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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1answer
30 views

Sanity check : zeroes of analytic function of 2 complex variables

Let $f$ be an analytic function defined on $\mathbb{C}^2$. Suppose it vanishes on a set of the form $U \times S$, where $U$ is a disk and $S$ is a countable set with an accumulation point. Is it true ...
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0answers
27 views

Show that $\delta =1$

I have a problem: Let $M \subset \Bbb C^2$ be a real analytic hypersurface: $$M=\left \{(z,w) \in \Bbb C^2 \colon \text{Im}\ w=|zw|^2+|z|^8+\frac{15}{7}|z|^2\text{Re}\ z^6 \right \}. \tag 1$$ ...
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2answers
94 views

Discrete set of zeroes of polynomials must be finite?

Let $F:\mathbb C^n\to\mathbb C^n$ be a polynomial mapping (i.e. $n$ polynomials in $n$ variables). Suppose that $Z = \left\{z \in \mathbb C^n : F(z) = 0\right\}$ is a discrete set (all points are ...
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1answer
69 views

Wirtinger derivatives and conjugate

I haven't found anywhere in the literature (that's available to me, at least) a proper explanation of the following relations for a function $f \in \mathcal{C}(\Omega)$, $\Omega$ domain of ...
6
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0answers
55 views

Methods for “recognizing” a polynomial of several variables.

If $f(z)$ is an entire function of a single complex variable, then the following are indirect methods for recognizing that $f$ is a polynomial. 1) Show that $f^{(n)}\equiv0$ for some $n\geq0$. 2) ...
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0answers
58 views

Doubts on Hartogs' lemma

I have the following version of Hartogs' Lemma (which is used to build up the proof of Hartogs' extension theorem). Let $\{\phi_\nu\}_\nu$ be a sequence of subharmonic functions which are uniformly ...
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1answer
30 views

Unclear result in characterization of subharmonic functions.

Let $\Omega$ be a region in $\mathbb{C}$ and let $\phi:\Omega \to [-\infty,+\infty)$ be an upper semicontinuous function. TFAE: \ i)$\phi$ is subharmonic in $\Omega$, ii) for any disc ...
4
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1answer
91 views

Polydisc is not biholomorphic to any strictly pseudoconvex domain

I want to prove the poly disc $P=\left\{z\in \mathbb{C}^2 : |z_1|<1,|z_2|<1\right\}$ is not biholomorphic to any strictly pseudo convex domain in $\mathbb{C}^2.$ Can any one provide a hint?
4
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0answers
65 views

Openness of a subset in complex 2-plane

Let $U$ be a subset of $\mathbb{C}^{2}$ containing the origin $0$. Assume that for any curve $C$ (an affine variety of dimension 1, maybe singular) passing through $0$ we have $U \cap C$ is ...
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0answers
42 views

Compute $[\Lambda,\ \bar{\Lambda}]$

I have a problem: We denote by $[X,\ Y]$ the commutator of $X$ and $Y$ defined by $$[X,\ Y]f(p)=X(Yf)(p)-Y(Xf)(p), \tag{1}$$ for any smooth function $f$ defined on a hypersurface $M$. ...
2
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1answer
52 views

A Submanifold $M$ of $\Bbb C^N$

I have a Proposition in my book, and I write here: For every $p \in M$, with $M$ be a hypersurface in $\Bbb C^N$ the following hold. \begin{align*} \mathcal V_p &= \left \{ X \in \Bbb C ...
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0answers
27 views

Domain of boundedness of a power series

In several texts about several complex variables, like Krantz for instance, domain of convergence and domain of boundedness of a given power series are defined, and the easy result that the former is ...
0
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1answer
43 views

Books for a beginner (Pseudoconvex Domains)

Can anyone recommend me a book on Pseudoconvex Domains with include definitions, as well as a few examples? I have some course notes on that subject, but it's really abstract and theoretical. I want ...
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2answers
52 views

Compute the Jacobian matrix of $f(z,w)=(ze^{i\alpha},\ w)$

I have a question: Let $f$ be a holomorphic and $f(z,w)=(ze^{i \alpha},\ w)$ with $z,\ w \in \Bbb C$. Compute its Jacobian matrix. I remember that: For a holomorphic function $f$, its Jacobian ...
0
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2answers
50 views

Separating a Complex Valued Function

Is there a formula (with mathematical reasoning) for separating a complex-valued function $f(z)=f(x+iy)$ into the form $ f(z)=u(x,y) + iv(x,y)$? Thank You, C.A
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1answer
51 views

Show that we can only find $6$ biholomorphic mappings $\phi$.

I have a problem: Let $U$ be a connected open subset of a point $(0,0)$, and $$ M= \{(z,\ w) \in \Bbb C^2: \text{Im}w = |z w|^2+|z|^8+\dfrac{15}{7} |z|^2 \text{Re}z^6 \}$$ Let $$\phi:\ U \to ...
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0answers
38 views

What is the motivation of the complex analytic spaces?

I wonder about the motivations of the complex analytic spaces, nevertheless they are too complicated and difficult. Is it just a generalization of analytic subset of complex manifolds? Or, they have ...
1
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1answer
52 views

winding number in several complex variables

Is there any analogue of the concept of winding numbers in the theory of several complex variables? If so, can anyone provide me references for studying it?
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0answers
32 views

Biholomorphic, Hypersurface

I'm learning the Hypersurface. And my teacher has a question: Find an example such that two Hypersurfaces are biholomorphic. I think that $$A=\{(x,y)\in \Bbb C,\ \rho(x,y)= x^2+y^2-1=0\}$$ and ...
1
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1answer
34 views

On Stein manifolds and constant functions

Stein manifolds are defined here: http://en.wikipedia.org/wiki/Stein_manifold#Definition Obviously, M is Stein implies that there is a non-constant holomorphic function defined in it. Is the converse ...
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0answers
123 views

Colimit and push forward of quasi-coherent sheaves in the analytic setting

I'm currently working on my master thesis and have some unresolved questions about quasi-coherent sheaves. Since I'm new to algebraic geometry, they might be rather trivial. I'm working in the ...
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1answer
75 views

Transformation property for classical Siegel modular forms of weight 2

Let $\mathbb{H}_g = \{ \tau \in GL_g(\mathbb{C}) | \; {^t\tau} = \tau, Im(\tau) >0\}$ be the Siegel upper half space. There are Eisenstein series $$ E_{2k}(\tau) := \sum_{\gamma\in (P_0\cap ...
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0answers
43 views

Submanifold of a Kobayashi hyperbolic manifold

Let $M$ be complex manifold which is Kobayashi hyperbolic. Let $N$ be a submanifold of $M$ obtained as the zeroes of an analytic submersion $f : M \rightarrow R$, $R$ complex manifold. Question : ...
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0answers
56 views

Explicit Weierstrass Preparation Theorem decomposition

Consider the function $f:\mathbb{C}^2\to\mathbb{C}$ given by $(z_1,z_2)\to z_1^3z_2+z_1z_2+z_1^2z_2^2+z_2^2+z_1z_2^3$. Find an explicit decomposition $f=h\cdot g_w$ as per the WPT, i.e. $h$ is ...
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1answer
83 views

Is the zero set of a holomorphic function nowhere dense?

Let $f:U\subset\mathbb{C}^n\to\mathbb{C}$ be non-trivial and holomorphic with $U$ open and connected. Is the zero set $Z(f)=\{z\in U\mid f(z)=0\}$ a nowhere dense set (i.e. is the interior of the ...
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1answer
208 views

Showing thin sets are Lebesgue measurable

I'm reading through Holomorphic Functions and Integral Representations in Several Complex Variables and have come across a proof I can't get through concerning thin sets. I'll include the definition ...
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0answers
46 views

Calculate all the local automorphisms

The Kohn - Nirenberg domain $\Omega_{KN}$ defined by $$\Omega_{KN}=\left\{(z,w)\in \Bbb C^2:\text{Re}\ w+|zw|^2+|z|^8+\dfrac{15}{7}|z|^2\text{Re}\ z^6<0\right\}$$ How to compute all the ...
2
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1answer
62 views

How do I compute this Holomorphically Convex Hull?

The Holomorphically Convex Hull is defined as $\hat{K}_\Omega= \{z \in \Omega: |f(z)| \leq \sup_K |f|, \forall f\in \mathcal{O}(\Omega)\}$, where $\Omega\underset{open}\subset \mathbb{C}^n$, ...
0
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1answer
49 views

On the hypothesis of the Additive Cousin Problem

The Additive Cousin Problem is the following: Assume that $D$ is a region (open, connected) of $\mathbb{C}^n$. Assume that the Dolbeault Cohomology Group of $D$, $H^1_{\bar{\partial}}(D)$ is equal to ...
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0answers
67 views

Several Complex Analysis

Let $a \in \mathbb{C} \setminus \mathbb{R}$. Show that if $f \in O(\mathbb{C}^{∗} \times \mathbb{C}^{∗})$ such that $f(X)=0$, then $f \equiv 0$, where $X=\{(e^z,e^{az})|z \in \mathbb{C}\}.$ $f$ is ...
1
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2answers
77 views

Uses of Jacobian of a map on $\mathbb{R}^n$.

For a map $f:\mathbb{R}^n\to\mathbb{R}^n$, Jacobian matrix of $f$ is defined as $$\begin{bmatrix} \frac{\partial f_1}{x_1}& \frac{\partial f_1}{\partial x_2}& \ldots \frac{\partial ...
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0answers
24 views

Holomorphic function extended to entire polydisc

How can I show that every holomorphic function in the border of a polydisc $\Delta \subset \mathbb C^n, n>1,$ has a extention to entire $\Delta$? I know this is just a consequence of Maximum ...
2
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1answer
59 views

Question regarding pluriharmonic function

A real valued function $f$ defined on an open subset $U$ of $\mathbb{C}^n$ is said to be Pluriharmonic if $$\frac{\partial^2}{\partial z_i\partial\bar{z_j}}f\equiv0,$$ for $1\leq i,j \leq n.$ I was ...
2
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1answer
100 views

Differential form on complex torus

Suppose $T= \mathbb{C}^n/\Gamma$ is a $n$-dim complex torus. How to prove that every exact $2$-form which has no $(0,2)$ component must be the image of a $(1,0)$ form? Is every torus Kahler? If the ...
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1answer
214 views

Formula for decomposing a form into $(p,q)$ forms

Let $L: \mathbb{C}^n \to \mathbb{C}$ be a real linear map. In other words, $L(a\vec{v}_1+b\vec{v_2}) = aL(\vec{v}_1)+bL(\vec{v}_2)$ for all $a,b \in \mathbb{R}$. Then $L$ decomposes uniquely into a ...
1
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1answer
85 views

Counterexample for Hartogs' Extension Theorem

I'll refer to Hartogs' Extension Theorem as it is stated in Wikipedia (https://en.wikipedia.org/wiki/Hartogs%27_extension_theorem#Formal_statement). I am trying to find a counterexample to show that ...
1
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2answers
240 views

Geometrical Meaning of derivative of complex function

What's the geometrical meaning of f'(z) in complex analysis, as we know in real analysis f'(x) has meaning ie. Slope of curve or gives max/ min. But what does derivative f'(z) has geometrical meaning ...
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0answers
21 views

doubt about definition of holomorphic polynomials

In a topic of several complex variable theory (in particular functions on $\mathbb{C}^2$), I came across a term homogeneous holomorphoic polynomial. By the word, I think it is a polynomial in complex ...
1
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1answer
172 views

zero set of an analytic functio of several complex variables

In one variable complex theory, we have the result that zeroes of a non-zero analytic function are isolated. In several variable theory, this result does not hold. I read it somewhere that this fact ...
6
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1answer
137 views

Level sets of holomorphic functions

It is a somewhat well known fact that any closed set (say in the plane) can be realized as the level set of a smooth ($C^\infty$) function, so level sets of smooth functions are as general as they can ...
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2answers
67 views

Dependence of roots on parameters

Let a function $f$ be holomorphic in a polydisk $U=U'\times U_n$,and suppose that for each fixed $z'\in U'$ it has a unique zero $z_n = \alpha(z')$ in the disk $U_n$. Then the function $\alpha(z')$ is ...
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1answer
81 views

Prove that each convex open subset of $\mathbb C^n$ is a domain of holomorphy. [closed]

Prove that each convex open subset of $\mathbb C^n$ is a domain of holomorphy. Please help. Thanks in advance!
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2answers
143 views

Why is the set of points where a complex polynomial does not vanish connected?

Let $p$ be a complex multivariate polynomial. Let $C$ be the set of those complex tuples where $p$ is nonzero. Then, $C$ is connected.
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0answers
21 views

Analytic semiconjugacy

Consider the following commutative diagram (semi-conjugacy): $$ X\;\; \stackrel{f}{\longrightarrow} \;\;X $$ $${\pi}\downarrow \;\;\;\;\;\; \;\;\;\;\downarrow {\pi}$$ ...
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1answer
81 views

Analyticity and composition of maps

Suppose $\Omega_3\subset \mathbb{C}^3$ and $\Omega_2\subset \mathbb{C}^2$ are two domains (open connected). Let $g:\Omega_3\to\Omega_2$ be a surjective analytic function and ...
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1answer
109 views

Constrained optimization with complex variables

Is there a theory of constrained optimization with complex variables, do you know any textbook on that topic? The typical textbooks on constrained optimization deal with real variables. I actually ...
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0answers
37 views

Intuitively what is it if making a modification of a torus?

It is well-known that if we have a equivalence relation in $\mathbb{R}^2$:$(z_1,z_2)\sim (z_1',z_2')$ iff $$\begin{pmatrix} z_1'\\ z_2' \\ \end{pmatrix}=\begin{pmatrix} 1&0\\ 0&1 \\ ...
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0answers
48 views

“Center of mass” of a complex hypersurface

Background: Suppose first that $f \in \mathbb{C}[z]$ and consider the "analytic hypersurfaces" $V(f)$ and $V(df)$, which are of course just sets of points with multiplicities. (That is, we think of ...
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1answer
249 views

How to imagine zeros of an analytic function of several variables

Let $f(z_1,\cdots, z_n)$ be a holomorphic function of several variables in an open subset of $\mathcal C^n$. Let $Z(f)=\{ (z_1,\cdots, z_n) \: | \: f=0\}$ be the zero set of $f$. If $n=1$, the zeros ...
2
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1answer
65 views

Can I switch the order of integration and “Real(z)” operation?

Let $f(z,\eta)$ be an entire function. I need to calculate (numerically) the integral: $$\int\limits _{0}^{\pi}\mbox{Re}\left(\int\limits _{0}^{\pi}f\left(z,\eta\right)d\eta\right)dz$$ Can I switch ...