# Questions tagged [several-complex-variables]

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130
questions

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115 views

### Oka-Grauert principle, up to the boundary

Let $Z\subset \mathbb{C}^n$ a domain of holomorphy with smooth boundary $\partial Z$ and closure $\bar Z$. There is a natural notion of holomorphic vector bundle over $\bar Z$, given in terms of ...

**3**

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124 views

### When holomorphic convexity implies polynomial convexity

For what follows I refer to Forstneric's Book "Stein Manifolds and Holomorphic mappings", Theorem 4.14.6 p. 168 (second edition).
I need some clarifications.
It starts talking about a ...

**4**

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65 views

### Restricted Perron-Bremermann envelopes

Consider an upper semicontinuous function $\phi: \Omega \to (-\infty, \infty]$, in the sense that $\phi = \phi^*$, where $\phi^*$ denotes the upper semicontinuous regularization
$$
\phi^*(z) = \...

**3**

votes

**1**answer

77 views

### Use of Invariant metric/distances to classify domains in $\mathbb{C}^n$

I am a graduate student in mathematics, who works usually in operator theory. Lately I had to read about about the Lempert’s theorem(a theorem regarding when some pseudometric/distances coincide) and ...

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137 views

### Is any proper subvariety contained in hypersurface

Suppose $A$ is a subvariety of an irreducible complex space(analytic variety) $X$. Is there an analytic hypersurface of $X$ containing $A$?

**2**

votes

**1**answer

68 views

### Fatou-Bieberbach domain in $\Bbb C^*\times\Bbb C^*$

According to Forstneric's book, pag 123, it is a long standing problem whether it exists a Fatou-Bieberbach domain in $\Bbb C^*\times\Bbb C^*$.
The idea is to search for an $F:\Bbb C^2\to\Bbb C^2$ ...

**3**

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39 views

### On strong $\mathbb{C}$-linear convexity

I would like to understand strong $\mathbb{C}$-linear convexity, as defined in the paper "Cauchy-type integrals in several complex variables" (a domain $D$ with $C^1$ boundary is strongly $\...

**2**

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91 views

### How to get the jet extension over the whole of $X$ in Popovici's article?

Recently, I am reading D. Popovici's article $L^2$ extension for jets of holomorphic sections of a Hermitian line bundle, https://arxiv.org/pdf/math/0409170.pdf where some parts possibly confuse me.
I ...

**3**

votes

**2**answers

259 views

### On a variation of Hartogs' separate analyticity theorem

Let $f(z_1,z_2,\ldots,z_n)$ be a function on $\mathbf{C}^n$ such that for all $i$, the restriction
$$
[z_i\mapsto f(z_1,z_2,\ldots,z_n)]
$$
is a "rational function".
(added: to be precise ...

**0**

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73 views

### About the definition of lineal convexity

I have been trying to understand the definition of lineal convexity. I am reading the article Duality of functions defined in lineally convex sets by Christer O. Kiselman. For a set $A\subset \mathbb{...

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48 views

### Extension to all dimensions of complex line integral

Let $\Gamma$ be a smooth curve in $\mathbb{C}^d$. Since $\mathbb{C}^d$ can be seen as $\mathbb{R}^{2d}$, one can define the line integral of functions $f:\Gamma\to \mathbb{C}$ using for instance ...

**4**

votes

**2**answers

119 views

### Why do we study biholomorphically invariant pseudodistances/metrics

It is said that pseudodistances/metrics which are invariant under biholomorphic maps are used to determine whether domains in $\mathbb{C}^n$ are biholomorphically equivalent or not.
Suppose $\Omega_1$ ...

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**0**answers

78 views

### A complex analytic interpretation of multiplicity on the special fiber of a flat family

Let $X$ be a variety over $\mathbb C$ and $\pi: X\to \Delta$ be a flat morphism over the unit disk $\Delta=\{z:|z|<1\}$. Let $Z$ be a component of $X_0=\pi^{-1}(0)$. The multiplicity of $Z$ is ...

**4**

votes

**1**answer

188 views

### Existence of plurisubharmonic functions on complex manifolds

Let $X$ be a noncompact complex manifold which contains no positive dimensional compact analytic sets.
Conjecture: There must be strictly plurisubharmonic functions on $X$ .
Is it true?

**0**

votes

**1**answer

142 views

### injective holomorphic mapping between unit disk and unit polydisk

In $\mathbb{C}^n,\ n\geq 2$, there is no bijection between unit disk $B^n(0,1)$ and unit polydisk $P^n(0,1)$. But if we wish to find injective holomorphic mapping from unit disk to polydisk(whose ...

**3**

votes

**1**answer

104 views

### 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, $\...

**5**

votes

**1**answer

154 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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83 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 $...

**3**

votes

**1**answer

181 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 ...

**2**

votes

**1**answer

110 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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28 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 ...

**5**

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**1**answer

145 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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39 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 ...

**6**

votes

**1**answer

198 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 ...

**3**

votes

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114 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' ...

**3**

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70 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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59 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 ...

**0**

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**1**answer

83 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 $dF(0) = I_n$ ...

**4**

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166 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. $G\circ G = F$?

**3**

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127 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 ...

**10**

votes

**3**answers

685 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)...

**2**

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226 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 $X$. ...

**2**

votes

**1**answer

117 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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85 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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63 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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210 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 ...

**1**

vote

**1**answer

145 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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38 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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109 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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55 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^...

**2**

votes

**2**answers

418 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 ...

**6**

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209 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 ...

**0**

votes

**1**answer

380 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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496 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?

**11**

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**1**answer

861 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 (\...

**13**

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**2**answers

896 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 ...

**2**

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**0**answers

71 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 ...

**4**

votes

**2**answers

662 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+...

**1**

vote

**1**answer

620 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 ...