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
58 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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0answers
31 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
61 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
votes
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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0answers
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 ...
0
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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
118 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, ...
2
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0answers
38 views

Local normal form of a (several complex variable) holomorphic map at a point?

Suppose $F\colon\Omega\subseteq\mathbb C^m\to\mathbb C^n$ is a holomorphic map. WLOG, $0\in\Omega$ and $F(0)=0$. I want to determine the local normal form of $F$, i.e. classifying $F$ up to local ...
2
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1answer
54 views

Proving that a holomorphic function is constant

I am attempting to prove the following: Let $X$ be a connected complex manifold, and $f\in \mathcal{O}(X)$. For any $x\in X$, there is a complex submanifold of $X$ which is biholomorphic to ...
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0answers
90 views

Relation between being holomorphic in $\Delta\times\Delta$ and in every relatively compact polydisk.

I asked to some professors of my university and no one was able to help me (the one who held the course is abroad for a period, otherwise I'd ask him, obviously). My problem is that simply I don't ...
0
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1answer
62 views

A necessary condition for a multi-complex-variable holomorphic function. [closed]

Let $\Omega\subset \mathbb{C}^n$ be an open unit ball, $f:\Omega \to\mathbb{C}$ is a bounded function. For $a \in \mathbb{C}^n$, define $$ ...
0
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1answer
58 views

Detail in proof of Hartog Theorem

I am stuck in the middle of what you can see below: when the book says "we can repeat the same construction [...] horizontal strip arbitrary close to $z_2=0$": We define different sets $E_l$ on any ...
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0answers
29 views

Is there a name for these sets of functions of several complex variables (most not analytic)?

For each $n \in \mathbb{N}$, I came up with the following sets that I found interesting; at least I've never seen them in the literature before. $S_n = $span{$z_1z_2 \cdots z_n, \bar{z}_1z_2\cdots ...
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2answers
36 views

The conditions that partial derivatives commute

State the conditions that partial derivatives commute, namely, $D_1D_2f = D_2D_1f$. I understand how to prove that these partial derivatives are equal but I don't understand what commute means. ...
3
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1answer
126 views

Interpretation of the Weierstrass Preparation Theorem

I am reading Griffiths and Harris' Principles of Algebraic Geometry but I am having trouble making sense of a statement following the Weierstrass Preparation Theorem (p.9 in my edition). The ...
4
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1answer
59 views

Does a left-invariant vector field on a complex Lie group preserve holomorphic functions?

Let $G$ be a (finite-dimensional) complex Lie group, and suppose $f : G \to \mathbb{C}$ is holomorphic. Let $X$ be a left-invariant vector field on $G$. Must $Xf$ be holomorphic? I think I have a ...
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2answers
30 views

About multivariable quadratic polynomials

Say one has a polynomial function $f : \mathbb{C}^n \rightarrow \mathbb{C}$ such that it is quadratic in any of its variables $z_i$ (for $i \in \{ 1,2,..,n\}$). Then it follows that for any $i$ one ...
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1answer
48 views

Can a product of a Stein manifold and a compact manifold be again Stein?

A Stein manifold is a manifold which is holomorphically separable and convex. It is well known that a product of two holomorphically convex (resp. Stein) manifolds is again holomorphically convex ...
2
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1answer
55 views

Is a connected Reinhardt Domain which containg $0$ necessarely a polydisc?

I'm studying several complex variables basics. Roughly speaking: call $D\subseteq\Bbb C^n$ the set of points in which a given power series $$ \sum_{\alpha\in\Bbb N^n}a_{\alpha}(z-z_0)^{\alpha} $$ ...
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3answers
115 views

Divergence theorem in complex analysis

I am revisiting my understanding of integration by parts in several complex variables, but I have run into an apparent contradiction. This shows my understanding is flawed, which is somewhat ...
2
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1answer
29 views

density on smooth boundary in several variables complex analysis

Assume bounded domain (open and connected) $\Omega\subset \mathbb{C}^n$, and a smooth function $\rho:\mathbb{C}^n\longrightarrow \mathbb{R}$ such that $\rho(x) = 0$ for all $x\in \partial \Omega$, ...
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0answers
44 views

Domain of convergence of series

Could you help me to find the domain of convergence of series : $$\sum\limits_{n,m=1}\frac{n}{m!}z_1^nz_2^m$$ in $\mathbb{C}^2$. The series is product of two series. I think the answer is ...
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0answers
26 views

A question about SCV

Suppose that if $f$ is holomophic in $C^n$ and f is a complex polynomial with respect to each variables separately.How to show that $f$ is a polynomial?
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1answer
25 views

Pluriharmonic is harmonic

I just started learning several complex variables and I'm a little bit confused. I just read that every pluriharmonic function is harmonic and I can't find any proof of that. Please help.
2
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1answer
48 views

Compact normal family

Let $D \subset \mathbb{C}^n$ be a bounded convex domain. Let $F_j : D \times D \rightarrow D$ be a sequence of holomorphic functions such that $F_j(q,q) = q$ for all $j$. Then $\{F_j\}_j$ is a ...
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1answer
33 views

Holomorphic funtions

Let $U$ be an open connected subset of $\mathbb{C}^n$, and $O(U)$ the ring of holomorphic functions on $U$. Prove that $O(U)$ is an integral domain. I have done If $fg\equiv0$ in $U$, then $f$ ...
2
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0answers
106 views

How to apply Fubini's theorem in proof of Osgood's lemma

In the proof of Osgood's lemma for seperate holomorphicity, at one step we get that $$f(z)=\frac{1}{(2\pi \iota)^n}\int_{|w_i-\zeta_i|=r_i}\sum_{v_1,v_2,\ldots,v_n}\frac{f(\zeta)z_1^{v_1}\ldots ...
1
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1answer
44 views

Sum of subharmonics is subharmonic (using the more general definition)

I want to prove that a sum of subharmonic is subharmonic using the following definition "Formally, the definition can be stated as follows. Let G be a subset of the Euclidean space ${\mathbb{R}}^n$ ...
3
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1answer
72 views

Irreducible polynomial and the zero set of its derivative

Let $P$ be a polynomial in $\mathbb{C}[z_1,z_2,...,z_n].$ Consider the derivative of $P,$ $D_{\mathbb{C}}P$, as a holomorphic map from $\mathbb{C}^n$ to $\mathbb{C}^n.$ I have the following question: ...
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0answers
82 views

Irreducibility of a polynomial and connectedness of its zero set

Let $P$ be a polynomial in $\mathbb{C}[z_1,z_2,...,z_n].$ Let $Z(P)$ denotes its zero set in $\mathbb{C}^n.$ I have the following question: Does the irreducibility of $P$ imply that $Z(P)$ is ...
0
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1answer
12 views

A function in $H^\infty(\Delta^2)$

Can you give an example of a function $f\in H^\infty(\Delta^2)$ with $f^{-1}\in L^\infty(T^2)$ but not inner? Here $H^\infty(\Delta^2)$ is the space of all bounded analytic functions defined on ...
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0answers
34 views

An iterative sequence of complex numbers

Consider a disk at center at $(0,0)$ of radius, $r$ $B_r(0)$ in the complex plane. Let $w_1$ and $w_2$ be two complex numbers belong to the disk $B_r(0)$. Consider a scheme, ...
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1answer
253 views

The germ induced by an irreducible polynomial

Let $P\in\mathbb{C}[z_1,z_2,\ldots,z_n]$ be an irreducible polynomial. Let $a\in\mathbb{C}^n$ be such that $P(a)=0.$ Consider the germ of holomorphic functions at the point $a,$ denoted by ...
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1answer
34 views

Is converse of Lewy theorem true?

In complex analysis, there is a result named Lewy's theorem, which states that: If $u=(u_1,u_2):\subseteq \mathbb{R^2}\to \mathbb{R}^2$ is one-one and harmonic in a neighborhood $U$ of origin ...
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0answers
32 views

Bounded-below multiplication operator on Hardy space

Let $H^2(\Delta^2)$ denotes the Hardy space on the bi-disc $\Delta^2$ and $M_f :H^2(\Delta^2)\rightarrow H^2(\Delta^2)$ be multiplication operator by $f\in H^\infty(\Delta^2)$ defined by ...
1
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1answer
23 views

Power series in $\mathbb{C}^2$.

In complex analysis of one variable, we have the series $\sum_nz^n$ is convergent for $|z|<1$. If i consider the same in $\mathbb{C}^2$, that the series $\sum_{n_1,n_2}z_1^{n_1}z_2^{n_2}$, for ...
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0answers
25 views

Is there a simple way to solve this system of equations?

Is there a simple way to solve easily the following system of equations in the unknowns: $ x_2 , y_1 , y_2 , z_1 , z_2 \in \mathbb{C} $ depending on fixed values $​​a, b$ and $c$ in $ \mathbb{C} $ ...
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0answers
39 views

Multiplication operator is bijective?

Let $H^2(\Delta^2)$ denote the Hardy space on the polydics and $M_f:H^2(\Delta^2)\rightarrow H^2(\Delta^2)$ be a multiplication operator by $f\in H^\infty(\Delta^2)$. Is the operator is bijective? ...
3
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1answer
110 views

Is the ring of entire functions coherent?

Call a commutative ring $R$ coherent if for each $n\in \{1,2,3,\cdots\}$ and each $n$-tuple $(r_1, ..., r_n)$ in $R^n$, the kernel of the map $R^n\owns (s_1, \cdots, s_n) \mapsto r_1 s_1 +\cdots + ...
2
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0answers
55 views

Something like the Weierstrass preparation theorem?

I'm reading Griffiths and Harris, On the Noether-Lefschetz Theorem and Some Remarks on Codimension-two Cycles, Math. Ann. 271, 31-51 (1985). Their proof of the Noether-Lefschetz theorem is based on ...
2
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0answers
64 views

Is the exponential map of GL(n,C) holomorphic?

Let $GL(n, \mathbb{C})$ be the complex general linear (Lie) group consisting of all invertible complex $n\times n$ matrices, and $gl(n,\mathbb{C})\cong C^{n^2}$ be its Lie algebra. The exponential map ...
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1answer
35 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
30 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
97 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 ...
1
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1answer
81 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 ...
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0answers
57 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
62 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 ...
0
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1answer
36 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 ...
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1answer
112 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?