Questions tagged [several-complex-variables]
For questions related to the study of functions of several variables, in particular the study of holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
685
questions
0
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1
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35
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How will I prove that the simultaneous limits for the following exist? [closed]
\begin{align}
\lim_{(x,y) \to (0,0)}f(x,y) = \lim_{(x,y) \to (0,0)} \dfrac{xy}{\sqrt{x^2+y^2}}
\end{align}
Actually I only know how to prove that simultaneous limits does not exist . So please give me ...
2
votes
1
answer
26
views
Is it sufficient to have a complex partial derivative for the complex partial derivative be continous?
Suppose we have $\phi(z,w)$ a function of two complex variables and that for each fixed $w$ the function $z\mapsto \phi(z,w)$ is holomorphic, that is, it exists $\frac{\partial}{\partial z}\phi(z,w)$. ...
- 301
0
votes
1
answer
39
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holomorphic function defined on $\mathbb{C}^n$
It is well known that if $D \subset \mathbb{C}$ is an domain then $f:D \to \mathbb{C}$ holomorphic implies $f \in C^{\infty}$. My question is, this property is still true for holomorphic functions $f: ...
- 536
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1
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25
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Show that this holomorphic function can be extended to $D_{2}((0,0) ;(2,2))$
consider a domain in $C^{2}$:$\Omega=D_{2}((0,0) ;(1,2)) \cup\left\{(z, w) \in \mathbb{C}^{2}:|z|<2 \text { and } 1<|w|<2\right\}$ and $f \in \operatorname{Hol}(\Omega)$, I want to show that ...
- 133
1
vote
1
answer
27
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Understanding difference between a distinguished boundary and 'normal' boundary in several complex variables
I am reading through Tasty Bits of Several Complex Variables and I come across the term distinguished boundary. It seems a distinguished boundary is different from a normal boundary as the author ...
- 184
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0
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24
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How the subject Several Complex Variables motivates to study Complex Manifolds?
How the subject Several Complex Variables motivates to study Complex Manifolds/ Complex Geometry?
How one can study Several Complex variables?
Please advise me a roadmap to learn this subject
- 47
1
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0
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32
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Does the residue map Res commutes with d?
Let $M$ be a complex manifold of dimension $n$, and $D$ be a smooth hypersurface. Let $\varphi$ be a $C^{\infty}$ $ k$-form on $M \backslash D$. We say that $\varphi$ has logarithmic singularities on $...
- 445
1
vote
1
answer
44
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Orthonormal Basis of Bergman Space
This is a problem (#39 or #40 depending on the edition) at the end of Chapter 1 in Krantz's book Function Theory of Several Complex Variables.
Let $\Omega\subset\mathbb{C}^n$ be a smooth and bounded ...
- 153
1
vote
1
answer
36
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Integral of characteristic function in three dimensions
I am having difficulty in evaluating the following integral:
$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x) f(y) f(z)\chi_{{(x,x,x): x\in \mathbb{R}}} dx dy dz$.
Could ...
1
vote
1
answer
33
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Domain of convergence in several complex variables
I am trying to understand how to calculate the domain of convergence for the series [From: Tasty Bits of Several Complex Variables] :
$$ \sum_{j,k} c_{j,k}\; z_1^j z_2^k $$
But I don't really ...
0
votes
0
answers
21
views
PDE and Several Complex variables
I am reading SCV from Grauert and Fritzsche. I have read that as an application one can use SCV in PDE (partial differential equations). I have some general questions:
Is this (SCV-PDE) different ...
- 13
0
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0
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10
views
How to determine the impact of a several factors evolution?
Let's take a simple example. Let's say I have :
A population, with size N
A sub-population, composed by people that goes to the hairdresser with size M (so M < N)
From the two previous points, I ...
1
vote
1
answer
75
views
Understanding arbitrary currents and currents of integration.
I've been doing some readings, and there are some notational issues with currents that I do not understand. Let $\Omega\subset \mathbb C^n$ be open. A current $T$ of bidegree $(p,q)$ is an element of ...
- 665
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0
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39
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Singularity set of coherent analytic sheaf
I am trying to understand section 5.5 of the standard reference Differential geometry of complex vector bundles (S. Kobayashi). Let me set up some notation: let $X$ be a complex manifold, $x_0\in X$ ...
0
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0
answers
49
views
$F(z,w) = F(e^{i \theta} z,e^{i \theta} w)$ then $F$ is a constant
I need a help with the following statement:
Let $D \subset \mathbb{C}^2$ be a domain containing $(0,0)$ and let $F:D \rightarrow \mathbb{C}$ be holomorphic such that
$$F(z,w) = F(e^{i \theta} z,e^{i \...
- 55
1
vote
0
answers
34
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Real life applications of severable complex variables functions theory and complex geometry
I'm studying several complex variables functions and complex geometry this semester, I knows that complex analysis has applications in electrodynamics and fluid mechanics, and differential geometry ...
- 133
5
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0
answers
160
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Poincaré's Theorem using Schwarz Lemma in $\mathbb C^n$
I have been asked to prove Poincaré's theorem using Schwarz Lemma. The statements are as follows:
Schwarz Lemma. Let $B^n_1(\mathbf{0}) := \{(z_1, \ldots, z_n)\in \Bbb C^n:
|z_1|^2 + \ldots + |z_n|^...
- 10.3k
0
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0
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20
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A Question about Cauchy Convergence Principle for multiple series...
In the book of Lectures on Several Complex Variable written by Paul M. Gauthier, it provided a theorem for discussing whether a multiple series is convergence. The multiple series is defined as $\sum_{...
- 85
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0
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16
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Integral Equality, proof of Cartan theorem
In the proof of Cartan's Uniqueness Theorem in Rudin's "Function Theory in the Unit Ball of $\mathbb{C}^{n}$ it's stated that homogeneity of maps $F_{s}$ implies: $$kF_{m}(z)=\frac{1}{2\pi}\int_{-...
- 458
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1
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49
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Holomorphic primitive of several complex variables function
Let $f:\Omega\subset\mathbb{C}^n\rightarrow\mathbb{C}$ a holomorphic function. For any $m\leq n$ I would like to find an $m-primitive$ of $f$, which is still holomorphic in each variable.
If we ...
- 407
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votes
2
answers
52
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Convergence of $\sum_{\alpha\in \Bbb N^n} \frac{z^\alpha}{\xi(\alpha)}$ on $\operatorname{int} U$ where $U \subset\Bbb C^n$ is bounded
Problem. Let $U\subset \Bbb C^n$ be bounded. For $\alpha\in \Bbb N^n$ and $z\in \Bbb C^n$, let $z^\alpha:= z_1^{\alpha_1} z_2^{\alpha_2} \ldots z_n^{\alpha_n} \in \Bbb C$. Prove that $$\sum_{\alpha\in ...
- 10.3k
1
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1
answer
63
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Several complex variables analogue of Cauchy's integral theorem
Cauchy's integral theorem says that the integral of any holomorphic function over a closed path in a simply connected domain is always $0$.
Is there a version of Cauch's integral theorem for several ...
- 407
0
votes
2
answers
51
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If $f\in\mathbb{C}[[z_1,\dots,z_n]]$ such that $f(0)\neq 0$ converges absolutely in a nbhd of $0$, does $\frac{1}{f}$ also converge in a nbhd of $0$?
Suppose $f\in R:=\mathbb{C}[[z_1,\dots,z_n]]$ is a formal power series in several variables such that $f(0)\neq 0$. As such, $f$ has a formal inverse $\frac{1}{f}$ in $R$. If it happens that $f$ ...
- 1,551
3
votes
1
answer
91
views
Is the following universal property of holomorphic functions true?
Suppose that we are given a smooth $f\in C^\infty(\mathbb{\mathbb{R}^2},\mathbb{C})$ and define $\iota$ to be the smooth embedding
$$
\begin{aligned}
\iota :\mathbb{R}^2&\to\mathbb{C}^2\\
(x,y)&...
- 356
4
votes
1
answer
82
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Convergence of $\sum_{\alpha\in \Bbb N^n} z^\alpha$ in $\{z\in \Bbb C^n: |z_i| < 1 \text{ for all } 1\le i\le n\}$
I am reading Introduction to Complex Analysis in Several Variables by Volker Scheidemann.
The convergence of power series in several complex variables is defined in the following way:
Let $\{c_\alpha\...
- 10.3k
1
vote
0
answers
44
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proof question about implicit function for $\mathbb{C}$
So I am stuck in where he says clearly at the end.
Proposition 1.10 (Implicit function theorem) Let $U \subset \mathbb{C}^m$ be an open subset and let $f:U \rightarrow \mathbb{C}^n$ be a holomorphic ...
- 1,543
1
vote
1
answer
63
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Holomorphic primitive of holomorphic function in several complex variables
Given a holomorphic function $f:\Omega\subset\mathbb{C}^n\rightarrow\mathbb{C}$,with $\Omega$ open and connected, when is it possible to find a holomorphic $m-$primitive, i.e. a holomorphic function (...
- 407
2
votes
0
answers
52
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When is the zero set of a polynomial in two complex variables disconnected?
For background, I'm just starting to learn a little bit about classical algebraic geometry, so I'm trying to look at some simple problems that are as concrete as possible to develop an intuition.
This ...
- 6,569
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0
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36
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A question about convex domain in$\mathbb{C}^n$
I'm reading Grauert's book：From Holomorphic Functions to Complex Manifolds.
The fourth problem in the exercise on page 8,it needs us to prove that a domain $G$ is convex if and only if for every $z\in\...
- 179
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votes
1
answer
64
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Two Cauchy-Schwartz type inequality on the norm of complex differential forms
Let $\Omega$ be a domain in $\mathbb{C}^n$, and let $f$ and $g$ be two complex differential forms on $\Omega$:
$$
f=\sum_{I, J} f_{I, J} d z^{I} \wedge d \bar{z}^{J},
$$
$$
g=\sum_{K, L} f_{K, L} d z^...
- 43
1
vote
2
answers
63
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Existence of $f\colon U\to V$ holomorphic non constant
Let $Y$ be a complex mainfold and $\Omega\subseteq\Bbb C^n$ domain, $n<\dim Y$. Take $z_0\in\Omega$ and $y_0\in Y$.
I need a non-constant holomorphic mapping $f\colon U\to V$ where $U\subset Y$ and ...
- 11.4k
0
votes
0
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37
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definition of uniformly bounded holomorphic function
I am reading the proof of Riemann removable singularity theorem in higher dimension. The statement is as followed:
If $X$ is a thin set in an open subset $D \subset \mathbb{C}^n$ and $f$ is a ...
- 405
1
vote
0
answers
36
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Several complex variables exercises with solutions
I am studying about several complex variables, and have found some great books regarding the topic, such as Several Complex Variables by Jaap Korevaar, Jan Wiegerinck, which also has exercises.
Can ...
- 11
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vote
2
answers
91
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$|f|\leq |g|$ in each $r\mathbb{T}^2$ implies on $\mathbb{D}^2$
Question: Let $f,g\in \mathcal{O}(\mathbb{D^2})$ (holomorphic functions on bidisc). Assume $|f|\leq|g|$ holds on each $r\mathbb{T}^2:=\{(z_1,z_2)\in \mathbb{D}^2: |z_1|=|z_2|=r\}$, $-\leq r<1$. And ...
- 1,384
1
vote
0
answers
37
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catlin multitype of a particular domain
This is a remark in a paper by Jiye Yu, "multitypes of convex domains".\
For a smoothly bounded domain $\Omega\subseteq \mathbb{C}^2$, and suppose that it is defined by the defining function ...
- 141
1
vote
1
answer
40
views
How "many" non-reduced spaces with smooth reduction are there?
Suppose $(M,\mathcal{O}_M)$ is an analytic subspace of $\mathbb{C}^n$, such that its reduction is simply some $\mathbb{C}^k\subseteq \mathbb{C}^n$. How "many" such structure sheaves are ...
- 554
0
votes
1
answer
14
views
image of pseudoconvex domain is not pseudoconvex?
Explicitly, how can we find functions and domains satisfy the following:
(i) Both $U$ and $V$ are domains of
$\mathbb{C}^n$.
(ii) $U$ is pseudoconvex.
(iii) there exists a surjective holomorphic map ...
- 91
0
votes
0
answers
60
views
Every extended meromorphic function is a rational function
I have some questions on the following proof that every extended meromorphic function is rational from Stein-Shakarchi. I believe I have some bad misconceptions which are preventing me from ...
- 1,294
0
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0
answers
39
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How do I prove e^w=z? (I wrote the proof process instead of photo)
Our motivation for the definition of the logarithmic function is based on solving the equation $$(1) e^w=z$$ for w, where z is any nonzero complex number. To do this, we note that when z and w are ...
- 1
3
votes
2
answers
66
views
Is any countable set in a bounded domain of $\mathbb{C}$ an analytic set?
Let $X$ be a complex manifold. A closed subset $Y$ of $X$ is called an analytic set if for each $y \in Y$, there is an open neighborhood $U$ of $y$ so that $Y \cap U$ is the zero set of finitely many ...
- 445
0
votes
0
answers
13
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A question related to domains of holomorphy and the boundary distance function
I foud the following definition of distance function:\
Definition A function $\delta\colon \mathbb{C}^n\rightarrow [0,\infty)$ is called a
distance function iff it satisfies
(i) $\delta(z)=0\...
- 91
2
votes
0
answers
40
views
Proof of Weierstrass Prep Theorem
From Huybrechts:
$\textbf{Proposition: Weierstrass Prep Theorem}$
Let
$$f: B_\epsilon(0) \rightarrow \mathbb{C}$$
be holomorphic on the polydisk $B_\epsilon(0)$. Assume $f(0)=0$ and $f_0(z_1) \neq 0$. ...
- 1,543
3
votes
1
answer
91
views
Extending a function which is smooth up to the boundary to a larger domain smoothly
Let $\Omega$ be a bounded open subset of $\mathbb{R}^{N}$ with smooth boundary, and $f\in\mathcal{C}^{\infty}(\bar\Omega)$ a function which is smooth up to the boundary. It is easy to use a cutoff ...
- 43
2
votes
1
answer
99
views
Hartog's Theorem proof clarification.
From Huybrechts:
Let $\epsilon=(\epsilon_1,...,\epsilon_n)$, $\epsilon'=(\epsilon_1',...,\epsilon_n')$be given such that for each $1 \leq j \leq n$, ($n \geq 2$) we have
$$\epsilon_j'< \epsilon_j.$$...
- 1,543
3
votes
0
answers
68
views
"residue theorem" for multivariate contour integral [duplicate]
I am playing with multivariate contour integrals, and I'm curious if there exists some sort of "residue theorem" for such integrals. Here is an example.
Consider the function $f(z,w) = \frac{...
- 509
1
vote
0
answers
159
views
Relation on torsion free sheaf and locally free sheaf in the analytic setting
It is well known that a torsion free sheaf is also locally free on a curve, in the algebraic geometry category. A further fact is that a torsion free sheaf is locally free outside a Zariski ...
- 445
0
votes
1
answer
51
views
Construc a complete metric which coincide with the original metric on a fixed small domain
As well known, $(\mathbb{C}^n, \omega=:\omega_{euclidean})$ is complete . Let $\hat{\omega}$ be a fixed noncomplete metric on $\mathbb{C}^n$.
My question: Can we construct a complete metric on $...
- 445
0
votes
0
answers
50
views
Which domains are contractible on $\mathbb{C}^n$?
Let $\mathbb{C}^n$ be the complex Euclidean space.
The wiki of contractible tells us that any Euclidean space is contractible, as is any star domain on a Euclidean space.
My question is:
1, Is a ...
- 445
3
votes
1
answer
175
views
Complement of zero set of holomorphic function is dense and connected
This is exercise 1.1.8 of Huybrechts' Complex Geometry.
Problem: let $U$ be an open connected subset of $\mathbb{C}^n$. Let $f:U\rightarrow \mathbb{C}$ be a holomorphic function. Show that the ...
- 1,703
4
votes
1
answer
178
views
Weierstrass Preparation Theorem, simple exercise
I am working with a specific version of the Weierstrass preparation theorem, trying to understand it with an example. I'll cite the theorem before stating the question:
Theorem
If $f:U\subset \mathbb{...
- 1,703