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
33 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 ...
0
votes
1answer
49 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 ...
5
votes
0answers
60 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
62 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
181 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
32 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
32 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
77 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
votes
0answers
71 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
115 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
41 views

Using an induction on $q$.

I have a problem (It's a Proposition in Baouendi'book) 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 $f$ ...
1
vote
0answers
89 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
37 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
vote
0answers
35 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
vote
0answers
44 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
44 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
17 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
18 views

Analytic semiconjugacy

Consider the following commutative diagram (semi-conjugacy): $$ X\;\; \stackrel{f}{\longrightarrow} \;\;X $$ $${\pi}\downarrow \;\;\;\;\;\; \;\;\;\;\downarrow {\pi}$$ ...
1
vote
0answers
33 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
39 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
vote
0answers
22 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
87 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
52 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
vote
0answers
31 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
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($\Delta$) ...
0
votes
0answers
13 views

Show that $\exists \beta(z,\bar{z},s)$ such that $\dfrac{1}{2i} [\Lambda, \bar{\Lambda} ]=\beta(z,\bar{z},s)\dfrac{\partial }{\partial s}$

I have a problem: We denote by $[X,\ Y]$ the commutator of $X$ and $Y$ defined by $$[X,\ Y]f(p)=X(Yf)(p)-Y(Xf)(p), \tag{*}$$ for any smooth function $f$ defined on a hypersurface $M$. ...
0
votes
0answers
24 views

Show that $\dim_\Bbb C \mathcal V_p=N-\text{rank}_\Bbb C\left ( \dfrac{\partial \rho}{\partial \overline{Z_j}}(p,\overline{p}) \right )$

I'm learning about Real Submanifolds in $\Bbb C^N$ (from Baouendi's book - page 7): For $p \in \Bbb C^N$ it is customary to denote by $T_{p}^{1,0}\Bbb C^N$ the space of holomorphic tangent vectors ...
0
votes
0answers
21 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
votes
0answers
25 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 ...
0
votes
0answers
34 views

Density of rational functions in open Stein sets

Lately I have been wondering on this problem: if $U \subset \mathbb C^n$ is an open Stein and I denote by $\mathcal R(U)$ the set of rational functions on $\mathbb C^n$ whose restriction to $U$ is ...
0
votes
0answers
16 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 ...
0
votes
0answers
65 views

Complex Quasi linear Differential Equation

So, I was reading some papers when I found a several complex variable quasi-linear differential equation. Let $f(z_1,z_2,z_3,z_4)$ be a complex holomorphic function, with $z_i \in ...
0
votes
0answers
44 views

holomorphic Lie group automorphisms of the complex Heisenberg group preserving the lattice

I don't understand Lie group,but I need an elementary result about it,so I ask for help for a simple question. Take a complex Heisenberg group $G$ $$ \left( \begin{array}{ccc} 1 & z_1 & ...
0
votes
0answers
17 views

inequality on integrals (2)

Is it possible find a constant $C>0$ such that $\displaystyle\sum_{i,j=1}^n\int_\Omega |u_{x_i}||v_{x_j}|dx\leq C \left(\sum_{i=1}^n\int_\Omega ...
0
votes
0answers
27 views

Prove that $\int_{{B(0,\epsilon)}\setminus \{z_1=0\}}\det\left(\text{Hessian}_u(z)\right)\mathrm{d}V=\infty$

I have a problem: For $u(z_1,z_2)=\left (-\log\left | z_1 \right | \right )^\alpha\cdot \left ( \left | z_2 \right |^2-1 \right )$, where $\alpha \in \left (0,1 \right )$. Prove that if $\alpha ...