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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33 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
66 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
73 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 ...
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1answer
11 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
31 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, ...
5
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1answer
252 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
31 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
30 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
22 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
34 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
107 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
48 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
57 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 ...
1
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1answer
31 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: $$M=\left \{(z,w) \in \Bbb C^2 \colon \text{Im}\ w=|zw|^2+|z|^8+\frac{15}{7}|z|^2\text{Re}\ z^6 \right \}. \tag 1$$ ...
2
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2answers
95 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
72 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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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
60 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
32 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
100 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?
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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
44 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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1answer
46 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
53 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 ...
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2answers
52 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 ...
2
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0answers
39 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
53 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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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
125 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
78 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
44 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
59 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
85 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 ...
0
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1answer
243 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 ...
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1answer
64 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
52 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
69 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 ...
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2answers
93 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 ...
2
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1answer
67 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
105 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 ...
10
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1answer
220 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 ...
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1answer
90 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 ...
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0answers
22 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 ...
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1answer
186 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 ...