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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2
votes
1answer
55 views

Book searching in Pluripotential theory

Can anyone recommend me a book on pluripotential theory with an intuitive approach? I have some course notes on that subject, but it's really abstract and theoretical. I want to understand why ...
4
votes
2answers
176 views

Is every holomorphic function in a neighborhood of the origin of $C^n$ necessarily bounded?

Is every holomorphic function in a neighborhood of the origin of $C^n$ necessarily bounded? In which case the statement can be true? Thanks in advance
4
votes
1answer
146 views

Prove that $A$ has Lebesgue measure $0$.

Suppose $G$ is a connected open set of $\mathbb{C}^n$. Prove that: (1). If $f \in$ PSH(G) and $f \not \equiv \infty$ then $A=\{z \in G: f(z)=-\infty\}$ has Lebesgue measure $0$. (2). If $f \in ...
1
vote
1answer
77 views

Prove that $\log(f+g)$ be a plurisubharmonic function

Suppose $G$ is an open set of $E$,($E$ is complex Banach space) and $f,~g :G \to \left[0,\infty \right)$ such that $\log f$ and $\log g$ be two plurisubharmonic(PSH) functions in $G$. Prove that ...
3
votes
1answer
103 views

Bergman-Shilov Boundary and Peak Points

Let $\Omega$ be a domain in $\mathbb{C}^n.$ Consider the Banach algebra $A(\Omega):=\mathcal{C}({\overline{\Omega}})\cap\mathcal{O}(\Omega).$ Denote the Bergman-Shilov boundary of $A(\Omega)$ by ...
2
votes
1answer
129 views

Proper holomorphic map from the unit disc to the bidisc

Let $F\colon \mathbb{D}\rightarrow\mathbb{D}^2$ be a proper, holomorphic map. It is not very difficult to see that every proper holomorphic map from $\mathbb{D}$ to $\mathbb{D}$ is a finite "Blaschke ...
1
vote
1answer
47 views

Proper holomorphic map from the unit disc to the polydisc of dimension greater than one .

Let $F:\mathbb{D}\rightarrow\mathbb{D}^n$ be a proper holomorphic map and $n\geq2.$ I have the following questions: Does F extends in a nbhd of $\overline{\mathbb{D}}$ holomorphically? If not what ...
1
vote
2answers
115 views

Prove that: $\sup_{z \in \overline{D}} |f(z)|=\sup_{z \in \Gamma} |f(z)|$

Suppose $D=\Delta^n(a,r)=\Delta(a_1,r_1)\times \ldots \times \Delta(a_n,r_n) \subset \mathbb{C}^n$ and $\Gamma =\partial \circ \Delta^n(a,r)=\left \{ z=(z_1, \ldots , z_n)\in ...
1
vote
1answer
99 views

Continuous extension of limiting Holomorphic function to the boundary.

Let $\Omega $ be a domain in $\mathbb{C}^n$. Consider the algebra $A(\Omega)=\mathcal{C}(\overline\Omega)\cap\mathcal{O}(\Omega).$ Let $\{F_\nu\}$ be a sequence in $A(\Omega)$ such that ...
5
votes
0answers
54 views

Branch locus along a smooth curve

Let $f : S \to X$ be a dominant morphism of smooth complex surfaces. Let $C \subset S$ be a smooth curve such that $df$ is of rank $1$ along $C$ and that in a neighborhood of $C$, $df_x$ is an ...
1
vote
1answer
133 views

What is the radius of convergence of a power series in two variables?

What is the radius of convergence of a power series in two real variables? If I were to fix one of the variables (i.e. make it a real constant), then would the radius of convergence simply be related ...
10
votes
1answer
289 views

Show that f is a polynomial

Suppose $f$ is an entire function on $\mathbb{C}^n$ that satisfies for every $\epsilon>0$ a growth-condition $$|f(z)|\leq C_{\epsilon}(1+|z|)^{N_{\epsilon}}e^{\epsilon | \text{Im}\,z|}$$ ...
6
votes
1answer
178 views

Convergence domain: $\{(z,w):|z|^2+|w|^2 < 1\}$

I would like a HINT for this: Exhibit a two variable power series whose convergence domain is the unit ball $\{(z,w):|z|^2+|w|^2 < 1\}$. ($z$ and $w$ are complex numbers.) I think that it ...
1
vote
1answer
49 views

Zeros of function $f: \mathbb{C}^{n}\to\mathbb{C}$

I'm having trouble understanding how this condition $(*)$ comes about. I understand the proof, but do not clearly understand how the lemma follows from the proof. Any clarification or additional ...
3
votes
1answer
118 views

Holomorphic extension of a function to $\mathbb{C}^n$

I am stuck at the following question : Let $f$ be a holomorphic function on $\mathbb{C}^n \setminus \{(z_1, z_2, \ldots, z_n) | z_1=z_2=0\}$. Show that $f$ can be extended to a holomorphic function ...
1
vote
1answer
37 views

Biholomorphic Equivalence in $\mathbb{C}^n$

I am stuck at the following problem : For $n \geq 2$, are the sets $\Delta_n$ and $H_n$ biholomorphically equivalent ? Here, $ \Delta_n = \{ ( z_1, z_2, \ldots z_n) : |z_i| < 1, i = 1, 2, \ldots, ...
1
vote
1answer
41 views

What does $\left< dz_j , \frac{\partial}{\partial z_j}\right>$ mean?

Here, in page 2 of Steven Krantz's book Function Theory of Several Complex Variables, what do those angle brackets mean? What kind of product is that between differentials and partial derivatives? ...
1
vote
1answer
47 views

What does $T_z\mathbb{R}^2\otimes\mathbb{C}$ in p. 2 of Huybrechts' book mean?

I apologize for my lack of imagination and the likely silliness of this question, but what does $T_z\mathbb{R}^2\otimes\mathbb{C}$ mean here (last paragraph)? And how does that extension work? Thank ...
2
votes
1answer
268 views

Zeros set of analytic functions over complex plane with several variables

I know that the zeros of analytic function (with one variable) over complex plane are isolated. However, I am not aware about the structure of the zeros set of analytic functions over complex plane ...
1
vote
0answers
29 views

Rational Singularities in dimenson 2 or highter and square integrability

I was wondering if there are any results available on square integrability of quotients of polynomials $P(z_1,z_2,\ldots,z_n)$ and $Q(z_1,z_2,\ldots,z_n)$ like $$\int_{\mathbb{T}^n} ...
6
votes
2answers
196 views

A ‘strong’ form of the Fundamental Theorem of Algebra

Let $ n \in \mathbb{N} $ and $ a_{0},\ldots,a_{n-1} \in \mathbb{C} $ be constants. By the Fundamental Theorem of Algebra, the polynomial $$ p(z) := z^{n} + \sum_{k=0}^{n-1} a_{k} z^{k} \in ...
4
votes
1answer
214 views

Weighted $L^2$ Estimates for Domains $\Omega\subseteq\mathbb{C}$

EDIT: After mrf's comment below and some discussion with my instructor for the course it was decided that the below was not really an issue. Namely, I went into reading this lecture with the notion ...
2
votes
2answers
81 views

Biholomorphism between an open set and $\mathbb C^n$

If $U$ is a polydisc in $\mathbb C^n$, that is, $U=\{z \in \mathbb C^n:|z_i|<1\}$, can we find a biholomorphic map from $U$ to $\mathbb C^n$?
2
votes
1answer
60 views

Biholomorphisms of the polydisk

Let $\mathbb{D}$ denote the unit disk in the complex plane, equipped with the Poincare metric. Let us denote the group of biholomorphisms of $\mathbb{D}$ by $Aut(\mathbb{D})$. Suppose $F: ...
13
votes
3answers
970 views

Why isn't several complex variables as fundamental as multivariable calculus?

One typically studies analysis in $\mathbb{R}^n$ after studying analysis in $\mathbb{R}$. Why can't the same be said $\mathbb{C}$?
1
vote
0answers
38 views

Analytic variety

Let $\Omega$ be a convex domain in $\mathbb{C^{n}}$ , $\Delta$ be unit disk in $\mathbb{C}$ and f: $\Delta \rightarrow b\Omega $ be a non-constant holomorphic map.Then the convex hull of f($\Delta$) ...
1
vote
1answer
74 views

Analytic extension in several variables

Let $D \subset \mathbb{C}^{n}$ be an open domain and $K \subset D$ be a compact subset such that $D - K$ is simply connected. Let furthermore $f$ be a analytic function defined on $D-K$. Is it ...
7
votes
2answers
481 views

Where to learn algebraic analysis

I have been studying categories, sheaf cohomology and complex analysis (the basics since I know just a little). Then recently I tried to find out more about algebraic analysis and these microlocal ...
1
vote
1answer
103 views

polynomial in several variable whose maximum modulus on the ball is known exactly

I'm interested in polynomials in several variables $p(x_1,\ldots,x_n)$, with complex coefficients, such that the maximum modulus of $p$ on the unit complex $n$-ball $$ \max \{ |p(z_1,\ldots,z_n)| : ...
1
vote
1answer
81 views

smooth domain in $\mathbb{C}^2$ and smooth bounday of bounded domain in $\mathbb{C}^2$

How can we define a smooth domain in $\mathbb{C}^2$ and a smooth boundary of bounded domain in $\mathbb{C}^2$? {where $\mathbb{C}^2$ := Cartesian product of complex plane }
2
votes
1answer
103 views

Derivative of composition with holomorphic function of several variables

Let $f : \mathbb C^n \rightarrow \mathbb C^m$ be holomorphic and $g : \mathbb C^m \rightarrow \mathbb C$ be smooth. I am looking for a simple formula for the mixed partials $\partial_i \partial_{\bar ...
3
votes
3answers
137 views

How to find $(-64\mathrm{i}) ^{1/3}$?

How to find $$(-64\mathrm{i})^{\frac{1}{3}}$$ This is a complex variables question. I need help by show step by step. Thanks a lot.
3
votes
1answer
169 views

Multivariate analytic function property

Suppose function $F: \mathbb{C}^n \to \mathbb{C}$ is analytic everywhere and in every coodinate; i. e. for any $q \in \mathbb{C}^n$ and for any $j \in \{1,2,\ldots,n\}$ function $f_{q,j}: \mathbb{C} ...
1
vote
1answer
155 views

Siegel upper-half space

In dimension 1, the Siegel upper-half space $\mathbb{H}=\{\tau\in\mathbb{C}:\Im\tau>0\}$ has the property that if $\lambda\in\mathbb{C}^\times$, then $a\lambda\in\mathbb{H}$ for some ...
1
vote
1answer
85 views

How to calculate $\Re\psi(\mathrm{i}y)=$? and $\Im\psi(\mathrm{i}y)=\frac{1}{2}y^{-1}+\frac{1}{2}\pi\coth{\pi y}$?

How to calculate $$\Re\psi(\mathrm{i}y)= ?$$ and how to proof $$\Im\psi(\mathrm{i}y)=\frac{1}{2}y^{-1}+\frac{1}{2}\pi\coth{\pi y}.$$ Here $\mathrm{i}^2=-1.\psi(s)$ is digamma function. Can you help ...
1
vote
1answer
197 views

formal partial complex derivatives and symmetry

This is probably a silly question but let me ask. As it is well known for a general function $f:\mathbb R^2\to \mathbb R$ which posesses partial derivatives of second order it is not necessarily true ...
1
vote
1answer
48 views

Sheaf on a Stein variety such that $H^{1}(X, \mathcal{F}) \neq 0$

I would like to find a non-coherent sheaf on a Stein variety $X$ such that $H^{1}(X, \mathcal{F}) \neq 0$. Does anyone know any example? Thank you!
6
votes
2answers
220 views

A complex valued continuous function which is holomorphic outside of its zeros

Let $D$ be a non-empty connected open subset of $\mathbb{C}^n$. Let $f$ be a complex valued continuous function on $D$. Let $Z$ be the set of zeros of $f$. Suppose $f$ is holomoprphic on $D - Z$. Is ...
2
votes
1answer
350 views

$n$-sheeted branched covering

Michael Artin's algebra let $f(x,y)$ be an irreducible polynomial in $\mathbb{C}[x,y]$ which has degree $ n>0$ in the variable $y$. The Riemann surface of $f(x,y)$ is an $n$-sheeted branched ...
2
votes
0answers
71 views

Holomorphic function vanishing in a real hyperplane

A real hyperplane of $\mathbb{C}^{n}$ is given by a equation $\textrm{Re}( \ell(Z)) = c$, where $c \in \mathbb{R}$ and $\ell(Z) = \sum_{j=1}^{n} a_{j}z_{j}, a_{j} \in \mathbb{C}$. How to prove that if ...
4
votes
2answers
266 views

Uniform convergence of the Bergman kernel's orthonormal basis representation on compact subsets

Consider the Bergman kernel $K_\Omega$ associated to a domain $\Omega \subseteq \mathbb C^n$. By the reproducing property, it is easy to show that $$K_\Omega(z,\zeta) = \sum_{n=1}^\infty \varphi_k(z) ...
6
votes
1answer
126 views

A sequence that tell us if a holomorphic function of several variables is identically zero

Is there any sequence $\{ Z_{\nu} \}_{\nu \in \mathbb{N}}$ in $\mathbb{C}^{n}$, $Z_{\nu} \rightarrow 0$, such that any holomorphic function in $\mathbb{C}^{n}$ which vanishes in $Z_{\nu}$ for all $\nu ...
2
votes
1answer
119 views

Easy way to calculate $(dd^c u)^n$

Let $u$ be a $C^2$ function from $\mathbb{C}^n$ to $\mathbb{C}$. Define $$ \partial u = \sum\limits_{i=1}^n \frac{\partial u}{\partial z_i}dz_i, \\ \overline{\partial} u = \sum\limits_{i=1}^n ...
4
votes
1answer
193 views

Explicit counter-example to corona problem

The corona problem is known to fail for the complex polydisk, for dimension greater than 2. Does anyone has an explicit example of such functions?
1
vote
1answer
71 views

understanding the definition of domains of holomorphy

can anyone give me an example and explain why any open set in $\mathbb{C}$ is a domain of holomorphy? I have understood the fact from here but not able to understand their explanation for $n=1$
1
vote
2answers
394 views

a suggestion for several complex variable book [duplicate]

Could any one tell me name of some books on several complex variable for some one who will start reading the subject for the first time in his life. he has back ground on Differential geometry,complex ...
3
votes
1answer
85 views

Extension of biholomorphic map to the boundary in higher dimension

Well I was thinking on this problem, any idea to progress will be gladly accepted, Let $f:\Omega\rightarrow V$ be a biholomorphic map where $\Omega$ and $V$ are bounded open set in $\mathbb{C}^n$ with ...
3
votes
1answer
166 views

Uniform convergent series

Let $\Omega$ be domain in $\mathbb C^2$. For each compact set $K_j$ define the holomorphic function $f_j$ on $\Omega$, such that $$\sup_{k_j}|f_j|<2^{-j}.$$ Define $$f= ...
2
votes
0answers
69 views

Eigenvalue of a form

I came across the following matrix while reading an article..Can you please help me to understand the following. We are defining following form: ...
1
vote
1answer
313 views

To show given function is smooth

Let $\phi$ be upper semi-continuous function defined on $\Omega\subset \mathbb C^2$. Let $\Omega_n= \{z \in \Omega: d(z,\partial \Omega) > \frac1n \}$. Let $\chi\in C_c^\infty$ of $|z_1|$, ...