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.
3
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0answers
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fundamental domains in H^2 containing large balls
I would like to construct a genus $g$ surface regularly tiled by triangles (for example by 238 triangles). Edmunds-Ewing-Kulkarni prove that the only obstruction to doing this is Euler characteristic ...
12
votes
1answer
466 views
Poincaré metric on the Riemann sphere minus more than two points
If we omit more than two points from the Riemann sphere, we will obtain a hyperbolic Riemann surface endowed with a canonical metric descending from its universal cover which is the Poincaré disk. Let ...
0
votes
1answer
139 views
Complex structures on topological surfaces
I am interested in the number of complex structures on a surface. More precisely, given a genus $g$ surface (topological manifold of real dimension 2) with $n$ punctures $X_{(g,n)}$, how many complex ...
4
votes
1answer
198 views
Stable extensions by line bundles on Riemann surfaces
Is there a compact Riemann surface $X$ and a line bundle $L$ of negative degree on $X$, such that for any nontrivial extension
$$ 0 \rightarrow L \rightarrow E \rightarrow L^{-1} \rightarrow 0, $$
$E$ ...
5
votes
0answers
125 views
Generalized Jacobians and modular units
Let $X$ be a proper algebraic smooth curve over a characteristic zero field $k$ and let $J$ be the Jacobian variety of $X$. Let $K$ be the function field of $X$. Assume that we are given $n$ distinct ...
1
vote
0answers
145 views
The Deligne-Mumford Compactification for Closed Surfaces
I am reading this note on super-Riemann surfaces. In the second paragraph of section 7.4.1 (page 87), there is a statement that I am trying to understand:
The compactified moduli space of closed ...
3
votes
1answer
147 views
English literature close to “Algébre et Théories Galoisiennes” by Régine and Adrien Douady
I'm currently working on my undergraduate dissertation. I'm working on covering sapces of Riemann surfaces so my supervisor asked me to read the book I mention in the title: "Algébre et Théories ...
6
votes
2answers
118 views
Embedding open connected Riemann Surfaces in $\mathbb{C}^2$
This question arises in the context of a question asked on MSE: Are concrete Riemann surfaces Riemann domains over $\mathbb{C}$. Part of the answer to that question is the question above which is ...
6
votes
2answers
250 views
Is every element of $Mod(S_{g,1})$ a composition of right handed Dehn twists?
Let $S_{g,1}$ be the surface of genus $g \geq 1$ and $1$ boundary component. Let $Mod(S_{g,1})$ be the mapping class group in which we allow isotopies to rotate the action on the boundary (...
0
votes
0answers
151 views
Towards recognizing St. Venant geometrical invariant
Using partial derivative notation we can express Gauss curvature $K$ in cartesian coordinates:
$$\quad p= \partial w/ \partial x, q= \partial w/ \partial y; r=\frac{\partial ^2w}{{\partial {x} ^2} },...
26
votes
3answers
962 views
References for Riemann surfaces
I know this question has been asked before on MO and MSE (here, here, here, here) but the answers that were given were only partially helpful to me, and I suspect that I am not the only one.
I am ...
7
votes
0answers
143 views
Maps between moduli of curves
Let $M_{g,n}$ be the moduli space of $n$-pointed curves, and $M_g[m]$ the moduli space of (unpointed) curves with $m$-level structure.
Fix $m>0$. Is it true that for $n$ large enough, there is a ...
3
votes
2answers
116 views
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 ...
2
votes
0answers
237 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 ...
12
votes
2answers
315 views
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 ...
14
votes
2answers
647 views
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
2answers
175 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 ...
2
votes
1answer
185 views
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....
2
votes
1answer
178 views
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
1answer
138 views
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
0answers
83 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
2answers
246 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$$ be its Harder-Narasimhan filtration: we have $V_i:=E_i/E_{i-1}$ ...
9
votes
0answers
97 views
Cluster algebra and Fenchel Nielsen coordinates
Certain cluster algebras arise from ideal triangulations of hyperbolic Riemann surfaces. The combinatorics behind their mutations can be understood in terms of "flips" in the triangulation, and the ...
2
votes
1answer
186 views
holomorphic sections of line bundles on Riemann surfaces
I have something elementary to ask. Let $E\rightarrow X$ be a holomorphic line bundle over a Riemann surface. Then in general a section of $E$ is a meromorphic function on $X$, since $O_{div(s)}\cong ...
1
vote
0answers
31 views
Is $O_{D}(X)$ a reflective Banach space?
Let $X$ be a Riemann surface and $O_{X}(D)$ be the line bundle associated with $D$. Let the metric on $O_{X}(D)$ be given by
$$
|1_{O_{X}(D)}(P)|=G(P,D)^2
$$
where $G(P,D)^2$ is the Green function ...
8
votes
0answers
101 views
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
0answers
30 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 ...
1
vote
1answer
51 views
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 ...
0
votes
0answers
73 views
How to measure the distance between two functions on a surface
Let $X$ be a compact Riemann surface. Let $x_{1},\cdots x_{n}$ be some distinct points on $X$. Let $E\rightarrow X$ be a holomorphic complex line bundle over $X$ and $f_{i}(e_{i})$ be functions ...
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vote
0answers
64 views
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
0answers
179 views
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
0answers
134 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 ...
4
votes
0answers
99 views
Exact sequence of Quillen metrics
Let $M$ be a compact Riemann surface equipped with its Arakelov metric. Let $\xi$ be a holomorphic line bundle on $M$ equipped with an admissble metric. For any point $P\in M$ we get an admissible ...
1
vote
0answers
107 views
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 ...
4
votes
1answer
103 views
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
0answers
39 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 ...
3
votes
1answer
115 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
1answer
132 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
0answers
40 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
0answers
64 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
1answer
108 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
1answer
139 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
0answers
91 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
1answer
192 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
2answers
135 views
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
1answer
149 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 ...
2
votes
1answer
169 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
0answers
115 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 ...
3
votes
1answer
269 views
Equivalence of the term “Divisor”
Throughout my university education, I have studied some theory of Riemann surfaces, focusing particularly on Miranda's Algebraic curves and Riemann surfaces. My current studies however are in the ...
5
votes
1answer
209 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 ...
