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.
676
questions
4
votes
2
answers
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Simple Closed Hyperbolic Geodesics on Punctured Spheres
Thinking of $\mathbb {CP^1}$ as the sphere $S^2\subset\mathbb R^3$, we can define the notion of a circle on it to be a subset that is got by a hyperplane section of $S^2$ inside $\mathbb R^3$. This ...
1
vote
0
answers
435
views
Path lifting property of holomorphic unbranched map
Suppose $X$ is a Riemann surface and $ a\in\ X $ suppose $ \phi\in\mathcal O_a $ is a holomorphic function germ at $a.$ According to the theorem 7.8 of Forster's book Lectures on Riemann surfaces on ...
13
votes
2
answers
471
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Geodesic current supported on a pencil?
Consider a geodesic current $\mu$ on a closed surface $\Sigma$, as defined by Bonahon ("The Geometry of Teichmüller space via geodesic currents"). These are $\pi_1(\Sigma)$-invariant measures on the ...
16
votes
2
answers
1k
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Are mapping class groups of orientable surfaces good in the sense of Serre?
A group G is called ‘good’ if the canonical map $G\to\hat{G}$ to the profinite completion induces isomorphisms $H^i(\hat{G},M)\to H^i(G,M)$ for any finite $G$-module $M$. I’ve had multiple academics ...
4
votes
2
answers
392
views
Unramified map of Riemann surfaces
Let $f:S \to T$ be a surjective, unramified, holomorphic map between connected Riemann surfaces. If $S$ is not compact is it always true that $f$ is a covering?
This is of course true if $S$ is ...
3
votes
1
answer
411
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non-existence of global coordinates
Assume we have a smooth manifold, $M$, of dimension $n$. (An example of interest is the case when $M$ is a compact and orientable Riemann surface of genus $g$, but the question is intended to be broad....
4
votes
3
answers
748
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Naive question on the Jacobian of a curve
Let $X$ be a smooth, projective curve of genus $g \ge 2$. We know that the Jacobian $J(X)$ of the curve is a principally polarized abelian variety. The principal polarization is induced by the ...
0
votes
1
answer
160
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How to classify a plane complex curve?
Let $p_1, p_2, t_1, t_2, a \in \mathbb{C}$ be constants. Consider the following plane complex curve in $\mathbb{C}^2$ ($c_1, c_2$ are indetermniates)
\begin{align}
& {p_1}^2 {p_2}^2 c_1 {t_1}^2 ...
3
votes
0
answers
282
views
Reference request: basics about modular curves
Where can I find a reference (with carefully written proofs) for basic facts about modular curves? Namely:
Congruence subgroups
The open modular curve $Y_\Gamma$ admits the structure of a Riemann ...
6
votes
2
answers
404
views
Harder Narasimhan filtration for the endomorphism bundle
Let $E$ be a vector bundle over a compact Riemann surface $X$, and let $$0=E_0\subsetneq E_1\subsetneq \ldots \subsetneq E_n=E$$ be its Harder-Narasimhan filtration: we have $V_i:=E_i/E_{i-1}$ ...
11
votes
0
answers
255
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Cluster algebra and Fenchel Nielsen coordinates
Certain cluster algebras arise from ideal triangulations of hyperbolic Riemann surfaces. The combinatorics behind their mutations can be understood in terms of "flips" in the triangulation, and the ...
2
votes
1
answer
886
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holomorphic sections of line bundles on Riemann surfaces
I have something elementary to ask. Let $E\rightarrow X$ be a holomorphic line bundle over a Riemann surface. Then in general a section of $E$ is a meromorphic function on $X$, since $O_{div(s)}\cong ...
1
vote
0
answers
41
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Is $O_{D}(X)$ a reflective Banach space?
Let $X$ be a Riemann surface and $O_{X}(D)$ be the line bundle associated with $D$. Let the metric on $O_{X}(D)$ be given by
$$
|1_{O_{X}(D)}(P)|=G(P,D)^2
$$
where $G(P,D)^2$ is the Green function ...
8
votes
0
answers
212
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Are spin Hurwitz numbers $r$-spin Hurwitz numbers?
(I think the answer is no, but I'm not sure.)
In Hurwitz theory, one counts $n$-fold branched covers $\Sigma'\to\Sigma$ of a Riemann surface $\Sigma$ with fixed
ramification data around each branch ...
2
votes
0
answers
40
views
Asymptotic flag in terms of geometry of the stratum of abelian differentials?
Let $C$ be a closed Riemann surface of genus $g\geq 1$. Fix a holomorphic 1-form on $C$; it endows $C$ with a flat structure (i.e. a metric of trivial holonomy which has conical singularities at a ...
2
votes
1
answer
106
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Equality on $\partial \mathbb{H}$ of lifts for isotopy to a conformal map
Let $\mathbb{H} \subset \mathbb{C}$ be the upper half plane. First recall the following statement: if $f^* \colon \mathbb{H} \rightarrow \mathbb{H}$ is quasi-conformal (qc), then there exists an ...
1
vote
0
answers
68
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For the geometric meaning of this value for complex curve with model over $\mathbb{Q}$
Let $X$ be a smooth projective algebraic curve defined over $\mathbb{Q}$ with genus $g$, we have an isomorphism $H^1_{dR}(X/\mathbb{Q})\otimes_\mathbb{Q}\mathbb{C}\cong H^1_{sing}(X,\mathbb{Z})\...
5
votes
0
answers
352
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Is there a relationship between the zeta function of a Laplacian and the Selberg Zeta Function?
Let us define the zeta function of an elliptic differential operator $H$ with eigenvalues $\lambda_n$ like so:
\begin{aligned}
\zeta_H(s)
:=
tr( H^{-s} )
\\
:=
\sum^\infty_{n=1} ...
0
votes
0
answers
209
views
Classifying $PGL(n,\mathbb{C})$-bundles over a compact Riemann surface
Let $X$ be a connected compact Riemann surface. How does one go on proving that the set of PGL($n,\mathbb{C}$)-bundles over $X$ is topologically classified by $\pi_1(PGL(n,\mathbb{C}))$? Is it true ...
1
vote
0
answers
116
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Differentials on tori realised as double of annuli
In this question it was described how to realise a torus as the double of an annulus Explicit construction of mirror surface and complex double for an annulus.
In short, the torus is realised ...
5
votes
1
answer
211
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Uniqueness of Mukai presentation of canonical model in genus 6
In his 92 paper, Mukai showed that a general genus $6$ curve may be represented in $\mathbb{P}^9$ as the intersection of the Grassmannian $G(2,5)$ (under the Plucker embedding), a plane $H\cong \...
1
vote
0
answers
57
views
Bound of analytic torsion for a line bundle
Let $E\rightarrow X$ be a line bundle over a compact Riemann surface. Let $\Delta_{E}$ be the Dolbeault Laplacian $\Delta_{\overline{\partial}}$ extended to sections of $E$. Let $g_1, g_2$ be two ...
5
votes
1
answer
813
views
Clarification on Beltrami Differentials
I have troubles with the theory of existence of quasi-conformal homeomorphisms realizing Beltrami coefficients. Let $X$ be a (compact) Riemann surface and $f \colon X \rightarrow \mathbb{C}$ be smooth....
0
votes
1
answer
371
views
The cohomology of meromorphic functions
Let $A$ be a sheaf such that $$A(U) = \{ f \in \mathbb M(U): f \in \mathbb{O}(U \backslash\{p_1,\ldots, p_n\}) \ \mbox{with at worst a simple pole at}\ p_i \} $$ where $\mathbb M(U)$ means the set of ...
1
vote
0
answers
46
views
Real section of moduli space of Riemann surfaces
In (https://www.sciencedirect.com/science/article/pii/002240499390049Y) it is mentioned the real section of the moduli space of Riemann surfaces of genus 0. It can be intuitively defined as a subset ...
1
vote
0
answers
110
views
Explicit construction of mirror surface and complex double for an annulus
My reference is Abikoff's book "Real analytic theory of Teichmuller spaces.
Let $X$ be a Riemann surface with two boundaries, we can construct a mirror surface $\bar{X}$ defined to be the same ...
5
votes
1
answer
141
views
Short basis in $\pi_1$ on a hyperbolic surface of bounded diameter
First, some terminology. Let $(S,x)$ be a compact surface of genus $g>0$. A standard collection of loops $\gamma_1,\ldots, \gamma_{2g}$ based at $x$ is a collection of loops that cuts $S$ into a ...
2
votes
1
answer
265
views
Build a Fuchsian group starting from punctures on a disk
Consider the moduli space of hyperbolic metrics on the disk with $n>3$ marked points on its boundary, $\mathcal{M}_{D,n}$.
$\mathcal{M}_{D,n}$ can be parametrised in terms of cross ratios of the ...
1
vote
0
answers
184
views
Cutting a circle from the hyperbolic plane
Let D be the Poincare' disk its natural hyperbolic metric and with at least 1 marked point on $\partial D$. Suppose I cut an hyperbolic circle of radius $r$ away from it, then I get a Riemann surface ...
3
votes
1
answer
549
views
What is a half cusp in hyperbolic geometry?
I already asked this question on math.stackexchange, but it was suggested that I post it here as well.
The paper Devadoss, Heath, and Vipismakul - Deformations of bordered Riemann surfaces and ...
4
votes
2
answers
247
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Metrics with fixed conformal structure and diameter
I have three questions.
I consider a sequence of metrics $h_n$ on a two-dimensional torus which all induce the same conformal structure. Suppose that the volume of $h_n$ is always $1$. Is it possible ...
3
votes
1
answer
200
views
Symplectic representation of modular group
The modular group $\Gamma_{g}$ of isotopy classes of diffeomorphisms of a genus $g$ surface $S$ acts on $H^1(S,\mathbb{Q})$ (or $H^1(S,\mathbb{Z})$) respecting the intersection pairing. This gives a ...
1
vote
2
answers
384
views
Defining “addition” on the Riemann surface of log(z)
The title of this question is a bit awkward, as adding two points together on a manifold is usually not considered possible, but in this case there appears to be a nice little hack.
Consider the ...
5
votes
0
answers
177
views
Projective unitary flat structures of $\mathbb{P}^1$-bundles on Riemann surfaces
Narasimhan and Seshadri proved a rather surprising result about vector bundles on a compact connected complex manifold $X$. That is
Two holomorphic vector bundles arising from unitary representations ...
4
votes
1
answer
352
views
The degree of the Gauss map of Theta divisor
Let $R$ be compact a Riemann surface of genus $g$ and $ J (R) $ be its Jacobian.
For a subvariety $X$ of $J(R)$ of dimension $d$, denote the set of non-singular points of $X$ by $X_{reg}$. Then the ...
15
votes
1
answer
623
views
Do $\mathbb{A}^1-S$ and $\mathbb{A}^1-\{0,1\}$ have a finite etale cover in common?
We work over the field of complex numbers. (But remarks in characteristic $p$ are very welcome.)
Let $S$ be a finite set of points in $\mathbb{A}^1$ containing $0$ and $1$. [Edit: Assume $S$ contains ...
1
vote
1
answer
269
views
Locus where a vector bundle is null
I am reading a paper in differential geometry, Hitchin's Langlands duality and G2 spectral curves (see the end of page 8 in the arxiv version), where $f: E \rightarrow F$ is a morphism of holomorphic ...
10
votes
2
answers
476
views
Riemann surfaces with an atlas all of whose open sets are biholomorphic to $\mathbb{C}$?
Is there a compact Riemann surface other than the sphere with an atlas consisting of open subsets biholomorphic to $\mathbb{C}$? Is there a compact Riemann surface other than the sphere which ...
1
vote
1
answer
234
views
Conjugate points and totally geodesic foliations
Suppose $(M,g)$ is a two dimensional simply connected compact Riemannian manifold with smooth boundary. I want to understand if in general there is a correlation between the following two statements ...
14
votes
3
answers
788
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Automorphisms of genus 6 surfaces
I have found in Wikipedia that the number of automorphisms of a Riemann surface of genus 6 does not exceed 150. The page offers neither a proof nor a reference. Can someone help me?
8
votes
2
answers
374
views
Image of boundary circle under map from punctured elliptic curve to ℂ
Let $E=\mathbb C/\Lambda$ be an elliptic curve,
and let $D\subset E$ be a very small disc.
($D$ is round for the usual flat metric on $E$)
By the main result of [1], there exists a holomorphic ...
5
votes
1
answer
190
views
Mapping-Class Groups of Subsurfaces of a Hyperbolic Surface
If $\mathcal{R}'$ is a closed subsurface of a hyperbolic surface $\mathcal{R}$, then there is an inclusion homomorphism between the mapping class groups:
$$\text{Mod}(\mathcal{R}')\longrightarrow \...
4
votes
1
answer
276
views
Shrinking the boundary of a Riemann surface
Let $X$ be a compact Riemann surface with boundary. Let us shrink each connected component of the boundary into a point. We get a closed topological surface $Z$ with several marked points (which came ...
11
votes
1
answer
711
views
Gluing Riemann surfaces
Let $X$ be a compact Riemann surface with boundary $\partial X$. Assume (for simplicity only) that $\partial X$ has a single connected component. Let us fix an orientation preserving diffeomorphism $\...
17
votes
2
answers
2k
views
Square root of the determinant line
Let $\Sigma$ be a compact Riemann surface equipped with a spin structure (a square root of $\Omega^1_\Sigma$, denoted $\Omega^{1/2}_\Sigma$).
Let $\Gamma(\Omega^1_\Sigma)$ be the space of holomorphic ...
12
votes
2
answers
1k
views
Universal covering of a 2-sphere without $n$ points
Let $X$ be the $\mathbb{C}\mathbb{P}^1$ with $n$ points deleted. Let $n\geq 3$. If I understand correctly, the universal covering of $X$ is isomorphic to the upper half plane as a complex analytic ...
4
votes
1
answer
285
views
Bergman kernel via Riemann Theta function
Let $\Sigma$ be a Riemann surface of genus $g$. A Bergman kernel $B(p,q)$ is a bilinear meromorphic form on $\Sigma \times \Sigma$ with poles of order $2$ along the diagonal $p = q$ and holomorphic ...
7
votes
1
answer
462
views
Why Green functions and not Neron functions?
Arakelov constructed a nice intersection theory on arithmetic surfaces. A key point is the notion of Green function for a Riemann surface, which will be involved in the ''part at infinity'' of the ...
7
votes
0
answers
272
views
How to interpret heat kernel at unit time on a Riemann surface?
Let $M$ be a compact Riemann Surface. Let $P$ be a fixed point on $M$ and let $\delta_{P}$ be the Dirac point distribution at $M$. Consider the fundamental solution of the heat equation
$$
(\partial_{...
2
votes
2
answers
283
views
Conformal hyperbolic metrics with mixed cone and cusp singularities
Let $X$ be a compact Riemann surface
and $D=\sum_{j=1}^n\,(\theta_j-1)\,P_j$ be a ${\Bbb R}$-divisor on $X$ such that $\theta_j\geq 0$ and $P_1,\cdots,P_n$ are $n$ distinct points on $X$.
We call $ds^...