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.

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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 ...
Mohan Swaminathan's user avatar
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 ...
Sumanta's user avatar
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13 votes
2 answers
471 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 ...
Dylan Thurston's user avatar
16 votes
2 answers
1k 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 ...
Tsein32's user avatar
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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 ...
Chitrabhanu's user avatar
3 votes
1 answer
411 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....
Wakabaloola's user avatar
4 votes
3 answers
748 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 ...
Jana's user avatar
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0 votes
1 answer
160 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 ...
Jianrong Li's user avatar
  • 6,101
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 ...
modular's user avatar
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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 $V_i:=E_i/E_{i-1}$ ...
Beth's user avatar
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11 votes
0 answers
255 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 ...
giulio bullsaver's user avatar
2 votes
1 answer
886 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 ...
Guest's user avatar
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1 vote
0 answers
41 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 ...
Guest's user avatar
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8 votes
0 answers
212 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 ...
Arun Debray's user avatar
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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 ...
user avatar
2 votes
1 answer
106 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 ...
Florian R's user avatar
  • 215
1 vote
0 answers
68 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})\...
Bonbon's user avatar
  • 806
5 votes
0 answers
352 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} ...
Nico A's user avatar
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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 ...
Hajime_Saito's user avatar
1 vote
0 answers
116 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 ...
giulio bullsaver's user avatar
5 votes
1 answer
211 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 \...
Hubokula's user avatar
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 ...
Bombyx mori's user avatar
  • 6,141
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....
Florian R's user avatar
  • 215
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 ...
J.Doe's user avatar
  • 81
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 ...
giulio bullsaver's user avatar
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 ...
giulio bullsaver's user avatar
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 ...
aglearner's user avatar
  • 14k
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 ...
giulio bullsaver's user avatar
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 ...
giulio bullsaver's user avatar
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 ...
giulio bullsaver's user avatar
4 votes
2 answers
247 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 ...
Selim G's user avatar
  • 2,626
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 ...
Klint's user avatar
  • 33
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 ...
Mike Battaglia's user avatar
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 ...
swalker's user avatar
  • 713
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 ...
Manoel's user avatar
  • 530
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 ...
Neeroen's user avatar
  • 153
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 ...
Raffaele C's user avatar
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 ...
Ali Taghavi's user avatar
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 ...
Ali's user avatar
  • 4,045
14 votes
3 answers
788 views

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?
S.Lia's user avatar
  • 141
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 ...
André Henriques's user avatar
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 \...
QGravity's user avatar
  • 969
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 ...
asv's user avatar
  • 21.1k
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 $\...
asv's user avatar
  • 21.1k
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 ...
André Henriques's user avatar
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 ...
asv's user avatar
  • 21.1k
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 ...
user113988's user avatar
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 ...
manifold's user avatar
  • 299
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_{...
Bombyx mori's user avatar
  • 6,141
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^...
Bin Xu's user avatar
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