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).
643
questions
3
votes
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
votes
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, ...
0
votes
1answer
37 views
1
vote
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 ...
-2
votes
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(...
1
vote
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
votes
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 $\langle\,,\mkern2mu\rangle$
$$B=\{x\in\mathbb{C}^n:\|x\|<1\}$...
0
votes
0answers
14 views
formula for expansion of eigenvalue with multiplicity $p$.
Suppose $A(z)$ is an $n \times n$ matrix with $A_{ij}(z)$ a complex polynomial in $z$ with eigenvalues $\lambda_j(z)$. If $\lambda_j(z)$ has multiplicity $p$ at $z_0$ show that
$\lambda_j(z)=\sum_{i=...
1
vote
0answers
80 views
Showing that two sets are not equal
I am struck in the following question :
For any real $p> 1$ show that it is impossible to find $a,b,c,d \in \mathbb{C}$ with $ad-bc\neq0$ such that
$$\{(z,w)\in\mathbb{C}^2:|z|^{2p}+|w|^2<1\}=\{(...
0
votes
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
votes
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
votes
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
$$w:D^* \to P $$
$$w(z)=\...
0
votes
0answers
40 views
Example based on the Hartogs's theorem.
I am reading an INTRODUCTION TO COMPLEX ANALYSIS (PART-2), FUNCTIONS OF SEVERAL COMPLEX VARIABLES all by myself.
I have came across an example which is based on Hartogs's theorem in Shabat and I am ...
1
vote
1answer
34 views
extension of plurisubharmonic functions across complex hypersurfaces
Let $U \subset \mathbb{C}^n$ be an open set and let $f: U \to \mathbb{R}$ be a continuous function. Moreover, assume that $f$ is smooth on the complement of a complex hypersurface $Z \subset U$ and ...
1
vote
0answers
28 views
Number of connected components of the polynomially convex hull of a compact $K\subset\Bbb C^n$
Let $K$ be compact in $\Bbb C^n$, without loss of generality $K$ connected.
We define its polynomially convex hull as
$$
\widehat{K}:=\{z\in\Bbb C^n\;:\;|f(z)|\le\|f\|_K\;\forall f\in\mathcal O(\Bbb C^...
1
vote
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 ...
2
votes
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 ...
0
votes
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
votes
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 \...
1
vote
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{...
1
vote
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. ...
0
votes
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
votes
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 ...
9
votes
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
votes
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 ...
1
vote
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
votes
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.
0
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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+\...
0
votes
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 $...
0
votes
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 ...
0
votes
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
vote
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
votes
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 ...
0
votes
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
votes
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 ...
0
votes
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 $A$?
1
vote
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
votes
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 !
0
votes
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
votes
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
votes
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
vote
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
votes
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 ...
