The riemann-surfaces tag has no wiki summary.
5
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1answer
352 views
Structure of the automorphism group of a Riemann surface
I was wondering if anything is known about the possible structure of $\mathrm{Aut}(S)$ for a Riemann surface $S$. More precisely, are there known obstructions for a finite group $G$ to be such an ...
1
vote
0answers
87 views
Torsion elements in the mapping class group
Let $S$ be an orientable surface of genus $g$ with $b>0$ boundary components, and let $\mathrm{Mod}(S)$ be its mapping class group, that is, the group of isotopy classes of its homeomorphisms ...
5
votes
1answer
100 views
Translation surfaces & integer multiples of $\pi$
Richard Schwartz, in Mostly Surfaces (Vol. 60. American Mathematical Soc., 2011),
defines (on p.14) a translation surface as "a Euclidean cone surface, all of whose 'angle errors' are integer ...
1
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0answers
46 views
An explicit description of the Torelli spaces of pointed genus 2 Riemann surfaces
In [N], there is a nice and very explicit description of the Torelli space ${\rm Tor}_{1,n}$ of $n$-pointed elliptic curves, for any $n\geq 1$:
$$
{\rm Tor}_{1,n}=\left\{
\big(\tau, ...
1
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1answer
263 views
Is there a toplogically trivial line bundle over a compact Riemann surfaces that isn't holomorphically trivial? [closed]
Is a complex line bundle over a compact Riemann surface topologically trivial iff it is holomorphically trivial? If so, how does one demonstrate that, and if not, what is a counterexample?
1
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0answers
30 views
Given a smooth 2-form $\varphi$ on a punctured Riemann surface, is there $\nu$ of type (1,0) such that $\varphi=d\nu$? [migrated]
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 ...
3
votes
1answer
163 views
When does a hyperelliptic Riemann surface admit a map of degree 3
Let $X$ be a hyperelliptic curve of genus $g>1$.
For which $g$ does $X$ admit a map $X\to \mathbb P^1$ of degree $3$?
I think a genus two curve $X$ admits a map of degree $3$.
Proof: Pick $P$ ...
0
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0answers
102 views
Coordinate charts on converging Riemann surfaces
Let $S$ be a $2-$dim manifold and $q \in S$. Furthermore, let $j_{n}$ be a sequence of complex structures on $S$ converging in $C^{\infty}_{\text{loc}}$ to a complex structure $j$ on $S$ as ...
11
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0answers
101 views
Canonical Immersion of the Double Torus
It is easy to check that the immersion $\mathbb{T}^2=\mathbb{S}^1\times \mathbb{S}^1\longrightarrow\mathbb{R}^4$, $(\alpha,\beta)\longmapsto(\cos\alpha,\sin\alpha,\cos\beta,\sin\beta)$ induces the ...
0
votes
1answer
154 views
existence of positive curved line bundles on a compact Riemann surface
Can anyone suggest a proof of the existence of positive line bundles on a compact Riemann surface, avoiding Hodge decomposition. (I am aware of the method in Dror Varolin's book, but I consider that ...
0
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0answers
50 views
Analytic functions space on Riemann surface
I have some questions about the analytic function space on Riemann surface and distinguished varieties:
Let S be a compact Riemann surface and $\Omega\subset S$ be a domain with piecewise smooth ...
1
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0answers
190 views
Ramification: Riemann surfaces vs Number fields
I am trying to understand the connection between Riemann surfaces and number fields. I am wondering if there an inconsistency in the definition of ramification in terms of Riemann surfaces vs number ...
4
votes
2answers
178 views
Projective curves of constant curvature
A nodal projective curve in $\mathbb{CP}^2$ inherits a Kähler metric from the Fubini-Study metric, and hence a Riemannian metric. In particular, with respect to this metric, a line has constant ...
0
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0answers
103 views
Kodaira-Spencer theory in d=1
Consider genus $g$ Riemann surfaces, and thier moduli space $\mathcal M_g$.
To determine dimension of $T\mathcal M_g$,
start with a complex structure, which in some coordinates can be written
...
2
votes
2answers
111 views
Dilatation of surface diffeomorphisms
Let $S$ be a higher genus surface, let $f\colon S\to S$ be a diffeomorphism and let $f_*\colon H_1(M)\to H_1(M)$ be the induced homology automorphism.
Define dilatation of $f_*$ as the largest ...
2
votes
0answers
108 views
Teichmuller geodesics vs. geodesics in the hyperbolic plane
Geodesics in $\mathbb H^2$ have the following properties:
For every two points in the plane there exists a unique geodesic joining them.
Every geodesic determines exactly two points on the ...
8
votes
4answers
354 views
Is there a non-abelian version of the Torelli map?
Let $C$ be a connected compact oriented real surface of genus $g$, let $G$ be a connected compact Lie group and let $G_\mathbb{C}$ be the complexification of $G$. One considers the moduli space $M ...
1
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0answers
139 views
Marten's proof of torelli theorem
I am trying to read the proof of torelli theorem by Henrik H.Martens "A new proof of torelli's theorem" Annals of mathematics vol78 no. 1 .The proof seems to me like using mysterious combination of 3 ...
3
votes
1answer
149 views
Is the fundamental group of an open arithmetic Riemann surface contained in $\Gamma(2)$
Let $X$ be a non-compact Riemann surface with universal covering $\mathbb H$ and suppose that the fundamental group of $X$ is an arithmetic subgroup of $\mathrm{Aut}(\mathbb H) = ...
1
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0answers
450 views
A metric on $S^{2}$ [closed]
Edit:Can this new version of this question be answered with the method of same comments to the previous version?
Let $p:S^{3}\to S^{2}$ be the Hopf fibration $p(z,w)= (\parallel ...
9
votes
3answers
386 views
Quotient of the hyperbolic plane with respect to commutator group of $\pi_1(\Sigma_g)$
Let $\Sigma_g$ be a Riemann surface of genus $g\geq 2$ and $G=\pi_1(\Sigma_g)$.
Let $\pi\colon \mathbb{H}\to \Sigma_g$ be the universal covering map. What kind of surface is $\mathbb{H}/[G,G]$?
...
1
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1answer
150 views
How to understand a rooting of a dessin d'enfant?
As I understand it, rooted maps on surfaces were first introduced in enumerative combinatorics because they are easier to count than unrooted maps, which can have non-trivial symmetries. A map is a ...
0
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0answers
188 views
Holomorphic vector bundles over $\mathbb{CP}^1$ and elliptic curves
Can anyone suggest a good exposition of the classification of holomorphic vector bundles over $\mathbb{CP}^1$? Also does there exist any analytic or more elementary proof of Atiyah's classification of ...
2
votes
1answer
106 views
Intersection of closed geodesics in hyperbolic surface
This question may be easy but I could not come up with a proof.
Let $F$ be a hyperbolic surface of finite type (with finitely many boundary and finitely many puncture). Let $\gamma$ be a closed ...
3
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0answers
139 views
different proofs of the fact that compact riemann surface has a non-trivial meromorphic function
I would have like to have a list of different proofs of the fact that compact riemann surface has a non-trivial meromorphic function.This is certainly one of the main results of compact riemann ...
3
votes
1answer
103 views
Connection between Strebel differentials, ribbon graphs, and Belyi maps
In this paper, a nice story is woven regarding the connection between quadratic differentials on Riemann surfaces, so-called 'ribbon graphs' drawn on those surfaces, and Belyi maps. However, I am ...
0
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0answers
208 views
Understanding a proof of a lemma in elliptic surfaces
In the following paper , The Kahler-Ricci flow on surfaces of positive Kodaira dimension (Sung and Tian) in page 621, I am trying to understand the proof of part 1 of Lemma 3.4.
In the part 1 of ...
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0answers
88 views
Finding Riemannian metric explicitly from the conformal structure on a surface
Each Riemann surface structure $S'$ on a topological surface $S$ can be obtained from a Riemannian metric which looks like $ds^{2} = g_{11}dx^{2} + 2g_{12}dxdy + g_{22}dy^{2}$ in local coordinates ...
1
vote
1answer
113 views
Singular leaf of Strebel differential
Let $R$ be a Riemann surface.Let $\gamma$ be a loop which is non-trivial in $H_{1}(R,\mathbb{Z})$. By the Jenkins–Strebel Theorem we know the following: there exists a holomorphic quadratic ...
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0answers
49 views
Riemann surface of $K_0(z)$
The question concerns the modified Bessel function of the second kind of order zero ($K_0(z)$).
How does its Riemann surface looks like? How can one evaluate its values on the "other" sheet(s)?
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0answers
111 views
The type of a Riemann surface arising from a polynomial vector field
Consider the planar polynomial vector field $$\begin{cases} \dot x=P(x,y)\\ \dot y=Q(x,y)\end{cases}$$
It defines a singular foliation on $\mathbb{C}P^{2}$. Assume that a complex leaf contains ...
1
vote
2answers
217 views
Finding an algebraic equation given divisors
I'm trying to find an algebraic curve that represents a specific Riemann surface and my question goes like this:
Given divisors
$(\omega_1) = P_1 + 5 P_2 + 2 P_3,$
$(\omega_2) = 5 P_1 + P_2 + 2 P_3,$
...
4
votes
1answer
146 views
Examples of hyperelliptic curves with hyperelliptic quotients that have more automorphisms
Does there exist a hyperelliptic curve $X$ of genus $g\geq 2$ over the complex numbers such that $X$ has a hyperelliptic quotient $X\to Y$ (in the sense that $Y$ is hyperelliptic and the morphism ...
3
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0answers
223 views
Complex structures on Riemann surfaces
This is cross posted from math.SE: http://math.stackexchange.com/q/876432/9
Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq (H^0(M;K^2))^*$. Considering $\alpha$ as a map ...
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2answers
798 views
History of the connection between Riemann surfaces and complex algebraic curves
As noted in the question "Links between Riemann surfaces and algebraic geometry", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces ...
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3answers
296 views
A special case of the uniformization theorem
I am interested in a proof of the following fact :
Suppose that $X$ is a Riemann surface homeomorphic to the Riemann sphere. Then $X$ is conformally equivalent to the Riemann sphere.
Of course, this ...
0
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1answer
124 views
Any no-zero homomorphism of holomorphic vector bundles over a compact Riemann surface factors through a maximal rank homomorphism
I was reading the paper "Stable and Unitary vector bundles on a compact surface" by Narashiman and Seshadri.
I quote from the paper-
Can someone please explain how does any non-zero homomorphism ...
1
vote
2answers
261 views
Holomorphic vector bundles over a Riemann surface does not satisfy $\mathbf{AB2}$ but satisfies $\mathbf{AB1}$
in Grothendieck's Tohoku paper (page 127), Grothendieck asserts that the category of holomorphic vector bundles $\mathbf{Bund}(X)$ over a fixed Riemann surface $X$ does not satisfy $\mathbf{AB2}$, but ...
4
votes
1answer
296 views
searching for an elementary proof a complex analysis result
Given a function $ g $ entire on the whole complex plane $ C $, it is possible to find an entire function $f $ such that $ f(z+1) -f(z)=g(z) $. The proof can be given using riemann ...
3
votes
1answer
140 views
Does every canonical decomposition of the intersection form come from a canonical homology basis?
Take a closed surface $X$ of genus $n$. By a canonical homology basis, I will mean a set of $2n$ homologically independent simple closed curves $\{\alpha_1,\ldots,\alpha_n,\beta_1,\ldots,\beta_n\}$, ...
1
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0answers
62 views
some intuition about the degree of a map
Consider a map
$$ f: \Sigma \to X/\sigma,$$
where $\Sigma=\Sigma_g/\Omega$ is a quotient of a Riemann surface by an antiholomorphic involution,
$\sigma:X\to X$ is an antiholomorphic involution
of some ...
1
vote
1answer
130 views
Is the structure constant additive on connected components?
Let $M$ be a Riemann surface and $\mu$ a metric on it, which could be non-compact. Moreover let $\Delta_{\mu,\,M}$ be the Laplacian on $M$ induced by $\mu$ and $\mathrm{det}^*(\Delta_{\mu,\,M})$ its ...
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0answers
76 views
Decomposition of the canonical flat connection on $\tilde M\times_{\rho} SL(n,\mathbb{C})$
I'm looking for a proof resp. reference for a statement of the following form:
Let $M$ be a compact Riemann surface, $\tilde M$ its universal covering, $\rho$ a semisimple representation of its ...
3
votes
3answers
181 views
Is it possible to find $h$ hermitian metric such that $Isom_{h}(X) \cong Aut(X)$?
I suspect this is true for some class of analytic manifolds (Riemann surfaces maybe), but my knowledge in differential geometry is very poor, so I could not conclude it. For complex manifolds, is it ...
1
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2answers
670 views
Spicing up Riemann surfaces course (revised)
I am a master's student planning to write a master's thesis on Riemann surfaces. I plan to study Forster's Lectures on Riemann surfaces. What side topics could one study to spice up the thesis? I am ...
1
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1answer
78 views
Generators for the affine automorphism group of the octagon
Consider an octagon $O$ with opposite edges identified. Lemma 3.2.4 of (1) (subscription link) claims that the affine automorphism group of $O$ is generated by $D_8$ and the shear $\sigma$ such that, ...
0
votes
1answer
107 views
A question for hyperbolic metric in the proof for Bohr's lemma
Recently I was reading an interesting proof for Bohr's lemma by the tool of hyperbolic metric, however I have a following question:
Given a holomorphic map $f$ on $D$, $f(0)=0$, and $|f|<1$ on ...
2
votes
2answers
274 views
Etale covers of a hyperelliptic curve
Let $X$ be a hyperelliptic curve of genus at least two.
Let $Y\to X$ be a finite etale morphism with $Y$ connected.Then $Y$ is a smooth projective connected curve.
Is $Y$ hyperelliptic?
More ...
2
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1answer
114 views
Uniform estimate for the Cauchy-Riemann equations on a hyperbolic Riemann surface
I have been trying to find the answer to this question in the literature, but have not succeeded. The question is as follows.
Suppose $X$ is a Riemann surface that admits a green's function (i.e. ...
3
votes
0answers
113 views
Criterion for a subgroup of $PSL_2(\mathbb R)$ to be Fuchsian
Let $\Gamma$ be a finitely generated subgroup of $PSL_2(\mathbb R)$.
I'm looking for effective criteria (idealy necessary and sufficient but just necessary would be a good start) ensuring that ...
