic atlases $\Phi|_{V \cap W}$ ...
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23 views

Topological structure of the Riemann sheets?

In complex analysis, for me, the most intriguing thing is the branch cut of a function. The multi-valuedness makes life very difficult. Presumably, for some functions, like $$ f(z) ...
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1answer
22 views

Canonical map from fundamental group to Fuchsian group?

Suppose we have a Riemann Surface $S$ of constant negative curvature $-1$. What is the canonical map from the fundamental group $\pi_1(S)$ to the discrete subgroup $\Delta \subset PSL_2(\mathbb{R})$ ...
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38 views

Alebgraic curve and Riemann surfaces

How do we prove that any smooth complex algebraic curve $C\subset\mathbb{P}^2$ is a Riemann surface? Does there exist a complex version of the implicit function theorem?
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Maximum principle of harmonic function on compact manifold

Thm . (Maximum Principle) Let h be a harmonic function on a domain D in C . (a) If h attains a local maximum in D then h is constant. (b) Suppose that D is bounded and h extends continuously to the ...
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“Pulling back a function from a neighborhood of 0 in $\mathbb{C}^2$ to $\mathbb{U}$ is analogous to computing the derivative of that function.”

I'm reading on Riemann Surfaces and after defining $$\mathbb{U} \equiv \{(z,\ell)\in\mathbb{C}^2\times\mathbb{P}_1;z\in\ell\},$$ and $B\ell_0:\mathbb{U}\to\mathbb{C}^2$, the author mentions that, ...
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26 views

Given a smooth 2-form $\varphi$ on a punctured Riemann surface, is there $\nu$ of type (1,0) such that $\varphi=d\nu$?

Let $X$ be a compact Riemann surface, $p\in X$, and $\varphi$ be a smooth 2-form on a $X-\{p\}$, and hence exact. I'm wondering if it is possible to find a form of type (1,0) whose differential is ...