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

learn more… | top users | synonyms (1)

1
vote
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 ...
0
votes
0answers
18 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 ...
0
votes
0answers
37 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 ...
1
vote
0answers
17 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 ...
2
votes
0answers
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 ...
4
votes
1answer
40 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 ...
0
votes
0answers
13 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 ...
8
votes
2answers
86 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 ...
2
votes
1answer
42 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 ...
2
votes
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 ...
1
vote
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 ...
2
votes
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 ...
2
votes
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 ...
0
votes
0answers
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?
0
votes
0answers
15 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 ...
1
vote
0answers
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 ...
3
votes
1answer
66 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 ...
1
vote
0answers
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 ...
2
votes
1answer
37 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?
3
votes
1answer
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 ...
1
vote
0answers
13 views

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 ...
0
votes
1answer
30 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 ...
0
votes
0answers
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 ...
0
votes
1answer
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} ...
0
votes
0answers
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 ...
1
vote
1answer
45 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). ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
7
votes
1answer
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 ...
0
votes
0answers
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 $L$ ...
0
votes
0answers
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
0answers
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 ...
0
votes
0answers
38 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 ...
1
vote
1answer
40 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 ...
2
votes
0answers
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 ...
2
votes
0answers
43 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 ...
0
votes
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 ...
0
votes
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 $\phi + c$. ...
0
votes
1answer
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 ...
3
votes
1answer
32 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 $\Phi|_{V \cap W}$ ...
0
votes
0answers
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) ...
1
vote
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 $\Delta \subset PSL_2(\mathbb{R})$ ...
0
votes
1answer
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?
1
vote
0answers
60 views

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 ...
3
votes
0answers
51 views

“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, ...
1
vote
0answers
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 ...
0
votes
0answers
27 views

tiling of Riemann surface of genus 2 by 12 heptagons

Let S be a Riemann Surface of genus 2. Is there a picture in the litterature for a tiling of S by 12 heptagons (I presume such a tiling exists; I am also aware of the beautiful pictures of the famous ...
4
votes
0answers
42 views

Compactness of the moduli space of bundles with fixed determinant

The moduli space of semistable holomorphic vector bundles of fixed rank and fixed determinant line bundle on a compact Riemann surface is known to be compact itself. (In particular, when the rank is ...