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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The Hessian quadratic form of a real function can't be complex

Let $\Omega\subseteq\Bbb C^n$ open, $z_0\in\Omega$, $r:\Omega\to\Bbb R$ twice real differentiable. We know that $\Bbb C^n\simeq\Bbb R^{2n}$ is an isomorphism of vect.sp. So we think $\Bbb C^{n}$ as ...
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20 views

change of complex variables

Suppose that $z_1 , z_2,\cdots ,z_k,\cdots ,z_d$ are complex variable numbers. Locally, and suppose $f_1, f_2 \cdots, f_k $ are $k$ holomorphic functions on $z_1, z_2 \cdots,z_d$. At ...
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17 views

holomorphic range of a zero variety

Let D be the unit disk, and $d\ge 2$. $F=(f_1,...f_d)$ is a holomorphic map over the closure of $D^d$ to $C^d$. What information can we get about the image of $\{0\}$x $D^{d-1}$ under $F$? Hope some ...
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3answers
41 views

Is the boundedness necessary to extend harmonically?

"If $u$ is harmonic and bounded in the punctured disk $0<|z|< \rho$, then $u$ can be extended harmonically to the disk $|z|<\rho$ harmonically." This fact has been shown here. My Question ...
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1answer
87 views
+100

Global Residue Theorem in CP^2.

Consider the following meromorphic form defined on $\mathbb{C}P^2$: Be $\phi_{1,2}$ the chart given by $\phi_{1,2}(Z_1,Z_2) = (1,Z_1,Z_2)$, in these coordinates define $\omega= \frac{1}{Z_1 Z_2} dZ_1 ...
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1answer
46 views

Variant of Riemann mapping theorem

Put $D=\{z\in \mathbb C: |z|<1\}$ (open disk) and let $\Omega$ be non empty open simply connected in $\mathbb C$ and $\Omega \neq \mathbb C.$ Then Riemann mapping theorem tells us that there exists ...
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7 views

Characterization of Multivariate Polynomials with Unique Critical Point

I would like information about $\{f\in \mathbb{C}[x_1,\ldots,x_n]:Z(\nabla(f))=\{0\}\}$. Above, the $Z$ denotes the vanishing locus of a function, i.e. the set of points where it vanishes, and ...
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3answers
55 views

Continuity of distance function

I wonder if this is obvious because it does not appear to me obvious at all: Reference: [Hormander: An introduction to Complex Analysis in Several Variables], page 37: Here is the quote Now, let ...
2
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0answers
32 views

Are there non-holomorphic or non-analytic polynomials of several complex variables?

Having no prior exposure to several complex variables, I am trying to read some papers involving this subject. I came upon the terms analytic polynomial and holomorphic polynomial. Do they simply ...
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24 views

Questions on Levi pseudoconvex domain

Here are some of the exercise questions which I am stuck: Question 1: Give an example of a real hypersurface $M\subset\mathbb{C}^n$ such that $0\in M$, such that $M$ has a polynomial defining ...
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1answer
30 views

Domain of holomorphy characterization

Let $U\subset\mathbb{C}^n$ be a domain of holomorphy. The following proposition is true? There is a holomorphic function $f\in H(U)$ such that for all $a\in\partial U$, $\lim_{z\to a}f(z)=\infty$ ...
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1answer
50 views

Zero Set of a complex polynomial of several variables

I'm reading over Scheidemann's Intro. to Several Complex Variables book and at the beginning on pg. 9 I got a little stuck on proving that the zero set of a complex poly in several variables is not ...
3
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1answer
37 views

How can a boundary measure of a function be absolutely continuous?

I'm studying firsts tools in several complex variables. In my book I found what follows: It can be proved that if $\varphi$ is strongly subharmonic and has a finite majorant in the unit ball, then ...
2
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0answers
22 views

Definitions of complxe singularity exponent

If $\Omega\subset\mathbb C^n$ is an open subset of $\mathbb C^n$ and $\varphi$ is a plurisubharmonic function in $\Omega$, for a point $x\in \Omega$, we define the complex singularity exponent of ...
4
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2answers
70 views

Proof or Counterexample:Is every open connected set $D \subset \mathbb C$ is a domain of holomorphy?

Def: An open set $D \subset \mathbb C^n$ is called a domain of Holomorphy if there exists a holomorphic function $f$ on $D$ such that $f$ cannot be extended to a bigger set. Is every non empty open ...
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0answers
12 views

What conditions are needed for cpt support of anti-gradient?

I'm reading through a paper by Michael Christ, "On the $\bar{\partial}_b$ Equation for Three-Dimensional CR Manifolds" found in the Proceedings of Symposia in Pure Mathematics, Volume 52, Part 3. One ...
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1answer
25 views

Relating holomorphic sections of a line bundle to holomorphic functions on the line bundle

I am in the following situation: $X$ is a complex manifold and $L$ a holomorphically trivial line bundle over $X$. My objective is to understand the space of holomorphic functions $\mathcal{O}(L)$ on ...
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2answers
30 views

Is it so easy to show a locally bounded function with holomorphic cross sections is holomorphic?

Let $f: U \rightarrow \mathbb{C}$ where $U$ is open in $\mathbb{C}^n$ and suppose every coordinate cross section of $f$ is holomorphic (I hope that's not too colloquial). I've heard it's a somewhat ...
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1answer
49 views

Finding an entire function $f$

Let $U\subset\mathbb{C}^n$ be a bounded domain. Give an example of an entire function $f:\mathbb{C}^n\longrightarrow\mathbb{C}$ such that: $$f[U]\subset D(0,1)$$ $$f[ext(U)]\subset ext[{D(0,1)}]$$ ...
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1answer
15 views

Convexity of the complex ellipsoid

Let $p_1,\dotsc,p_n$ be positive integers. Define the complex ellipsoid $$\Omega(p_1,\dotsc,p_n)=\left\{(z_1,\dotsc,z_n)\in\Bbb C^n:\sum\limits_{i=1}^n{\left|z_i\right|^{2p_i}}<1\right\}.$$ I ...
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50 views

what would be the formula of $\phi$ in this question?

Suppose $\phi:\mathbb{C}^2\longrightarrow\mathbb{C}^2$ be an entire map (i.e, the components of $\phi$ are entire in each variable separately) with $\phi_1$,$\phi_2$ as its components satisfying ...
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2answers
37 views

Is any simply connected domain conformally equivalent to Cartesian product of unit disks?

By Riemann mapping theorem, any simply connected domain is conformally equivalent to the unit disk. Is any simply connected domain in the complex plane conformally equivalent to the Cartesian product ...
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1answer
31 views

Understanding a Wermer's counterexample.

I am reading some lecture notes on holomorphic functions of several complex variables, see page 105. The part I am struggling with is a proof by Wermer I have asked about runge domains, and ...
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1answer
51 views

Notation for the set of holomorphic differential forms

I used to write $\Omega^p(M)$ for the set of $p$-forms on $M$ and $\Omega^{p,q}(M)$ for the set of $(p,q)$-forms on a complex manifold $M$. Now some authors use $\Omega^p(M)$ to denote the set ...
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43 views

Complex Variables1

Let $f: G \to C$ be analytic and suppose that $G$ is bounded. fix $z_0$ in $\{G\}$ and suppose that $\lim _{z\to w}\sup|f(z)| \leq M$ for w in $\{G\}$, $w \neq z_0$. Show that if $\lim _{z\to ...
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34 views

Proof of Weierstrass Preparation Theorem

In Griffiths and Harris, Principles of Algebraic Geometry, on page 8, near the end of the proof of the Weierstrass Preparation Theorem, he states that $h(z,w)$ has only removable singularities in the ...
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1answer
16 views

Can a complete pluripolar set be a single point?

Let $f:\mathbb{C}^n\rightarrow\mathbb{R}\cup\{-\infty\}$ be a plurisubharmonic function which is not identically $-\infty$.The set $\mathcal{P}:=\{z\in\mathbb{C}^n:f(z)=-\infty\}$ is called a complete ...
4
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1answer
44 views

Calculating integral using Cauchy integral formula in two variables

I want to compute the integral: $\iint_{\partial_0P}\frac{1}{1-4zw}dzdw$ (or any similar integral) using Cauchy integral formula for two complex variables over polydiscs. The distinguished boundary is ...
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1answer
29 views

Multivariate Complex Function

Suppose $f(x,w)\not=0$ for all $x,w\in H^+\cup H^-$ (open upper and lower half planes) and $f$ is a multivariate entire function. Must there exist univariate entire functions $\phi_1$ and $\phi_2$ ...
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22 views

Notions of convergence for maps of complex manifolds

Suppose that $X_t$ and $Y_t$ are two families of complex manifolds, parametrized by maps $\sigma : \mathcal X \to T$ and $\tau : \mathcal Y \to T$. Suppose too I have holomorphic maps $f_t : X_t \to ...
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1answer
25 views

Analytic continuation in several variables

Suppose we have a function $f : U \to \mathbb{R}$, where $U = (0,1)^n \subset \mathbb{R}^n$ is the open box, and that $f(x_1,x_2,\cdots,x_n)$ is separately real analytic in each $x_i$. Does there ...
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3answers
58 views

Ideal of Ring of holomorphic functions

Can you tell me a non trivial ideal of ring of holomorphic functions from C to C.
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1answer
32 views

sub mean value property of plurisubharmonic function

It is well known that a plurisubharmonic function $\varphi$ defined in a domain $\Omega\subset \mathbb C^n$ satisfies the sub mean value property. Now if $\varphi$ is defined on a complex manifold ...
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3answers
44 views

Holomorphic curve with unit norm

Is there an open set $U \subset \mathbb{C}$ and a holomorphic function $\gamma: U \to \mathbb{C}^{2}$ such that $\forall z \in U: \|\gamma(z) \| =1$? if the answer is yes, can the method of unit ...
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1answer
26 views

Radii problem in a power series

I was studying some basic matters of several complex variables (here $\Omega\subseteq\Bbb C^n$, open): After this, before the proof, the author pointed what follows: So I'm going to tell you ...
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1answer
41 views

Cauchy formula in Polydisks

I don't understand a remark after the proof. Here's the theorem: The proof is done by induction on $n$; starting from $n=1$ on the unitary disk in $\Bbb C$, which is the well known Cauchy formula. ...
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1answer
34 views

How is the boundary in product spaces defined?

The general question: how is the boundary defined in product spaces? Given two topological spaces $X,Y$, I'd say that $\partial(X\times Y)=\partial X\times\partial Y$. But looking at what follows it ...
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1answer
105 views

Complex hypersurface globally defined

Let $A$ be a pure one-codimensional analytic subset of a domain $D \subset \mathbb{C}^n$. Is it true that $A$ is defined by one single holomorphic equation $f(z)=0$ if $D$ is bounded and pseudo-convex ...
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1answer
64 views

$f$ is holomorphic iff $df$ is $\Bbb C$-linear

Let $\Omega\subseteq\Bbb C^n$ open connected, $f:\Omega\to\Bbb C$ differentiable in the real sense. We know that $f$ is holomorphic iff $\partial_{\bar z_j}f=0\;\;\forall j=1,\dots,n$ . We know also ...
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1answer
38 views

Gluing together holomorphic functions on $\mathbb{P}^n$

The problem Let $U_j$ for $0\leq j\leq n$ denote the standard coordinate charts of the complex manifold $\mathbb{P}^n$. Fix $d\geq 1$ and assume we are given holomorphic functions $f_j:U_j\to ...
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27 views

Commutative Operators from QM

In Theoretical Chemistry, there seems to be a lot of assumptions about mathematics that are incorporated without justification. One example that I found questionable is this: $$\int \Psi_1^*\ ...
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1answer
28 views

How can I prove that $\operatorname{hess}r(iu,iu)\neq-\operatorname{hess}r(u,u)$?

Let $\Omega\subseteq\Bbb C^n$ open, $z_0\in\Omega$, $r:\Omega\to\Bbb R$ twice real differentiable. We know that $\Bbb C^n\simeq\Bbb R^{2n}$ is an isomorphism of vect.sp. So we think $\Bbb C^{n}$ as ...
0
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1answer
54 views

How can I prove the hypersurface $M$ is neither convex nor concave?

The question is so simple: how can I prove that $M$ (as defined below) is neither convex nor concave (in the real sense)? The point here is to define a new notion of convexity, the complex convexity, ...
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30 views

Complex Hessian Signature

It' all, simply, about the signature of a matrix. Let $\Omega\subseteq\Bbb C^n$ open, $r:\Omega\to\Bbb R$ twice differentiable (real differentiable, not necessarely complex differentiable, i.e. not ...
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1answer
30 views

What are the real and imaginary parts of the complex function?

So, it is asked to find the real and imaginary parts of the specific complex function: $f(z)=sin(z)+i(3z+2) $ So I use $z$ as $z=x+iy$ everything seemed clear till I met Mr. Sinus: $u+iv= ...
3
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1answer
55 views

How can I prove the hypersurface $M$ is neither convex nor concave?

The question is so simple: how can I prove that $M$ (as defined below) is neither convex nor concave (in the real sense)? The point here is to define a new notion of convexity, the complex convexity, ...
2
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1answer
64 views

Does in this case exist necessarely an eigenvalue equal to $0$?

I pasted more than I refer, hoping to be more clear. Look at the claim of the theorem: it states we can change coordinates untill we reach a "good" form for the equation of $r$, which defines the ...
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34 views

What is the definition of the conformal class of a bilinear form?

In the fourth line you can see. It talks about conformal class of a Levi form. I know what a conformal map is, but I can't deduce what a conformal class is from this. Can somebody help me? Many ...
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1answer
28 views

Is this the hessian bilinear form or $1/2$ of it?

Look at: the author writes "full hessian", but it's clearly $\frac12\operatorname{hess}r$. Or not? Other question: why does the author mean by "real harmonic polynomial"? And why it should be ...
2
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1answer
106 views

Chain rule in several complex variables: Wirtinger derivatives

Let $\Omega,\Omega'\subseteq\Bbb C^n$ open, $F:\Omega\to\Omega'$ holomorphic invertible function; it's a variable change, so let's call $F(z)=\tilde z$. Let $r:\Omega\to\Bbb R$ twice differentiable, ...