# Questions tagged [several-complex-variables]

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### 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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### 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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### 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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### $(-2)$-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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### 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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### 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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### 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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### 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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### “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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 (...