All Questions
39 questions
18
votes
2
answers
1k
views
Proving algebraicity of compact Riemann surfaces without Chow's theorem
I am trying to write a report for a complex analysis class where I prove Riemann-Roch and apply it to prove algebraicity of compact Riemann surfaces. While writing this, I found that Riemann-Roch ...
- 283
16
votes
1
answer
1k
views
Is a one-dimensional compact complex analytic space necessarily projective?
Let $X$ be a compact complex analytic space with singular locus $X^{\mathrm{sing}}$. Suppose that $X\setminus X^{\mathrm{sing}}$ is a Riemann surface. If $X^{\mathrm{sing}} = \emptyset$, then $X$ is ...
user94803
14
votes
2
answers
613
views
A "holomorphic" Peano curve?
A Peano curve is a continuous map $[0,1]\to [0,1]^2$ whose image is the whole square.
I would like to know if on can obtain "holomorphic" Peano curves. Namely, is it possible to find a continuous ...
- 14.3k
11
votes
1
answer
752
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 $\...
- 21.8k
10
votes
2
answers
492
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 ...
- 356
9
votes
3
answers
927
views
Are real-analytic functions in $\mathbb{R}^2$ holomorphic after suitable change of coordinates?
I'm not sure this is a research-level question, but I couldn't find an answer after a bit of searching, so here goes.
Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a real-analytic function. Can we always ...
- 5,590
9
votes
1
answer
321
views
Notational question about quadratic differentials in Strebel's book "Quadratic differentials"
In Kurt Strebel's book "Quadratic Differentials", in Chapter 2, $\S4$, he begins by saying:
"Every analytic function $\varphi$ is a domain $G$ of the $z$-plane defines, in a natural way, a field of ...
- 7,527
8
votes
2
answers
528
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 ...
- 1,566
8
votes
2
answers
401
views
Holomorphic maps from a Riemann surface of infinite genus
Let $X$ be a Riemann surface of infinite genus and let $n$ be an arbitrarily large natural number.
Do there always exist a closed Riemann surface $Y$ of genus greater than $n$ and a nonconstant ...
- 81
8
votes
1
answer
273
views
Self homeomorphism of $\mathbb CP^1$ holomorphic a.e
Suppose $\varphi:\mathbb CP^1\to \mathbb CP^1$ is a homeomorphism holomorphic on a connected open subset $U\subset \mathbb CP^1$ with $\mathbb CP^1\setminus U$ of zero measure.
Is it true that $\...
- 14.3k
7
votes
1
answer
474
views
Embed a bordered Riemann surface into punctured Riemann surfaces?
Let $U$ be a bordered Riemann surface of genus $g$ with $n -1$ punctures and one hole (i.e., the border has one connected component). For any punctured Riemann surface $\Sigma$ of genus $g$ with $n$ ...
- 71
7
votes
1
answer
279
views
Riemann uniformization theorem (limit case)
Let $\mathbb D_r=\{z\in\mathbb C:|z|\le r\}$ be the closed unit disk of radius $r$,
let $\mathring {\mathbb D}_r=\{z\in\mathbb C:|z|< r\}$ be its interior,
and let $\mathbb A_r=\mathbb D_r\setminus ...
- 43.2k
6
votes
1
answer
485
views
A basis of holomorphic differentials on Fermat curves
I am currently reading the paper "Holomorphic Differentials of Generalized Fermat Curves" by Rubén Hidalgo. The case I am interested in is that of a classical Fermat curve $F_k$, which in ...
- 63
5
votes
1
answer
519
views
Branched covers of the sphere branched over few points
Let $X$ be a compact Riemann surface of genus $g\geq 2$. By the Riemann-Roch theorem, $X$ is a branched cover of the sphere, branched over finitely many branched values. What is the smallest number of ...
- 6,548
4
votes
3
answers
687
views
Finite covers of punctured Riemann surfaces
Let $X$ be a compact Riemann surface, i.e. compact smooth complex analytic (hence automatically algebraic) curve. Let $A\subset X$ be a finite subset, and $X_0:=X\backslash A$.
Let $Y_0$ be a smooth ...
- 21.8k
4
votes
2
answers
1k
views
Is complex analytic extension of real-analytic diffeomorphism a diffeomorphism ?
Hi, my question is :
Let $D$ be the open unit disk in $\mathbb{C}=R^2$ and $f:D\to D$ be a real-analytic diffeomorphism. Let us think of the canonical embedding : $\mathbb{C}=R^2\subset \mathbb{C^2}. ...
- 1,471
4
votes
1
answer
279
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 ...
- 21.8k
4
votes
1
answer
424
views
Reverse residue theorem without using Serre's duality
In V. Talovikova's text "Riemann-Roch Theorem", a key part of the proof of Riemann-Roch theorem is the following proposition (4.6 in the text):
Let $\{a_1, \dots,a_n\}$ be a set of points in ...
4
votes
0
answers
229
views
Real part of a holomorphic section of a vector bundle
Let $F\to M$ be a holomorphic vector bundle over a complex manifold $M$ and let $s:M\to F$ be a no-zero section. Let $E$ be the complexification of $F$, and suppose that $E$ admits a holomorphic ...
- 163
3
votes
2
answers
618
views
Schwarz Lemma in terms of conformal surfaces or holomorphic curves?
Scharwz Lemma in its general form says that any holomorphic map between hyperbolic surfaces is contracting.
Noting that Riemann surfaces admit a unique metric of constant curvature -1, I wonder if we ...
user2529
2
votes
2
answers
417
views
Stoilow Theorem
I want to see the precise statement and a proof for a theorem of Stoilow on "inner" functions (I do not know what this exactly means, I suppose it is an open map with other natural properties). A ...
- 85
2
votes
1
answer
150
views
Branched covering maps between Riemann surfaces
What is an example of a branched covering map between Riemann surfaces of infinite degree? i.e. something like a branched version of the exponential map $exp: \mathbb{C} \to \mathbb{C}^*$.
Thanks!
- 357
2
votes
1
answer
216
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. ...
- 174
2
votes
0
answers
358
views
Triangulating Riemann surfaces by using non-constant meromorphic functions
Let $X$ be a connected Riemann surface, i.e. $X$ is a one dimensional connected complex manifold (Hausdorff and second-countable as a topological space). The following is a classical result:
Theorem (...
- 395
2
votes
0
answers
154
views
Algebra of meromorphic functions on a Riemann surface
Let $C$ be compact Riemann surface of high genus. Let $p$ be a general point on $C$ and $z$ a local coordinate around $p$.
Given a meromorphic function on $C$, regular outside $p$, we can look at its ...
- 2,384
2
votes
0
answers
127
views
Vector bundles over Riemann surfaces of infinite genus
Has any work been done on the description of (finite rank) holomorphic vector bundles over Riemann surfaces of infinite genus?
Is there a theory of moduli spaces of such objects?
This question is ...
- 23.3k
2
votes
0
answers
441
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 ...
- 2,106
1
vote
2
answers
1k
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 ...
- 2,106
1
vote
1
answer
372
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 ...
- 5,230
1
vote
1
answer
87
views
Disk with punctures and convex geodesical hull of the punctures isomorphic?
Consider a unit disk with marked points $z_i$, $i=1, \dots , n$ on its boundary.
Let us call this surface $X$.
As it is well known, the disk can be equipped with an hyperbolic metric and is then ...
- 1,435
1
vote
1
answer
386
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 ...
- 2,106
1
vote
0
answers
119
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 ...
- 1,435
1
vote
0
answers
215
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 $n\...
- 11
0
votes
2
answers
2k
views
Motivation behind defining the Ramification Divisor
I would like to understand what exactly is the motivation for defining the notion of a ramification divisor of a function.
As I see the definition,
If $f$ is a meromrophic function between two ...
- 3,541
0
votes
1
answer
921
views
immersion: submanifold of complex manifold
Let $\alpha : \mathbb{C} \rightarrow M$ be an immersion and $M$ a $n$ dimensional complex manifold with complex structure $I$. Does then follow that $\alpha (\mathbb{C})$ is a one dimensional ...
- 23
0
votes
1
answer
159
views
Teichmüller theory for open surfaces?
I have a rather straightforward and perhaps somewhat naive question: Is there a Teichmüller theory for open surfaces?
My motivation basically is that I would like to find out more about the "...
- 7,127
0
votes
1
answer
310
views
Inverse "Riemann mapping" [closed]
The Riemann mapping theorem states, that any simply connected domain $U \subset \mathbb C$ can be conformally mapped to the open unit disk $D$. I.e. there is a Diffeomorphism $\Psi: D \to U$ such that ...
- 126
0
votes
0
answers
66
views
Uniformization and constructive analytic continuation of Taylor-Maclaurin series
Context. In their paper, "Uniformization and Constructive Analytic Continuation of Taylor Series", Costin and Dunne present a constructive method to greatly increase the accuracy of a ...
0
votes
0
answers
204
views
Expansion around a singular point of a multivalued meromorphic function (due to Riemann/Cauchy)
In Riemann's publication about Abelian functions
'Theorie der Abelschen Functionen' (Here the original paper in german)
at the end of Chapter 4, part 2 is clamed that for every Riemann
surface $T$ and ...
- 5,998