# Questions tagged [several-complex-variables]

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118
questions

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### Is a domain biholomorphic to the unit ball a Runge domain?

Let $\Omega \subset \mathbb C^n$ be a bounded domain which is biholomorphic to the unit ball $B^n=\{|z|<1 \mid z\in \mathbb C^n\}$. Can we show $\Omega$ must be a Runge domain? By definition, $\...

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36 views

### Is the $L^2$ or $L^p$ space of global sections of a holomorphic line bundle with respect to a singular metric a Hilbert space?

Let $(X,\omega)$ be a Kahler manifold and $L$ be a holomorphic line bundle over $X$ with a singular metric $e^{-\phi}$. We refer the definition of singular metric (weight) to Demailly 92' Singular ...

**5**

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131 views

### Non-constant holomorphic map onto a smooth curve

Let $\Gamma$ be a smooth projective curve in $\mathbb{P}^2$ and let $U$ be an open neighborhood of $\Gamma$. Denote by $\Gamma_1,\Gamma_2,\ldots,\Gamma_n$ a finite collection of smooth curves ...

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75 views

### How to regard negative PSH function with neat analytic singularities as a generalization of Green-type function?

I am reading this paper:A SIMPLIFIED PROOF OF OPTIMAL L2-EXTENSION THEOREM AND EXTENSIONS FROM NON-REDUCED SUBVARIETIES by Hosono. https://arxiv.org/pdf/1910.05782.pdf.
The setting is as follows.Let $...

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168 views

### Proper analytic embedding of $\overline{\Bbb C}$ minus a Cantor set into $\Bbb C^2$

I am a PhD student in several complex variables.
I am reading this paper by Orevkov proving that there exists a proper analytic embedding of $\overline{\Bbb C}$ minus a Cantor set into $\Bbb C^2$.
I ...

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votes

**1**answer

96 views

### Decomposition of a real analytic variety

Is the following true? If so, I would be grateful for a reference that contains such a result and its proof.
Let $f:\mathbb{R}^d\rightarrow \mathbb{R}$ be a real analytic function, and $V:=\{\mathbf{...

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26 views

### Example of constant Levi rank pseudoconvex

It is known that near a strongly pseudoconvex point, the Levi rank at any boundary point is a constant, which is equal to $n$, the dimension of the domain.
I am looking for a bounded pseudoconvex ...

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136 views

### Space of holomorphic embeddings of open unit ball in ${\mathbb C}^n$

Let $B$ be the open unit ball in $\mathbb C^n$. Consider the space $\mathcal F$ of holomorphic embeddings of $B$ in $\mathbb C^n$ equipped with the compact-open topology. (A holomorphic embedding of $...

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65 views

### Square root of “simple” holomorphic automorphism

Let $f: \mathbb{C} \to \mathbb{C}$ and $g: \mathbb{C} \to \mathbb{C}$ be holomorphic functions s.t. $f(0) = 0$ and $g(0)=0$. Let $F: \mathbb{C}^2 \to \mathbb{C}^2$ be defined $F(z,w) = (z+f(w), w)$ ...

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35 views

### Constructing certain Global section with prescribed zero locus over Stein manifold

Let $X^n$ be a Stein manifold (complex submanifold in $\mathbb{C}^N$ for some large $N$). Let $D = \{(z,z)\in X\times X: z\in X\}$ be the diagonal in $X\times X$. I'm looking for some holomorphic ...

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195 views

### $(-2)$-curves in complex $3$-folds

Let $X$ be a smooth complex $3$-fold,
and let $C \subset X$ be an embedded smooth rational curve whose
normal bundle $N_{C/X}$ is isomorphic to $\mathscr{O}(-1) \oplus \mathscr{O}(-1)$.
Is it true ...

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95 views

### Clarification of Shabat's proof of Hartogs' lemma

I posted the question on math stackexchange, but my earlier questions there on SCV got no responses, so maybe I'll get some input here.
I have trouble with understanding Shabat's proof of Hartogs' ...

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63 views

### Does there exist a Runge Fatou-Bieberbach in each Fatou-Bieberbach domain?

A Fatou-Bieberbach domain $\Omega \subseteq \mathbb{C}^n$ is a domain that is a proper subset of $\mathbb{C}^n$ and is biholomorphic to $\mathbb{C}^n$. A domain is said to be Runge if for each ...

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57 views

### Bounding injective holomorphic mappings on $\mathbb{C}^n$ in the spirit of Andersen-Lempert

I'm hoping the following is true.
Let $Aut_0^I(\mathbb{C}^n)$ denote the set of holomorphic automorphisms $\phi:\mathbb{C}^n \to \mathbb{C}^n$ s.t. $\phi(0)=0$ and $d \phi(0) = I_n$ where $I_n$ is ...

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80 views

### About maxima of injective holomorphic maps on $\mathbb{C}^n$

I am hoping the following is true. Mention of related ideas/topics are appreciated.
Suppose $F:\mathbb{C}^n \to \mathbb{C}^n$ is a injective holomorphic mapping such that $F(0)=0$ and $dF(0) = I_n$ ...

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136 views

### “Square root” of a holomorphic automorphism

Suppose $F \in Aut(\mathbb{C}^n)$. Does there exist a $G \in Aut(\mathbb{C}^n)$ s.t. $G\circ G = F$?

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118 views

### Milnor Number of real and imaginary parts of holomorphic germs?

By performing some computations using the Singular software, I've noticed the following pattern: if $\mu$ is the Milnor Number of a holomorphic germ $f\in \mathcal{O}_n$ at the origin, then the Milnor ...

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428 views

### Complex manifold with boundary

My question is of local nature.
Let $$f:\mathbb C^n\to\mathbb R$$ be a $C^\infty$ function that vanishes at $0\in \mathbb C^n$, with non-zero derivative.
Then, around $0\in \mathbb C^n$, $$M:=f^{-1}(0)...

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217 views

### Is there a by-hand prove that $\Gamma(\mathbb{C}P^n,E)$ is finite dimensional for a holomorphic vector bundle $E$?

Please let me know whether this question is suitable for Mathoverflow.
Let $E$ be a finite holomorphic vector bundle (or more generally a coherent analytic sheaf) on a compact complex manifold $X$. ...

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**1**answer

108 views

### Modulus bounded by Nevanlinna characteristic in several variables

Let $f:\mathbb{C}^n\to\mathbb{C}$ be an entire holomorphic function of $n$ complex variables. Then its Nevannlinna characteristic equals
$$
m_f(r)=\int_{\partial B(r)}\log^+|f(z)|d\eta(z),\quad\forall ...

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80 views

### Automatic plurisubharmonicity for a non-negative function

I feel confused about a point in this very short paper. On the top of page 3, it is claimed that:
If $S$ is a totally real submanifold in a compact almost complex manifold $(X,J)$, then any function ...

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59 views

### Equations needed to define a normal complex surface singularity

This questions is highly related with this other question of mine: Irreducible surface singularity that is not a local set-theoretical complete intersection I just thought that a different look at the ...

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188 views

### Irreducible surface singularity that is not a local set-theoretical complete intersection

I have been looking for a criterion for the germ of an irreducible complex surface singularity $(X,x)$ to be a set-theoretical complete intersection.
A germ $(X,x)$ of an isolated complex singularity ...

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140 views

### Computing the convex hull of a region of $\mathbb{C}^2$

Consider a function $f(z, w)$ of two complex variables. The function is symmetric with respect to $z$ and $w$. When $\Re(z)>0$ and $\Re(w)>0$, the function is analytic in its two variables. When ...

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36 views

### On Remmerts reduction

Let $(X,0)$ be a normal surface singularity. An let $\pi: \tilde{X} \to X$ be the minimal resolution. Now, we can apply a result of Oliveira (exploiting previous work by Laufer) and obtain a 1-...

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97 views

### Interpretation of deformation of complex structure

Let $X$ be a smooth complex analytic space and let $D$ be the unit disk in $\mathbb{C}$. Let $\omega:Y \to D$ be a deformation of complex structures of $X$ in the sense that (1) $\omega^{-1}(0) \simeq ...

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51 views

### Analytic continuation of a Dirichlet series with several complex variables

For $w_1,w_2,z_1,z_2\in\mathbb{C}$ with $\operatorname{Re}(w_1)>0$ and $\operatorname{Re}(w_2)>0$, define
\begin{equation*}
U(w_1,w_2;z_1,z_2):=\prod_{p}\left(1-\frac{e^{z_1}}{p^{1+w_1}}-\frac{e^...

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388 views

### Does Bergman metric induce the standard topology?

I am a physics student and am interested in the study of invariant metrics. I have searched several textbooks, including those fat books of Krantz, but the following concern seems not to be mentioned ...

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198 views

### Bezout theorem for germs of holomorphic functions

UPDATE.
It was pointed out by @Dmitri that two smooth curves given by $f=y$ and $g=y+x^k$ in $\mathbb C^2$ provide a simple counterexample.
Let $f_1, \ldots, f_p, g_1, \ldots, g_q$ be germs of ...

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294 views

### Exponential Sequence of Sheaves

Let $(X, \mathcal{O}_X)$ be a complex analytic space in the sense of Grauert, i.e., a $\mathbb{C}$-analytic ringed space which is locally isomorphic to a local model. We may assume that $X$ is a ...

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486 views

### a question on Hodge and Atiyah's paper “integrals of the second kind on an algebraic variety”

I have a question on Hodge and Atiyah's paper "Integrals of the second kind on an algebraic variety". It is about the exact sequence below formula (14) and above formula (15) on page 71:
$$H_{2n-q}(S)...

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75 views

### A question on f.g. ideals of $\textrm{Hol}(\mathbb{C}^2,\mathbb{C})$

Suppose that $I$ and $J$ are finitely generated ideals of the ring $\textrm{Hol}(\mathbb{C}^2,\mathbb{C})$ of all entire functions in two complex variables.
Then is $I\cap J$ finitely generated too?

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799 views

### Dual of the space of all bounded holomorphic functions

Let $\mathbb{B}$ be the open unit ball in $\mathbb{C}^n, n\geq 1$ and let $H^\infty (\mathbb{B})$ be the space of all bounded holomorphic functions on $\mathbb{B}$. It is well known that $H^\infty (\...

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645 views

### Why only $\bar\partial$ but not $\partial$ in Dolbeault cohomology

While I learn about $\partial$ and $\bar{\partial}$ operators, I had some questions about the reason why people prefer $\bar\partial$ over $\partial$. Specifically,
When defining Dolbeault ...

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68 views

### Notation and geometry facts in a paper on the Diederich-Fornæss index

I am reading this article by Bingyuan Liu on the Diederich-Fornæss index.
I am having some problems with both the notation and the geometrical side.
1)I don't know what kind of objects $N,L$ are ...

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492 views

### zeros of holomorphic function in n variables

Conjecture: Let $f:{\mathbb C}^n\rightarrow{\mathbb C}$ be an entire function in $n$ complex variables. Assume
that for every $x\in{\mathbb R}^n$ there exists a $y_x\in{\mathbb R}^n$ such that
$f(x+...

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**1**answer

534 views

### A question about openness theorem

The openness theorem says that:
If $\varphi$ be a negative plurisubharmonic function
in the unit ball $B(0,1)$ in $\mathbb{C}^{n}$ satisfying
$$
\intop_{B(0,1)}e^{-\varphi}<\infty,
$$
then there ...

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406 views

### Let $h(z) = g(f(z))$. If $f$ and $h$ are non-constant holomorphic function on domains in $\mathbb C^n$, then is $g$ holomorphic?

Suppose there exist functions $f,g,h$ such that $h(z) = g(f(z))$. If $f$ and $h$ are non-constant holomorphic function on domains in $\mathbb C^n,\, n>1$ and $g$ is continuous, then is $g$ ...

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323 views

### Equivalence of the term “Divisor”

Throughout my university education, I have studied some theory of Riemann surfaces, focusing particularly on Miranda's Algebraic curves and Riemann surfaces. My current studies however are in the ...

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199 views

### Can any plurisubharmonic function be represented as a sum of non-positive plurisubharmonic functions?

Let $f$ be a plurisubharmonic function, $f < 0$ in $\Omega$. Can we always find two negative plurisubharmonic functions $u$ and $v$ and real numbers $a,b\in(-1,1)$ such that
$$-f=(-u)^{a}+(-v)^{b}$...

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130 views

### What do we necessarily need for the image of a domain of holomorphy to be a domain of holomorphy

I posted this on Math.Stack.Exchange with no luck, so I thought it would be perhaps better suited for this site.
We recall that a domain of holomorphy is a domain in $\mathbb{C}^n$ that is ...

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298 views

### Holomorphic Sard's theorem 2

My previous question on this topic had a negative answer, but Tom Goodwillie in the comments suggested a statement, which may be true, and even a strategy of how to prove it. I haven't been able to ...

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209 views

### Practically Calculating the Domain of a Power series for function of several complex variables

For simplicity, let us consider a function $f$ holomorphic on a domain $D \subseteq \mathbb{C}^2$. We may therefore write $f$ as a sum of power series $$f(z) = \sum_{\nu_1 \nu_2 =0}^{\infty} c_{\nu_1 \...

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151 views

### Real solution of a complex equation with complex solution

Assume that $(M, [\lambda, \mu])$ defines an embeddable 3 dimensional CR structure where $\lambda$ is a real form and $\mu$ is a complex 1-form.
Because $M$ is embeddable, $\mu=dz$ for some ...

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296 views

### Understanding Remmert-Stein extension theorem

I'm trying to study the Remmert-Stein theorem in analytic geometry. This is an important result which can be used to prove the Proper Mapping theorem.
A preliminary result is stated in various books (...

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319 views

### English reference for Fischer-Grauert theorem and its generalization by Schuster

From this MSE question and its answer, and from this MO question I have learned of the following remarkable theorem of Wolfgang Fischer and Hans Grauert.
Theorem. A proper holomorphic submersion with ...

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**1**answer

351 views

### Holomorphic Sard's theorem?

I have originally posted this question on math.SE, but it received little attention, so I repost it here.
Let $U\subset \mathbb{C}^{n}$ and $V\subset \mathbb{C}^{m}$ be open and connected. Let $\Phi:...

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71 views

### Discrete set of critical points of a holomorphic map

I have originally posted this question on math.SE, but it received no attention, so I repost it here.
Let $U$ be an open domain in $\mathbb{C}^{n}$. Let $m\ge n$ and let $F:U\to C^{m}$ be a ...

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173 views

### The approximation property for some spaces of holomorphic functions

I am reading a circle of papers which use arguments based on Fredholm determinants of nuclear operators to compute numerical quantities associated to real-analytic and holomorphic dynamical systems. ...

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185 views

### Kähler metric on compact complex manifolds with simple normal crossing divisor

Let $X$ be a reduced compact complex analytic space of $\dim_{\mathbb{C}}X\ge2$; by [KJ] definition 3.29, remark 3.44 and theorem 3.45, it admits a strong resolution $R(X)$ which is smooth, $E=\pi_X^{-...