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Questions tagged [riemann-surfaces]

For questions about Riemann surfaces, that is complex manifolds of (complex) dimension 1, and related topics.

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Proof of residue theorem (residue formula) for differential forms on curves over an arbitrary closed field.

I have been reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. They give a residue formula like this: Let $\mathbb{k}$ be an algebraically closed field and $X$ ...
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$\tau$ structure of the sixth Painlevé equation

I am studying the isomonodromic deformations theory, which leads in the case of a $\mathcal{C}_{0,4}$ Riemann surface to the sixth Painlevé equation. I read that this equation had a $\tau$-structure,...
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Geometric meaning of an integral on a compact Riemann surface

Let $X$ be a compact Riemann surface and fix a volume form $\Omega$ on $X$ such that $\int_X\Omega=1$. Now let's fix a function $g:U\subset X\to\mathbb R$ on $X$ with the following properties: $U=X\...
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How does the topology of the graphs' Riemann surface relate to its knot representation?

Let me give a worked-out example: The following cubic planar non-simple graph $\hskip2.3in$ has the adjacency matrix $A=\pmatrix{0&3\\3&0}$. The graph has three faces, so the rank of $G$ is $...
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How to write holomorphic line bundles on compact Riemann surfaces in coordinates

I would like to write down coordinate charts for holomorphic vector bundles on a smooth complex (compact) curve $C$. For example, if $C = \mathbb C \mathbb P^1 = \{ [x_0 : x_1 ] \}$, let \begin{...
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Given a curve over $\mathbb{P}^1$ how can I determine the monodromy around a ramification point?

Given a smooth curve of the form $$ f(y) - g(x) $$ over $\mathbb{A}^1_x$ with a smooth compactification $C \to \mathbb{P}^1$, how can I determine the monodromy of the points of the fibers? Are there ...
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Where can I find precise examples of ramified coverings of $\mathbb{C}P^1$?

One of the definitions of simple Hurwitz number $h_{g,\mu}$ is that it counts up to automorphisms the number of ramified coverings of $\mathbb{C}P^1$ such that covering space is a connected surface of ...
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Relationships between the field of meromorphic functions and the fundamental group

Let $X$ be a connected compact Riemannian surface. Then the ring of meromorphic functions on $X$ is a field and we denote it as $M(X)$. Let $E$ be a finite extension over $M(X)$. Then we can show that ...
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$\{(x,y)\in \mathbb C^2|y^2=\sin x\}$ as interior of compact Riemann Surface with Boundary

A takehome exam problem for my Riemann Surfaces class, which used Griffith's Introduction to Algebraic Curves, was the following: Show that $S=\{(x,y)\in \mathbb C^2|y^2=\sin x\}$ is not interior ...
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Polynomials for Bicubic Planar Dessin d'Enfants

A dessin d'enfant is a graph, with its vertices colored alternately black and white, embedded in ... a plane. For the coloring to exist, the graph must be bipartite. ... The ... embedding may be ...
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How to tell if a simply-connected curve is the complex plane or disk

Suppose I have an analytic function $f: \mathbb{C}^2 \to \mathbb{C}$ whose zero locus $V=\{(z,w) \in \mathbb{C}^2 : f(z,w)=0\}$ is smooth and simply connected. By uniformization, $V$ is conformally ...
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Is there a natural ring structure on $\operatorname{Pic}(\mathbb{CP}^1)$?

The set of isomorphism classes of holomorphic line bundles on a complex manifold $X$ is a group under tensor product. This group is called the Picard group and is denoted $\operatorname{Pic}(X)$. We ...
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Fixed Point Involutions

In recent reading on Riemann surfaces and complex manifolds (primary Miranda with a few random finds online), I encountered the notion of involutions, in particular fixed point involutions. We recall ...
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lift of antiholomorphic involution of Riemann surface to its Jacobian's cohomology

Start from a connected closed Riemann surface $\Sigma_g,$ obtained as the (symmetric) covering of an open and/or unoriented surface $\Sigma,$ namely $\Sigma=\Sigma_g/\Omega,$ where $\Omega$ is an ...
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Can a pair of pants inherit a negatively curved Riemann metric from 3-space?

It is well-known that a pair of pants can be given a constant negative curvature metric. Indeed, one way to see that the genus $g$ surface can be given a constant negative curvature metric for $g \geq ...
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How to show that the Weierstrass function is meromorphic in whole of $\mathbb{C}$

The Weierstrass function is defined as: $$\wp(z) = \frac{1}{z^2}+ {\sum\limits_{\omega\in\Gamma'}} \left (\frac{1}{(z-\omega)^2}-\frac{1}{\omega^2}\right).$$ Fix $r>0$. For $\left|z\right| \leq r,$...
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Riemann surfaces and folding

I am learning about Riemann surfaces as covering, and was interested to know how Riemann first thought of them. Looking at his collected papers, I found the following definition in his lecture called “...
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Understanding Complex Differentials (forms)

In the study of Riemann surfaces, many books bring in their discussions, the complex differentials or differential forms, and there my understanding gets stopped. I personally interacted with many ...
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Computing dim of $\check{H}^1(X, \mathbb{C})$ where $X$ is a compact Riemann surface

Suppose $X$ is a compact Riemann surface of genus $g$, $\mathcal{O}$ represents the sheaf of holomorphic functions and $\Omega$ the sheaf of holomorphic 1-forms I want to show the dimension of $H^1(X,...
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Obtaining a single-valued branch of $\ln \left( \frac{z-a}{z-b} \right)$ with a branch cut

It is rather easy to see that the function $$f(z) = \ln \left( \frac{z-a}{z-b} \right)$$ has branch points at $z=a$ and $z=b$, My question is why considering a branch cut "connecting" $a$ and $b$ ...
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Not taking holomorphic bundles for granted

When we say something to the effect of, "Consider a holomorphic bundle $V$ on a complex manifold $X$...", we are saying that the transition functions of $V$ are holomorphic functions with respect to ...
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Basic question: Condition for a map associated to a linear series to be an immersion

I am reading this set of lectures of a class by Prof. Harris. There is a theorem. Let $X$ be a Riemann surface and $\phi:X\rightarrow\mathbb{P^r}$ be the map defined by a linear series without ...
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Complex structures on Riemann surfaces

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 $T^{0,1} M \to T^{1,0} M$, the bundle $$ \{v + \alpha(v) \mid v \in T^{0,1} M\} \...
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Mittag-Leffler Problem

We have: $X$ a compact Riemann surface defined by $y^{2}=1-x^{6}$ and $P=(0,1) \in X$ a point given in local coordinates $(x,y)$. Furthermore, we have a meromorphic function $f(x,y)=y/x$ such that $f \...
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Fundamental solution to Laplace equation on arbitrary Riemann surfaces

So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper (...
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Miranda Pag. 66 Plugging Holes

Let $X$ be a Riemann surface. A hole chart on $X$ is a complex chart $\phi: U \mapsto V$ on $X$ such that $V$ contains an open punctured disc $D_0=\{z: 0 < ||z-z_0 || < \epsilon \}$ with the ...
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Flat embedded surfaces in arbitrary 3-manifolds.

It is generically known that any surface (at least embedded in $\mathbb{R}^3$ - and see edit below) can be deformed so that it is flat - i.e. has a metric of zero curvature - as long as one adds some ...
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de Rham cohomology on moduli of curves

I do not know the constructions of Deligne-Mumford; so let us suppose that the moduli space $\mathcal{M}_g$ of Riemann surfaces of genus $g$, with $g>1$, is constructed using the moduli of abelian ...
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Visualizing the domain of the square root

I would like to show someone the domain of the complex square root function (the 2-sheeted riemann surface). Is there a good interactive visualization software for this? I would like some sort of ...
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Computing global monodromy of branched covering of Riemann sphere

Given an irreducible polynomial $F(X,Y)\in\mathbb{C}[X,Y]$, there is an associated Riemann surface $S$ obtained from the zero set of $F$. Projection onto the second coordinate gives a map $\pi:S\to \...
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Homology of Fermat curve

Let $C(n):X^n+Y^n=Z^n$ be the plane projective Fermat curve of degree $n$ over $\mathbb{C}$. Shorter version of the question: How can I describe explicit representatives for a basis for the singular ...
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Line bundle on a curve is positive iff it has positive degree

Let $C$ a complex curve and $L$ a holomorphic line bundle on it. I want to show that $L$ is positive iff it has positive degree. Here the degree is defined as $\int_C c_1(L)$ and positive means that ...
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Automorphism of a compact Riemann surface with finitely many points removed maps punctured disc to punctured disc

I wonder if the following is true: If we have a compact Riemann surface $X$ where finitely many points are removed, so $X'=X - \{x_1,...x_k\}$ and an automorphism $f$ of $X'$, is it true that $f$ maps ...
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Sphere fibrations over Riemann surfaces

In a physics application, there's a 5-dimensional space which consists of a fibration of $S^3$ over a genus-$g$ Riemann surface $\Sigma_g$. In fact this nontrivial fibration is part of a bigger where ...
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Roots of a canonical line bundle on a compact Riemann surface

Suppose we have a compact Riemann surface $X$ of genus $g$. Let $K$ denote the canonical line bundle on $X$, it's well known that $deg\ K=2g-2$. A square root of $K$ by definition is a holomorphic ...
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Riemann Surfaces for Graphs and vice versa…

They draw nice pictures of Riemann Surfaces in "Automatic Generation of Riemann Surface Meshes" by Matthias Nieser, Konstantin Poelke, and Konrad Polthier: Identify each layer with a vertex, then for ...
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Explain example in Elkies's ABC Conjecture paper

The reference: The ABC's of Number Theory (PDF) On page 17 of the PDF, or 72 of the scan, he solves a Putnam problem. In the solution he uses a special case of Mason's theorem, for $F$ a polynomial, ...
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structural (algebraic) sheaf on a Riemann surface “inside” the sheaf of holomorphic functions

This question consists of $3$ points, and each of them deals with the relationship between the "algebraic" structure on a projective curve and its "analytic" structure. I know that this argument has ...
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Order of point in divisor

Let $y^2=f(x)$ be hyperelliptic curve over $k$ and $(a,\sqrt{f(a)})$ point on the curve and let $b\in k$. I would like to prove that divisor of the function $g(x,y)=\frac{x-b}{x-a}$ is equal to $$...
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Compactness of the moduli space of bundles with fixed determinant

The moduli space of semistable holomorphic vector bundles of fixed rank and fixed determinant line bundle on a compact Riemann surface is known to be compact itself. (In particular, when the rank is $...
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On a method to compute dimension of moduli space of Riemann surfaces

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 $$J=\...
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Triple Cover of the Riemann Sphere

I have the triple branched covering $X$ of $\mathbb{P}^{1}$ defined by $y^{3}=x^{6}-1$. I want to show the following: (i) The canonical embedding $\phi: X \rightarrow \mathbb{P}^{3}$ can be given in ...
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Probably Riemann surface integral

Here is the integral: May you please suggest some beautiful idea on using Riemann surface, or some Gauss-Ostrogradsky at the beginning. Also, the initial integral looks really symmetric, so maybe ...
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SU(2) Lefschetz decomposition of Jacobian of Riemann surfaces

Start with a (closed and oriented) Riemann surface with $g$ handles $\Sigma_g.$ I'm interested in (the cohomology of) its Jacobian $Jac(\Sigma_g)=T^{2g},$ in particular how the $SU(2)$ or $SL(2,\...
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Reference about Moduli Spaces of Riemann surfaces

I'm looking for some (introductory, and in any case not too technical) reference (book, lecture notes, papers) regarding moduli spaces $\mathcal{M}_g$ and $\mathcal{M}_{g,n}$ of (punctured) Riemann ...
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Are there Galois covers of curves branched at 1 point?

Let $G$ be a finite group, not necessarily abelian. Is there any smooth algebraic curve $C$, with an action of $G$ on $C$, such that the natural quotient map $C \to C/G$ is branched at precisely one ...
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Spinors on algebraic plane curves

I'm interested in parameterizing spinors on Riemann surfaces. For my purposes, it's best to represent the Riemann surfaces as immersed in $\mathbb{C}P^2$, i.e. as algebraic plane curves. Apparently, ...
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How do we define a complete metric on a Riemann surface with punctures?

This question is related to another question. If we have a Riemann surface with punctures of negative Euler characterstisc, how can one define a complete hyperbolic metric? I know that in this ...
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Weierstrass $\wp$-Function Addition Property

Consider the function $$ \det\left( \begin{array}{ccccc} &1 &\wp(z) &\wp'(z) \\ &1 &\wp(w) &\wp'(w) \\ &1 &\wp(-z-w) &\wp'(-z-w) \end{array} \right)=f(z) $$ I'm ...
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Constructing normalizations of algebraic curves vs constructing Riemann surfaces of functions

This question is sort of a further extension to this question I have been asking, Relation between n-tuple points on an algebaric curve and its pre-image in the normalizing Riemann surface It seems ...