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.

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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 ...
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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 $\frac{\partial}{\partial z}\phi(z,w)$. ...
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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: ...
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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 ...
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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 ...
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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
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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, $x_0\in X$ ...
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$F(z,w) = F(e^{i \theta} z,e^{i \theta} w)$ 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 \...
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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 ...
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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|^...
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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_{...
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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_{-...
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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 ...
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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 ...
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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 ...
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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 $f$ ...
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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)&...
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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\...
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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 ...
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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 (...
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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 ...
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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\...
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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^...
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2 answers
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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 ...
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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 ...
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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 ...
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2 answers
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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3 votes
2 answers
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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 ...
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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\...
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2 votes
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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 $f_0(z_1) \neq 0$. ...
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3 votes
1 answer
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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 ...
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2 votes
1 answer
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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 $$\epsilon_j'< \epsilon_j.$$...
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3 votes
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"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{...
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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 ...
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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 $...
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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 ...
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3 votes
1 answer
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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 ...
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4 votes
1 answer
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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{...
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