# Tagged Questions

**1**

vote

**1**answer

135 views

### A family of examples of (Brody) hyperbolic surfaces

Let $P(x)$ and $Q(x)$ be two general polynomial of the same degree $d$ (d can be arbitrary large). Consider the surface $S : z^2 = P(x) Q(y)$ in the projective space $\mathbb{P}^3$. I want to prove ...

**-1**

votes

**1**answer

125 views

### Order of the zero of a meromorphic function under the action of $Gal(\mathbb{C},\mathbb{Q})$

Let's take $X$ a Riemann surface as an algebraic curve in $\mathbb{P}^n$.
The group $Gal(\mathbb{C},\mathbb{Q})$ (automorphisms of $\mathbb{C}$ which act as the identity on $\mathbb{Q}$) acts on $X$ ...

**6**

votes

**1**answer

263 views

### interpolation with derivative of rational fraction

Studying a problem in conformal geometry, I am facing to the following interpolation problem.
Let $P$ and $Q$ two coprime polynomials. Then let $A$ and $B$ two coprime polynomials such that
...

**0**

votes

**1**answer

306 views

### Question on Hartogs's Extension Theorem

Does Hartogs's extension theorem hold if one replaces the word holomorphic by analytic (of course still in several variables)?
For Hartogs's Extension Theorem see here:
...

**4**

votes

**1**answer

239 views

### infimum of the Calabi energy in a given Kahler class

Given a compact Kahler manifold $M^n$ and a Kahler class $\Omega$. We have the Calabi energy (or Calabi functional)
$$\textrm{Ca}(\omega)=\int_Ms^2(\omega)\omega^n,\qquad \forall \omega\in\Omega.$$
...

**1**

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**1**answer

145 views

### Can a locally defined holomophic function which vanishes on a subvariety $V$ be written in terms of globally defined polynomials vanishing on $V$?

Given a complex algebraic, affine variety $X$ and a subvariety $V$, let $I_V$ indicate the ideal of polynomials on $X$ with vanish on $V$.
Given an open subset $U$ of $X$, is it true that the ideal ...

**2**

votes

**1**answer

268 views

### Constructing the imaginary part of a holomorphic function

Hallo,
Let $f: U \rightarrow \mathbb{R}$ be a analytic function, where $U \subset \mathbb{C}^{n}$ is a open set (paracompact, starshaped or convex i.e. sufficiently nice). Does there exist a function ...

**11**

votes

**2**answers

394 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 ...

**3**

votes

**2**answers

386 views

### Corona Theorem in several variables

Hallo,
I have read about the Corona Theorem (see link:http://en.wikipedia.org/wiki/Corona_theorem). From this one ca deduce that: Let $f_{1}, ..., f_{n}$ be holomorphic bounded functions on the unit ...

**1**

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**2**answers

186 views

### k-Hyperbolic manifolds

A complex manifold $N$ is $k$-hyperbolic ($\dim N \geq k$) if any holomorphic map from $\mathbb C^k$ to $N$ has rank strictly less than k. Brody hyperbolic manifolds are $1$-hyperbolic for example. ...

**3**

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**3**answers

360 views

### Interesting results for open Riemann surfaces

As far as I know, interesting results for open Riemann surfaces are quite rare. One of them is the theorem of Gunning and Narasimhan, which asserts that every connected open Riemann surface admits a ...

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**2**answers

203 views

### Dimension of the full automorphism

Let $\mathbb P_1$ be the one dimensional complex projective space.
What is the connected component of the full automorphism of
$\mathbb C^*\times \mathbb P_1$.
Is it a complex Lie group? I mean is it ...

**-1**

votes

**1**answer

308 views

### holomorphic extension of a function [closed]

hi,
I have the following question: let $U \subset \mathbb{C}^{n}$ be some open set containing zero. let $\tilde{U} = U \cap \mathbb{R}^{n}$. assume we have a real-valued analytic function $f : ...

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**3**answers

384 views

### Is there an explicit formula for the modulus of an annulus given a parameterization of the inner and outer boundries?

Every open set in the complex plane homeomorphic to an annulus is biholomorphic to exactly one annulus whose inner radius is 1 and whose out radius is $r>1$. Each value of $r$ gives a different ...

**1**

vote

**1**answer

172 views

### ( finite ) Blaschke product in higher dimensions ?

Hello, as we know, the (finite) Blaschke product $P$ in $\mathbb{C}$ or in $ \mathbb{R}^2 $ is defined by $\prod_{j=1}^{k} \frac{z-a_j}{1-\bar{a_j}z}, a_j \in \mathbb{D}$. I was wondering whether ...

**1**

vote

**1**answer

360 views

### How to study the nonregular part of a finite branched holomorphic covering?

A finite branched holomorphic covering is a holomorphic map $f : V \to W$
between holomorphic varieties $V$ and $W$ such that
$f$ is a finite branched covering (in the topological sense)
There is a ...

**4**

votes

**0**answers

247 views

### Etymology of the O-notation for algebras of holomorphic functions

The notation $O(X)$ seems to be a quite standard notation for the algebra of all holomorphic functions on some connected domain in $\mathbb{C}^n$ (or a complex manifold). I would like to know where ...

**2**

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**1**answer

222 views

### Little Picard for (open) complex manifolds?

"Little Picard" states that if a complex function $f(z)$ is entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point. The ...

**0**

votes

**2**answers

752 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 ...

**2**

votes

**3**answers

279 views

### Holomorphic map from a neighborhood in $\mathbb C$ to S^3

Does there exist a holomorphic map from a neighborhood of $\mathbb C$ to $S^3 \subseteq \mathbb C^2$?

**1**

vote

**2**answers

208 views

### On bounded homogeneous connected domains of C^n

So let $D\subseteq \mathbb{C}^n$ be a bounded connected open set with a transitive action of its group of biholomorphisms (which we denote by $Hol(D)$). Note that I'm not assuming that $D$ is ...

**1**

vote

**0**answers

214 views

### glue together a sequence of holomorphic forms

hallo,
my problem is the following: i have a finite sequence of holomorphic $k-$forms $\alpha_{k}$, each defined on open subsets $U_{k} \subset M$, where $M$ is a complex $n$-dimensional manifold, ...

**0**

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**0**answers

314 views

### covariant derivative complex manifold

Assume we have $X$ a complex manifold and $Y = Y^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ and $Z = Z^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ two vector fields on $X$. Let $\nabla$ be the ...

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**1**answer

201 views

### biholomorphism complex manifold induced structure

Let $X$ be a $n$ dimensional complex manifold with complex structure $I$ and assume one has a diffeomorphism $f : \mathbb{C} \rightarrow X$ of some open set $U$ in $\mathbb{C}$ into its image $f(U)$. ...

**1**

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**1**answer

250 views

### how to prove the relationship between pseudoconvexity and the monge-ampere matrix?

In several complex variables , to determine the pseudoconvexity of a domain in $C^n$ is very important . There are various criterion to decide whether a domain is pseudoconvex . In particular ,if ...

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**3**answers

867 views

### Conformal Mappings for hyperbolic polygon

I am searching for a conformal mapping from the upper halfplane onto a hyperbolic polygon, i.e. the sides of the polygon have to be geodesics.
The classical Schwarz Christoffel theorem does the job ...

**2**

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**1**answer

204 views

### Is a compact subset of a Stein space admitting a fundamental system of Stein neighbourhoods necessarily holomorphically convex?

Let X be a Stein manifold and let K be a compact subset of X. Suppose that K possesses in X a fundamental system of neighbourhoods which are Stein spaces. Then, it is a result by Rossi that such a ...

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**1**answer

262 views

### Is the closure of an open holomorphically convex subset of a Stein space holomorphically convex?

Let X be a Stein manifold and U an open, connected, relatively compact, holomorphically convex subset of X. Is the closure of U in X holomorphically convex?
Also, if X is a Stein space with a finite ...

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votes

**1**answer

499 views

### Why can't subvarieties separate?

I'm posting my answer to this question as its own question:
Let $V$ be an irreducible projective variety over $\mathbb{C}$. Let $U$ be a Zariski open set in $V$. I'll use $V(\mathbb{C})$ and ...

**4**

votes

**3**answers

1k views

### Most important domains, extension theorems, and functions in several complex variables

For a new learner of several complex variables, the many domains (eg holomorphically convex, pseduconvex, Stein) and the many extension theorems (eg Riemann) and the many functions (plurisubharmonic) ...