3
votes
1answer
38 views

Irreducibility of holomorphic functions in a neighborhood of a point

Let $D \subset \mathbb C^n$ be a domain and let $f \in \mathscr O(D)$, $f \not\equiv 0$ be a holomorphic function. Define $$ V_f = \bigl\{ z \in D : f(z) = 0 \bigr\}. $$ Let $p \in V_f$. Suppose ...
1
vote
1answer
55 views

Riemann surface from $x^2 + y^2 = 1$ for $x,y \in \mathbb{C}$

I am reading Edward Frenkel's book Love and Math. In Chapter 9, it is talked about the one-to-one correspondence of solution of algebraic function of complex numbers and Riemann surfaces. can anyone ...
-4
votes
0answers
44 views

The riemann hypothesis [closed]

I'd like to speak the zeta functions that gives raise to the famous riemann hypothesis telling about the proprerties of its non trivial zeros. Last week i have finished my complex analysis course and ...
0
votes
2answers
43 views

Multiplicity of an holomorphic map between Riemann surfaces

I need help understanding the meaning of multiplicity in a point of an holomorphic map between Riemann sufaces. So $F\colon X \to Y$ be an holomorphic, not constant map between Riemann surfaces and ...
1
vote
1answer
60 views

Reading Griffiths Harris: Quick question

Why is a meromorphic section without zeros and poles on a compact Riemann surface necessarily a constant? Thank you very much.
1
vote
1answer
46 views

maple plot of Belyi function

I would like to understand how to construct Figure 5 of the paper Composition is a generalized symmetry by Alexander Zvonkin: The hypermap/dessin d'enfant of Figure 4 is while the Belyi function ...
4
votes
0answers
64 views

Writing the roots of a polynomial with varying coefficients as continuous functions?

Consider the monic polynomial $$p_{\zeta}(z) = z^n + a_{n-1}(\zeta)z^{n-1} + \dots + a_0(\zeta), $$ where the $a_{i}$'s are continuous functions defined over $\mathbb{C}$. As is well known, the ...
1
vote
1answer
30 views

non-singular Riemann surface implies irreducible polynomial without connectedness?

Let $$ F(w,z) = \sum_{i=0}^n a_i(z)w^{n-i}$$ be a polynomial in $z,w$. Define a Riemann surface as the set $$\Gamma:= \left\{ (z,w)\in \mathbb C^2 \mid F(z,w)=0 \right\} $$ and call it non-singular if ...
0
votes
0answers
16 views

The normalization of a product of varieties

Let $X,Y$ reduced complex analytic spaces, $X^{'}$ and $Y^{'}$ the normalizations of $X$ and $Y$, respectively. Let $(X \times Y)^{'}$ the normalization of $X \times Y$. Is true that $(X \times Y)^{'} ...
2
votes
2answers
83 views

Discrete set of zeroes of polynomials must be finite?

Let $F:\mathbb C^n\to\mathbb C^n$ be a polynomial mapping (i.e. $n$ polynomials in $n$ variables). Suppose that $Z = \left\{z \in \mathbb C^n : F(z) = 0\right\}$ is a discrete set (all points are ...
2
votes
0answers
27 views

Blow up in holomorphic dynamics

Someone could explain the concept of blow up used in holomorphic dynamics? Especifaclly in the context of iteratation of holomorphics functions. This concept could be taken to some of the deformation ...
2
votes
0answers
41 views

Rank, degree and slope of a general coherent sheaf

Let $(X,\mathcal O_X)$ be a ringed space and $\mathcal F$ be a coherent sheaf of $\mathcal O_X$-modules on $(X,\mathcal O_X)$. Are there the definitions of rank, degree and slope of $\mathcal F$ in ...
2
votes
0answers
24 views

Analytic cohomology on Zariski site vs analytic cohomology on analytic site

If I have an affine algebraic complex manifold (in fact it is Stein), what is known relating the cohomology of analytic sheaves using only Zariski opens vs the cohomology of analytic sheaves using the ...
1
vote
2answers
124 views

roots of a polynomial inside a circle

I am asked to show that for $n$ larger or equal to $2,$ the roots of $1 + z + z^{n}$ lie inside the circle $\|z\| = 1 + \frac{1}{n-1}$ Attempt1: Induction for the case $n = 2,$ the roots of $1 + z + ...
0
votes
1answer
66 views

function meromorphic on C

Good evening I have a doubt: let $f$ and $g$ are two functions meromorphic on $\mathbb{C}$ such that $g(w) =f(\frac{1}{w})$. Now g is defined for $w = 0$ (because of all meromorphic $\mathbb{C}$).Can ...
1
vote
1answer
45 views

Base-point-free linear systems (elementary?) property

I'm having troubles solving exercise K on page 167 of the book "Algebraic curves and Riemann surfaces" of Miranda. The question is the following one : Let Q be a base-point-free linear system, let ...
4
votes
2answers
94 views

Applications of Stein spaces in Algebraic Geometry

I want to know where are essential applications of the theory of Stein spaces in algebraic geometry. I heard Cartan's theorem A & B were used in Serre's GAGA, but are there any other applications? ...
2
votes
1answer
73 views

Question about divisors

Let $\Lambda=<2\omega_1,2\omega_2>$ be a lattice in $\mathbb{C}$, $C=\mathbb{C}/\Lambda$ an elliptic curve. Let $O(0,0)$ (neutral element of the lattice), and $A \in \mathbb{C}/\Lambda$ point of ...
5
votes
1answer
81 views

Proving the Uniformization Theorem for Elliptic Curves (An Exercise from Silverman's Advanced Topics on Elliptic Curves )

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves there are two demonstrations of the Uniformization Theorem for the Elliptic Curves (It says that, given an Elliptic Curve $E$, ...
4
votes
2answers
60 views

Holomorphic functions on algebraic curves

I have been asked to solve the following problem, but I really need some help... How are the holomorphic functions $f:C\to D$, where $C,D$ are nonsingular algebraic curves of genus 1? I know that I ...
4
votes
1answer
115 views

What's the intuition for the fact that $\mathscr{O}(-k)$ and $\mathscr{O}(k)$ are so different?

maybe this question makes no sense and I just cannot accept the fact that dual the line bundle is different from the respective line bundle itself. Since it looks like that manifolds are more ...
3
votes
1answer
79 views

Weird definition of Kodaira-Spencer map (What's a relative Kähler differential on a manifold?)

When I was reading "Advances in Moduli Theory" by Shimizu Yuji, I´ve found a weird way of writing the Kodaira-Spencer map $\rho$. For a given analytic family of complex compact manifolds $\pi ...
4
votes
1answer
53 views

Do I correctly understand the constructions involved in definition of Cartier divisor?

Let $(X,\mathcal O)$ be a ringed space, where $\mathcal O$ is a sheaf of unital commutative integrity domains. Let $\widetilde{\mathcal M}_U$ be the field of fractions of ring $\mathcal O_U$ for any ...
1
vote
0answers
28 views

zeros and poles of meromorphic section of $\mathcal{O}_{\mathbb{P}^2}(-1)$

I'm a bit confused regarding the following example: it is stated that $$ \frac{x^3+y^3+z^3}{x^2yz}$$ is a meromorphic section of $\mathcal{O}_{\mathbb{P}^2}(-1)$, and then one is asked to find poles ...
0
votes
0answers
39 views

relation between quotient field of holomorphic functions and meromorphic functions

Let $U \in \mathbb{C}^n$ open connected. Is it true that the quotient field of $\mathcal{O}_{\mathbb{C}^n,z}$ for $z \in U$ is the stalk of the sheaf $\mathcal{K}$ where $\mathcal{K}(U)$ is the field ...
1
vote
1answer
66 views

Prove that each convex open subset of $\mathbb C^n$ is a domain of holomorphy. [closed]

Prove that each convex open subset of $\mathbb C^n$ is a domain of holomorphy. Please help. Thanks in advance!
3
votes
2answers
62 views

Elliptic Operators and Continuity

I am reading a book on Hodge theory (Ref: http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/hodge-smf.pdf) or for english ...
3
votes
0answers
75 views

Computing Riemann surfaces of a given algebraic function

I've never seen written in a book a way or an algorithm for computing Riemann surfaces of a given algebraic function. I would like to know how to construct such Riemann surface using intuitive cutting ...
0
votes
2answers
126 views

Why is the set of points where a complex polynomial does not vanish connected?

Let $p$ be a complex multivariate polynomial. Let $C$ be the set of those complex tuples where $p$ is nonzero. Then, $C$ is connected.
7
votes
1answer
223 views

Projective closure of an algebraic curve as a compactification of Riemann surface

Assume $f \in \mathbb{C}[x,y]$ a polynomial such that the affine algebraic curve $X=V(f)$ has no singular points. Then there is a natural structure of non-compact Riemann surface on $X$, which can be ...
1
vote
0answers
308 views

Position and nature of singularities of an algebraic function (Ahlfors)

I want to solve the following exercise, from Ahlfors' Complex Analysis text, page 306: Determine the position and nature of the singularities of the algebraic function defined by $w^3-3wz+2z^3=0.$ ...
3
votes
2answers
165 views

Prove that a complex valued polynomial over two variables has infinitely many zeroes

This is a homework question that I am struggling with. Given a polynomial over the complex numbers in two variables, show that the polynomial has infinitely many zeroes. So let's say that the ...
0
votes
0answers
22 views

Functions on $K(X)$ and DVR.

In our definition a variety is an integral and separated scheme $X$ and we denote with $K(X)$ the fild of rational functions on $X$. Let $X$ be a normal variety. Let $D$ be an integral codimension-one ...
1
vote
1answer
58 views

Affine algebraic curve is Riemann surface

The problem: Let $P\in\mathbb{C}[z]$ be a non-constant polynomial with simple zeros. Show that the affine algebraic curve $X=\{(z,w)\in\mathbb{C}^2\,:\,p(z)=w^2\}$ is a (connected) Riemann surface. ...
1
vote
0answers
47 views

Question on a statement about analytic variety irreducible at $0$.

I am trying to understand this statement, it is in "Principles of Algebraic Geometry" by Philip Griffiths and Joe Harris. In page 13, third point, they are trying to prove that an analytic variety ...
1
vote
1answer
88 views

Singular points of an analytic variety

I am struggling with a question that was posed here before, about why a reducible analytic variety $V=V_1\cup V_2$ must be singular in $V_1\cap V_2$. I must say I didn't really figure out the ...
5
votes
1answer
70 views

Partial derivatives in $\mathbb{C}^n$

I'm trying to figure out an equality from a proof by Griffiths and Harris to the holomorphic inverse function theorem (in Principles of Algebraic Geometry). They state: $$\frac{\partial}{\partial ...
4
votes
1answer
49 views

Dimension of a meromorphic differentials space

What is the dimension $d_{ \large k,n}$ of the space of the degree $k$ meromorphic differentials on the sphere with fixed residues ($\alpha_i$) at $n$ points $z_i$ ? The question is asked in this ...
2
votes
2answers
429 views

Generalization of Cauchy Residue theorem to Multi-dimensional holomorphic functions

We know Cauchy Residue theorem from the Complex analysis. however I wonder if there is a kind of Generalization of Cauchy integral and Residue theorem to the complex multidimensional holomorphic ...
2
votes
0answers
81 views

Visualize a projective curve $X^3+Y^3=Z^3$ in $P_2(C)$ as a torus

Let $P_2(C)$ be the 2 dimensional complex projective space, I want to prove that the projective curve defined by $M=\{[X,Y,Z]\in P_2(C)|X^3+Y^3=Z^3\}$ is a torus. I know that there is a theorem saying ...
9
votes
4answers
328 views

Complex analysis book for Algebraic Geometers

I know that there exist many questions on this site on complex analysis books but my question is more specific than that. I am looking for recommendations for a concise complex analysis book but with ...
13
votes
1answer
190 views

Real points of a complex curve

Since the "real points" of a complex curve can mean a couple of different things, bear with me while I'm annoyingly formal here. Consider first a cubic curve $y^2 = x^3 + a x + b$. Write $$S := \{ ...
5
votes
0answers
63 views

Branch locus along a smooth curve

Let $f : S \to X$ be a dominant morphism of smooth complex surfaces. Let $C \subset S$ be a smooth curve such that $df$ is of rank $1$ along $C$ and that in a neighborhood of $C$, $df_x$ is an ...
18
votes
0answers
400 views

Is there an exposition of complex analysis firmly separating the algebra, analysis, and topology?

Complex analysis seems to work because of the interplay between algebraic geometry over $\mathbb{C}$, and analysis and topology exploiting the fact that $\mathbb{C}/\mathbb{R}$ happens to be a ...
5
votes
0answers
127 views

What is the complex *algebraic* moduli of elliptic curves?

It's well-known that $SL_2(\mathbb{Z}) \backslash \mathfrak{h}$ is a coarse moduli space for complex elliptic curves. Thus, I would expect this to be related to the pullback of $\mathcal{M}_{ell} ...
0
votes
1answer
82 views

Show that $f(z)= \frac{-1}{z}$ maps each circle of the form $|z+ti| = (t^2-1)^{1/2}$ onto itself.

Show that $f(z)= \frac{-1}{z}$ maps each circle of the form $|z+ti| = (t^2-1)^{1/2}$ onto itself
6
votes
0answers
165 views

When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ for $\Gamma$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least ...
0
votes
0answers
61 views

$C^{\infty}$ 1-form on a Riemann surface is unique.

Let $X$ be a Riemann surface and $\mathcal{A}$ be a complex atlas on $X$. Suppose that $C^{\infty}$ 1-forms are given for each chart of $\mathcal{A}$, which transform to each other on their common ...
2
votes
0answers
173 views

Algebraic curves and riemann surfaces

I am a physics undergrad with no formal background in complex analysis. I have done complex analysis at the level of the first 4 chapters (till Complex integration) from Churchill and Brown. I am very ...
3
votes
2answers
68 views

Why function $j(\tau)$ has degree 1?

We have $$ j(\tau)=\frac{1}{q}+\sum_{n=0}^{\infty}a_nq^n, a_n\in\mathbb{Z},q=e^{2\pi i\tau} $$ Then it is said that because $j$'s only pole is simple, $j$ has degree 1 as a map ...