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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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) ...
5
votes
0answers
71 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 ...
4
votes
0answers
66 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 ...
4
votes
0answers
210 views

What's the difference between analytic singularity and algebraic singularity?

Let $f$ be a holomorphic function defined in $U\in \mathbb{C}^n$, and $f=0$ gives an analytic variety. Suppose $f$ is irreducible as a germ of holomorphic function in $\mathcal{A}_z$, here $z$ is a ...
2
votes
0answers
20 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 ...
2
votes
0answers
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 ...
2
votes
0answers
25 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
votes
0answers
81 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 ...
2
votes
0answers
50 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
votes
0answers
59 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 ...
2
votes
0answers
45 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
votes
0answers
40 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 ...
2
votes
0answers
92 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 ...
2
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0answers
72 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: ...
2
votes
0answers
130 views

Uniform Convergence of Bergman Kernel

In my complex analysis class we've shown that if $\Omega \subset \Bbb C^n$ is compact and if $H^2(\Omega)$ is the set of square integrable holomorphic functions on $\Omega$, then the Bergman kernel is ...
1
vote
0answers
33 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 ...
1
vote
0answers
88 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 ...
1
vote
0answers
34 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 ${\mathbb{R}}^n$ ...
1
vote
0answers
35 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? ...
1
vote
0answers
128 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 ...
1
vote
0answers
45 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 : ...
1
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0answers
66 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 ...
1
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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 ...
1
vote
0answers
71 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
vote
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 ...
1
vote
0answers
21 views

Analytic semiconjugacy

Consider the following commutative diagram (semi-conjugacy): $$ X\;\; \stackrel{f}{\longrightarrow} \;\;X $$ $${\pi}\downarrow \;\;\;\;\;\; \;\;\;\;\downarrow {\pi}$$ ...
1
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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 \\ ...
1
vote
0answers
50 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 ...
1
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0answers
30 views

Rational function on polydisc

Who has idea to prove this: Let $\mathbb{U}^n$ be a polydisc, $f\in \mathcal{O}(\mathbb{U}^n)\cap C(\bar{\mathbb{U}}^n)$, if $|f|=const$ on the skeleton of $\mathbb{U}^n$, then $f$ must be a rational ...
1
vote
0answers
145 views

The Wirtinger theorem proof

Let $M$ be a complex hermitian manifold with symplectic form $\Omega$ and $N \subseteq M$ be its smooth $2n$-dimensional (real dimensions, $2n \geq 2$) compact submanifold. I want to show the ...
1
vote
0answers
65 views

A functional power series equation

I am interested in solving the following functional equation: $$(1-w-zw)F(z,w)+zw^nF(z,w^2)=z-zw\,.$$ Here $n\geq2$ is a fixed integer and $F(z,w)$ is a power series in two variables with complex ...
1
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0answers
33 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} ...
1
vote
0answers
41 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($\Delta$) ...
0
votes
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
65 views

Hurwitz's theorem for a system of functions (corrected version)

This is the corrected version. First, let me define a notation of $H(G_1\times G_2 \times \ldots \times G_m)$. We say that $f\in H(G_1\times G_2 \times \ldots \times G_m)$ if $$f:G_1\times G_2 \times ...
0
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0answers
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^*\ ...
0
votes
0answers
29 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 ...
0
votes
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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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 ...
0
votes
0answers
42 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 ...
0
votes
0answers
25 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?
0
votes
0answers
78 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
votes
0answers
33 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, ...
0
votes
0answers
31 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 ...
0
votes
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} $ ...
0
votes
0answers
27 views

Show that $\delta =1$

I have a problem: Let $M \subset \Bbb C^2$ be a real analytic hypersurface: $$M=\left \{(z,w) \in \Bbb C^2 \colon \text{Im}\ w=|zw|^2+|z|^8+\frac{15}{7}|z|^2\text{Re}\ z^6 \right \}. \tag 1$$ ...
0
votes
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 ...
0
votes
0answers
33 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 ...