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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60 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
80 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
49 views

entension of a holomorphic map on projective algebraic manifolds

I'm reading a paper.It says if $M$ is a projective algebraic variety and $f:N-S\rightarrow M$ is a holomorphic map,where $N$ is a connected complex manifold and $S\subset N$ is a proper analytic ...
4
votes
0answers
67 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
256 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 ...
3
votes
0answers
79 views

vanishing theorem in algebraic geometry

This is a general question: As we know there are lot of vanishing theorem like Fujita vanishing, kodaira Nakano vanishing, vanishing for big nef line bundle, Kollár vanishing, etc. Those just for ...
3
votes
0answers
43 views

When a current is actually a holomorphic form?

If a current $f$ of bidegree $(p,0)$ (acting on forms of bidegree $(n-p,n)$) satisfies $\bar{d}f=0$, is it true that $f$ is a holomorphic differential form? In general, do we have any standard ...
2
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0answers
24 views

$(P,H)$ Euclidean Hartogs figure

Let $(P,H)$ be a Euclidean Hartogs figure in $\mathbb{C}^n$ and $f:H\to \mathbb{C}^n$ a holomorphic injective map, then we know that $f$ extend holomorphically to polidisc $P$ (i.e. there is a ...
2
votes
0answers
28 views

change of complex variables

Suppose that $z_1 , z_2,\cdots ,z_k,\cdots ,z_d$ are complex variable numbers. Locally, and suppose $f_1, f_2 \cdots, f_k $ are $k$ holomorphic functions on $z_1, z_2 \cdots,z_d$. At ...
2
votes
0answers
72 views

Are there non-holomorphic or non-analytic polynomials of several complex variables?

Having no prior exposure to several complex variables, I am trying to read some papers involving this subject. I came upon the terms analytic polynomial and holomorphic polynomial. Do they simply ...
2
votes
0answers
44 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
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0answers
23 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
59 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
179 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
74 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
79 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
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0answers
55 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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0answers
42 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
50 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 : ...
2
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0answers
213 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 ...
2
votes
0answers
115 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
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0answers
149 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
24 views

Uniqueness set for analytic functions of several variables

Is there a simple (and not so restrictive) condition for a set to be an uniqueness set for the space of holomorphic functions defined on some open subset $U \subseteq \mathbb{C}^n$? By uniqueness set ...
1
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0answers
36 views

Every convex domain in $\mathbb{C^n}$ is a weak domain of Holomorphy

I am trying to prove that every convex domain in $\mathbb{C^n}$ is a weak domain of Holomorphy. Let $G$ be a convex domain. I pick a point $p \in \partial{G}$. Then By Hahn-Banach Separation ...
1
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0answers
37 views

Every Holomorphic function on a Hartog's figure can be extended holomorphically to the while of $P^n$

Let $f$ be a holomorphic function on the Euclidean Hartog's figure that is $$H=\{(z,w)\in P^2 : 1 \gt|z| \gt q_1 \text{or} |w| \lt q_2 \}$$ where $0 \lt q_i \lt 1$. I need to show that it has a ...
1
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0answers
27 views

Power Series in Poly Discs (Assuming Cauchy Integral Formula)

Suppose that $f \in H(\Omega)$ and suppose that $\Omega$ contains the closure of some polydiscs $D(p;r)$. By polydiscs I means $D(p;r)=\{z=(z_1,z_2,..,z_n) \in \mathbb{C^n}| |z_i-p_i| \lt r_i \}$ for ...
1
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0answers
67 views

Multivariate Residue Theorem

Let $G(s,t)$ be a complex valued function in two variables that converges absolutely for $Re(s), Re(t)>1$. Suppose we can analytically continue $G$ in such a way that $$G(s,t) = f(s)g(t)H(s,t)$$ ...
1
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0answers
17 views

$U:$ domain of holomorphy, $d_{\infty}(K,\partial U)=d_{\infty}(\hat{K}_U,\partial U)$.

Let $U\subset\mathbb{C}^n$ be a domain of holomorphy, then we know that $$d_{\infty}(K,\partial U)=d_{\infty}(\hat{K}_U,\partial U)$$ for each compact subset $K\subset U$, also that ...
1
vote
0answers
85 views

The Hessian quadratic form of a real function can't be complex

Let $\Omega\subseteq\Bbb C^n$ open, $z_0\in\Omega$, $r:\Omega\to\Bbb R$ twice real differentiable. We know $\Bbb C^n\simeq\Bbb R^{2n}$ is an isomorphism of vect.sp. So we think $\Bbb C^{n}$ as an ...
1
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0answers
22 views

holomorphic range of a zero variety

Let D be the unit disk, and $d\ge 2$. $F=(f_1,...f_d)$ is a holomorphic map over the closure of $D^d$ to $C^d$. What information can we get about the image of $\{0\}$x $D^{d-1}$ under $F$? Hope some ...
1
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0answers
34 views

Questions on Levi pseudoconvex domain

Here are some of the exercise questions which I am stuck: Question 1: Give an example of a real hypersurface $M\subset\mathbb{C}^n$ such that $0\in M$, such that $M$ has a polynomial defining ...
1
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0answers
72 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
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0answers
91 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
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0answers
55 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
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0answers
31 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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0answers
152 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
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0answers
48 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
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0answers
26 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
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0answers
22 views

Analytic semiconjugacy

Consider the following commutative diagram (semi-conjugacy): $$ X\;\; \stackrel{f}{\longrightarrow} \;\;X $$ $${\pi}\downarrow \;\;\;\;\;\; \;\;\;\;\downarrow {\pi}$$ ...
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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 \\ ...
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0answers
57 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 ...
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0answers
32 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
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0answers
73 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
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0answers
43 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
10 views

Any convex Reinhardt domain is logarithmically convex

I have the following question in Shabat p.59: Prove that any convex Reinhardt domain is logarithmically convex. I think I have a good idea about how to show this, but need to be clear on the ...
0
votes
0answers
21 views

global nullstellensatz

Let J be an ideal of analytic functions in several variables on the open unit ball. If Z(J), the analytic set of J (common zero set in the ball) is a compact subset of the ball does a global ...
0
votes
0answers
67 views

Zero set of an analytic function

If $f$ is a holomorphic function on $C^n$ which vanishes on $R^n$, it is easy to see that it vanishes everywhere. But if the zero set of $f$ is contained in $R^n$, can I deduce that $f$ vanishes ...
0
votes
0answers
37 views

To show $\bar{D}u=f$ where $u(z)=\frac{1}{2\pi i} \int_{\Omega} \frac{f(\lambda)}{\lambda-z} d\lambda \wedge d\bar{\lambda}$,

Let $\Omega \subset \mathbb{C}$ be a bounded open set. Suppose $f \in C^{1 }(\Omega)$, $f$ is bounded and $$u(z)=\frac{1}{2\pi i} \int_{\Omega} \frac{f(\lambda)}{\lambda-z} d\lambda \wedge ...