Questions tagged [several-complex-variables]

Use this tag for questions related to the study of analytic functions of strictly more than one complex variables. For the single complex dimension case, use (complex-analysis).

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3
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1answer
50 views

Problem book recommendations on complex manifolds

I came across the book on Cauchy Riemann manifolds, "CR manifolds and tangential Cauchy Riemann complexes". The book does not have a problem section. I would be grateful if anyone recommends ...
4
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1answer
97 views

Kunneth formula for Čech cohomology

I want to inquire if there exists a Kunneth formula of underlined form: Let $X$ and $Y$ be two compact complex manifolds with two holomorphic vector bundles $E$ and $F$ on $X$ and $Y$ respectively, ...
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1answer
37 views

Triangle inequality for Bergman metric

Please help to prove triangle inequality for d(z,w) Where
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0answers
32 views

Trying to find whether a set is complex analytic

I haven't been able to decide whether the image $X$ of the holomorphic function $f: \mathbb{C} \rightarrow \mathbb{C}^2$ given by $f(z) = (z^2 - 1, z^3 - z)$ is analytic or not. Here is what I have ...
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1answer
83 views

Map from $\{w\in \mathbb{C}\mid \Im(w)\geq 0\}$ to $\{s\in \mathbb{C}\mid \Re(s)\geq \frac{1}{2}\}$

Question Find an onto tranformation (non mobius) from $\{w\in \mathbb{C}\mid \Im(w)\geq 0\}$ to $\{s\in \mathbb{C}\mid \Re(s)\geq \frac{1}{2}\}$ Attempt: Define, $w=i(s-\frac{1}{2})$ Then it maps $\Im(...
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1answer
10 views

Are finitely generated ideals of holomorphic functions closed in the Frechet topology of uniform convergence?

Let $D$ be a polydisc in $\mathbb{C}^n$ and $\mathcal{O}_D$ the sheaf of holomorphic functions on $D$. Further let $J\subset \mathcal{O}_D$ be a finitely generated ideal, since $D$ is Stein it follows ...
2
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1answer
110 views

Linear mappings between two Sets

For $n\ge 2$ consider the following two subsets of the inner product space $\mathbb{C}^n$ with usual Euclidean inner product denoted by $\langle\,,\mkern2mu\rangle$ $$B=\{x\in\mathbb{C}^n:\|x\|<1\}$...
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0answers
14 views

formula for expansion of eigenvalue with multiplicity $p$.

Suppose $A(z)$ is an $n \times n$ matrix with $A_{ij}(z)$ a complex polynomial in $z$ with eigenvalues $\lambda_j(z)$. If $\lambda_j(z)$ has multiplicity $p$ at $z_0$ show that $\lambda_j(z)=\sum_{i=...
1
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0answers
80 views

Showing that two sets are not equal

I am struck in the following question : For any real $p> 1$ show that it is impossible to find $a,b,c,d \in \mathbb{C}$ with $ad-bc\neq0$ such that $$\{(z,w)\in\mathbb{C}^2:|z|^{2p}+|w|^2<1\}=\{(...
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0answers
44 views

There is no invertible linear mapping which maps complex ellipsoid onto the unit disc in $\mathbb{C}^2$

I am trying to solve the following question: For each real $p\geq 1$ consider the set $$D_p=\{(z,w)\in \mathbb{C}^2:|z|^{2p}+|w|^2<1\}.$$ Then there is no invertible linear operator on $\mathbb{C}^...
2
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0answers
56 views

Distinct ellipsoids are not biholomorphic in $\mathbb{C}^2$

I am stuck on the following problem: For each real $p\geq 1$ consider the the set $$D_p=\{(z,w)\in\mathbb{C}^2:|z|^{2p}+|w|^2<1\}.$$ Let $p\neq q$, then, there does not exist any biholomorphism ...
2
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1answer
64 views

Map $w(z)=\frac{1}{1+z^2}$ for $0<|z|<1$ is Conformal

Question Map from punctured disc $$D^*:=\lbrace z\in\mathbb{C}:0<|z|<1\rbrace$$ to a domain $P$ $$P:=\lbrace w\in\mathbb{C}:1/2<\Re(w), w\neq 1\rbrace$$ defined as $$w:D^* \to P $$ $$w(z)=\...
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0answers
40 views

Example based on the Hartogs's theorem.

I am reading an INTRODUCTION TO COMPLEX ANALYSIS (PART-2), FUNCTIONS OF SEVERAL COMPLEX VARIABLES all by myself. I have came across an example which is based on Hartogs's theorem in Shabat and I am ...
1
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1answer
34 views

extension of plurisubharmonic functions across complex hypersurfaces

Let $U \subset \mathbb{C}^n$ be an open set and let $f: U \to \mathbb{R}$ be a continuous function. Moreover, assume that $f$ is smooth on the complement of a complex hypersurface $Z \subset U$ and ...
1
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0answers
28 views

Number of connected components of the polynomially convex hull of a compact $K\subset\Bbb C^n$

Let $K$ be compact in $\Bbb C^n$, without loss of generality $K$ connected. We define its polynomially convex hull as $$ \widehat{K}:=\{z\in\Bbb C^n\;:\;|f(z)|\le\|f\|_K\;\forall f\in\mathcal O(\Bbb C^...
1
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1answer
71 views

Solving 2D Poisson equation using complex analysis

In 2D, the Poisson equation $$ \nabla^2 \Phi(\vec{x}) = \delta^{(2)}(\vec{x}) $$ admits the solution $$ \Phi(\vec{x}) = \frac{1}{2\pi}\log(|\vec{x}|) + \text{constant}\,. \tag{1} $$ By using Fourier ...
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0answers
35 views

Contour integrals in complex manifold

We know that for contours $\gamma$ in $\mathbb{C}$, we can define winding numbers as $$ \frac{1}{2\pi i} \oint_\gamma \frac{dz}{z} $$ Are there similar formulation for $\mathbb{C}^2$ or even for all ...
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0answers
30 views

Generalizing holomorphic functions to complex Banach spaces

Recall that a function $f:\mathbb{C}^2\to \mathbb{C}$ is a holomorphic if the Cauchy Riemann equations hold for each variable of $\mathbb{C}^2$. However we can have a function $f:\mathbb{C}^2\to \...
5
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1answer
78 views

Is this result true? Uniform convergence and inverses

Running through some geometry papers, I found some authors use the following idea: Let $f_n : \Bbb C^m \to \Bbb C^m$ be a sequence of holomorphic functions converging uniformly to $f : \Bbb C^m \to \...
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0answers
16 views

Does the Bochner-Kodaira-Nakano inequality hold w.r.t. some singular metric?

The following contents are copied from Demailly's e-book Chapter VII-(2.7)-Complex Analytic and Differential Geometry. Let $(X, \omega)$ be a compact hermitian manifold, $\operatorname{dim}_{\mathbb{...
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1answer
64 views

Why could we use the complex basis for the cotangent space and what does Cauchy Integral Formula in this basis say?

I previously studied some differential forms on real manifolds from Introduction to Smooth Manifolds by Lee and I just started reading Principles of Algebraic Geometry by Griffith & Harris. ...
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1answer
35 views

Biggest (centered) polydisc of holomorphy

I have trouble defining the biggest centered polydisc of holomorphy (where I can apply cauchy's inequality) of a multivariate complex holomorphic function. As an example, suppose a function of 2 ...
4
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1answer
63 views

Does analytic imply mixed paritials exist?

Let $f(z,w)$ be defined on $\Bbb{C}\times\Bbb{C}$. Assume for each fixed $z$, $f(z,w)$ is analytic for all $w$. Likewise, assume for each fixed $w$, $f(z,w)$ is analytic for all $z$. Is it the case ...
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0answers
167 views

When exactly is the principal AGM equal to the optimal AGM?

Definitions Following the terminology of SageMath, let the principal arithmetic-geometric mean, $\operatorname{AGP}$, of $(a,b)\in\mathbb{C}^2$ for $a\ne 0$, $b\ne 0$, $a\ne\pm b$ be defined as ...
2
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1answer
14 views

Example of Quasi-circular domain

Can someone provide me with an example of a quasi-circular domain? A domain $D\subset\mathbb{C}^n$ is said to be m-quasi-circular, (where $m=(m_1,m_2,..,m_n); m_i$ being positive integers), if D is ...
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0answers
24 views

A function with $10$ variables and I want to prove that is bounded from above

I have a function with $10$ variables and I want to prove that is bounded from above. I know about the cases of $2$ and $3$ variables where critical points play an important role. However, I cannot ...
0
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1answer
32 views

there does not exist no-where vanishing kahler form on S^2n [closed]

To show that there does not exist no-where vanishing kahler form in S^{2n} I am puzzled at how to prove. It seems to be very easy but I failed.
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0answers
21 views

Conditions on $X,\Omega$ such that $g(K)$ is a Stein compactum in $X$ $\forall g:\Omega\to X$ holomorphic and $\forall K\Subset\Omega$.

Let $\Omega\subset\Bbb C^n$ open bounded and $X$ complex manifold. I am searching for some condition on $X,\Omega$ such that $g(K)$ is a Stein compactum in $X$ for every $g:\Omega\to X$ holomorphic ...
3
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0answers
89 views

Envelope of a real function consisting of a complex function and its conjugate

For a real function $f(x)=A(x)e^{ix}+\overline{A}(x)e^{-ix}$, where $\overline{A}$ represents complex conjugate of $A$. Note that $A(x)$ itself is a complex function $A(x)=A_r(x)+i A_i(x)$. It seems ...
0
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0answers
47 views

Doubt on computation of a integral

Assume $ k < n$. Consider the Lebesgue measure $v $ in $\mathbb{C}^n$ (normalized so the unit ball $B_n$ of $\mathbb{C}^n$ has measure 1). I want to integrate a function $f\circ P$, where $f$ is ...
0
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0answers
39 views

Hartog's theorem idea as presented in Huybrecht

I have two questions. First if I get the main idea of the proof. Secondly I want to ask why do we need the condition $|z_{i \neq 1}| < 1$ and $|z_{i \neq 2}| < 1$. The main idea is that if we ...
2
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0answers
85 views

How can we justify the following passage in Volker Scheidemann's book “Introduction to complex analysis in several variables”?

I'm trying to read a proof of Laurent expansions for holomorphic functions in a polyannuli in $\mathbb{C}^n$ and I'm having trouble justifying the passage outlined in green in the screenshot below. He ...
0
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0answers
25 views

Characterization of a domain of holomorphy

I need to show that the property of being a domain of holomorphy is the same as being a holomorphically convex domain (this result is known as Cartan-Thullen theorem). However, the proofs I found in ...
1
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1answer
102 views

Hartog's extension theorem for codimension 2

I need to use a version of Hartog's extension theorem, that I did not find by googling around. However I think I found a solution myself, and wanted to ask if this is correct. Any feedback would be ...
-1
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1answer
26 views

A question regarding pluriharmonic functions

A real-valued function $u$ that is defined on a domain $D$ of $\mathbb{C}^n$ is pluriharmonic if $u$ is of class $C^2$ and for all $a\in D$ and $b\in\mathbb{C}^n$ the function $\lambda\mapsto u(a+\...
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0answers
26 views

Definition of pluriharmonic functions

In the litterture a real-valued function $u$ that is defined on a domain $D$ of $\mathbb{C}^n$ is pluriharmonic if 1) it is of class $C^2$ and 2) for all $a\in D$ and $b\in\mathbb{C}^n$ the function $...
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0answers
24 views

Analyticity at boundary for function of two variables.

This may be a basic question, but suppose I have a function $f(z,\epsilon)$ such that: $f(z,\epsilon)$ is analytic in the cut, complex $z$ plane for all $\epsilon>0$. All cuts in $z$ are located ...
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0answers
12 views

Is the disjoint union of totally real manifolds still TR?

Consider a family of disjoint totally real smooth manifolds $M_r\subset\Bbb C^2$ continuosly parametrized by $r\in[a,b]$, each of them having real dimension 1. Is their union $M$ still totally real? ...
1
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1answer
28 views

Analog to the identity theorem for holomorphic functions for functions of two complex variables

Consider a function $f:\mathbb{C}^2\rightarrow\mathbb{C}$ that is holomorphic and vanishes on a non empty bounded open set $S$. If $f$ were a function from $\mathbb{C}$, by the identity theorem, $f$ ...
0
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0answers
76 views

system defined by one space with solutions on another: $x∈{\mathbb{C}},y∈{\mathbb{C}} ↣(x,y)∈{\mathbb{R}^2}$

Here is system of equations defined from one domain~codmain but presented onto another (in this case, a proper subset, the reals out of all complex numbers). Is it notated sensically and properly? The ...
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0answers
53 views

Convexity in the $n$ dimensional space $\mathbb{C}^n$?

In the $n$ dimensional space $\mathbb{C}^n$, the word convexity has different definitions (geometric convexity, polynomial convexity, holomorphic convexity...). What is the difference between them? ...
3
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1answer
62 views

Contractivity of a projection on $C(\mathbb{T}^2)$

Consider the $C^\star$ algebra of all complex-valued continuous functions on the 2-torus $\mathbb{T}^2$ with the sup norm. Let us consider the set $\{z^m w^n\}_{m,n\in \mathbb{Z}}$, whose span is ...
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0answers
60 views

Is any proper analytic subvariety contained in an analytic hypersurface?

Suppose $A$ is subvariety of an irreducible complex space(analytic variety) $X$. Is there an analytic hypersurface of $X$ containing $A$?
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0answers
46 views

Minimum of a complex function with various constraints

Let $a\in \mathbb{C}$ be a fixed parameter. We condider the domain in $\mathbb{C}^3$ : $$S = \{ (z_1, z_2, z_3) \in \mathbb{C}^3 , \mathrm{Re} (z_i) \leq \frac 12 \}$$ I would like to prove that ...
0
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1answer
25 views

Product of two simply connected compact sets in $\mathbb{C}$ is polynomially convex in $\mathbb{C}^2$?

Is it true that the product of two simply connected compact sets in $\mathbb{C}$ is polynomially convex in $\mathbb{C}^2$ ? Thanks !
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0answers
45 views

Function $x/y-y/x$ for complex $x,y$

I am trying to understand some properties of the function $f(x,y)=x/y-y/x$, where $x,y\in\mathbb{C}$. Does $f$ have a name? I understand that when $|x|=|y|=1$, $\frac{1}{2i}f(x,y)$ measures the sine ...
0
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1answer
40 views

Exercise of Stein manifold and analytic subset

Let $\Omega$ be a Stein manifold and $V\subset \Omega$ an analytic set. Show that para every $z\in \Omega\setminus V$ there is a holomorphic function $f$ on $\Omega$ vanishing on $V$ and $f(z)\neq0$. ...
4
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1answer
59 views

Name and reference for “ultra holomorphic” functions

Any holomorphic function $f : U\to \mathbb{C}$ from a domain $U\subset\mathbb{C}$ induces a real-analytic mapping $f(x+iy)=u(x,y)+iv(x,y)$ which as such induces by complexification of $x,y$ and $u,v$ ...
1
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0answers
35 views

A uniqueness property of several complex variables

according to this website the following theorem holds true for functions of several complex variables: If $f(z)$ is an analytic function on a domain $D\subset C^n$ that vanishes in a real ...
0
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0answers
29 views

How to show that a random variable is $\pi/2$ circular symmetric analytically?

Suppose I have a random variable $Y$ such that $Y \in \left[ {1 + 1j,1 - 1j, - 1 + 1j, - 1 - 1j} \right]$. Now by looking at $Y$, I can easily say that $Y$ has a $\pi/2$ circular symmetric ...

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