Questions tagged [several-complex-variables]

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Show that this holomorphic function can be extended to $D_{2}((0,0) ;(2,2))$

consider a domain in $C^{2}$:$\Omega=D_{2}((0,0) ;(1,2)) \cup\left\{(z, w) \in \mathbb{C}^{2}:|z|<2 \text { and } 1<|w|<2\right\}$ and $f \in \operatorname{Hol}(\Omega)$, I want to show that ...
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Regarding equality of two infimums

Let $M$ be a domain in $\mathbb{C}^n$. Let $\mathbb{D}$ be the open unit disc in $\mathbb{C}$. Let $H(\mathbb{D},M)$ denote the space of all holomorphic functions from $\mathbb{D}$ to $M$. Let $m_1, ...
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A question on the proof of Bedford-Taylor theorem in Demailly's book

I am trying to understand a proof of the Bedford-Taylor theorem on the weak convergence of Monge-Ampere operators of decreasing sequences of plurisubharmonic functions. I am reading a proof in the ...
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Starlike sets in $\mathbb{C}^n$

Let $S$ be a bounded domain in $\mathbb{C}^n$. $S$ is called starlike about the point $x_0\in S$ if for every point of $S$, the segment of the straight line from the point to $x_0$ lies in $S$. If $S$ ...
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Relation between polynomial convexity and Runge-Stein neighborhood basis

I am searching for some reference about the relation between polynomial convexity and Runge-Stein neighborhood basis for a compact set $K$ inside $\Bbb C^n$. I read on this paper, Remark 3.1, that ...
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Number of roots of a Vandermonde like complex determinant

I am originally interested in the determinant $$ \left|\begin{array}{cccc}\exp(i\lambda_1\cdot x_1) & \exp(i\lambda_2\cdot x_1) & ... & \exp(i\lambda_n\cdot x_1) \\\exp(i\lambda_1\cdot x_2)...
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2 answers
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$\frac{\partial f}{\partial \overline{z}}=0$ in distributional sense implies $f$ is holomorphic

Let $f=(u,v)\in \mathscr{D}'(U,\mathbb{C})$ be a distribution, where $U\subset\mathbb{C}=\mathbb{R}^2$ is an open set and $u$ and $v$ are the projection of $f$ onto the real and imaginary axis (ie $\...
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Carathéodory metric on product domain

Let $G\subseteq \mathbb{C}^n$ be a bounded domain. Consider the Carathéodory metric $C_G$ on $G$. If $G=\mathbb{D}^n$ (unit polydisc), then $C_G(a,z)=\max_{1\leq j\leq n}p(a_j,z_j)$, where $p$ denotes ...
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Do we have the Oka coherence theorem for finite group actions?

We first consider the sheaf of holomorphic functions $\mathcal{O}(\mathbb{C}^n)$ on $\mathbb{C}^n$. By Oka coherence theorem, $\mathcal{O}(\mathbb{C}^n)$ is coherent over itself. Now we consider a ...
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What domains are there other than unit ball and polydisc, on which Caratheodory metric is known?

What are few (bounded)domains in $\mathbb{C}^n$ on which the explicit expression of Caratheodory metric is known. For example, unit ball and unit polydisc.
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Milnor fibration and Runge pair

Let $f:\mathbb{A}^3\to \mathbb{A}^1$ be a polynomial map. Let $0\in \mathbb{A}^1$ be critical value. If $c \in \mathbb{A}^1$ is very close to zero (c is a regular value), then for Milnor fibration we ...
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Catlin multitype example

I asked this over question over stackexchange but did not get any answer. This is a remark in a paper by Jiye Yu, "Multitypes of Convex Domains", p. 838 If (a smoothly bounded domain) $\...
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Analogue of Grauert's upper semi-continuity for Bott–Chern cohomology

In Coherent analytic sheaves, one has the following theorem due to Grauert: Let $f: X \rightarrow Y$ be a holomorphic family of compact complex manifolds with connected complex manifolds $X, Y$ and $V$...
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Holomorphic connectedness in several complex variables

Let $\Omega$ be domain in $\mathbb{C}^n$. Suppose we have taken two distinct points from $\Omega$. Does there exist a domain $U$ in $\mathbb{C}$ such that there is a holomorphic function from $U$ to $\...
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Holomorphic mapping on a manifold approximating a constant map

Let $X,Y$ be complex manifold, $Y$ Stein. It sounds quite reasonable to formulate the following claim: given $y_0\in Y$, for every $\epsilon>0$ and $M\subset X$ compact, there exists an holomorphic ...
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Geometric meaning of Catlin multi types

Can someone working in the area of several complex variables explain the geometric idea behind the Catlin multitype. I have seen the technical definition, but unable to grasp the idea behind this. ...
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On extension of Monge-Ampere masses

It is known (evidently due to Bedford and Taylor) that if $u$ and $v$ are bounded plurisub-harmonic functions on an open domain in $\mathbb{C}^n$, then $$\mathbb{1}_{\{u>v\}}(dd^cu)^n = \mathbb{1}_{...
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2 votes
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Regular exposable points on the boundary of compacts in Stein manifolds

Given a Stein manifold $Y$, there exists $\rho$, a $\mathscr C^2$-smooth strictly plurisubharmonic exhausting function for $Y$, such that the set of critical points $C=\{z\in Y\;:\;d\rho(z)=0\}$ is ...
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Precise definition of locally closed complex curve

In Stein Manifold and Holomorphic Mappings, by Forstnerič, I refer to Definition 8.9.9: An exposed point is a point belonging to a certain subset $\Sigma$ of $\Bbb C^2$, enjoying certain properties. ...
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Are open subsets of a $\sigma$-compact LCH space $\mathcal{K}$-analytic?

I'm reading Guedj and Zeriahi's Degenerate Complex Monge-Ampère Equations Chapter 4 which talks about capacities. Specifically Corollary 4.13 claims that when $X$ is a locally compact Hausdorff $\...
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43 votes
3 answers
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Putnam 2020 inequality for complex numbers in the unit circle

The following simple-looking inequality for complex numbers in the unit disk generalizes Problem B5 on the Putnam contest 2020: Theorem 1. Let $z_1, z_2, \ldots, z_n$ be $n$ complex numbers such that ...
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3 votes
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Perturbations of Neumann $\bar \partial$-Laplacian

Let $\Omega \subset \mathbb{C}^n$ be a smooth, bounded, strongly pseudoconvex domain. Let $\square =\bar \partial \bar \partial^*+\bar \partial^* \bar \partial $ and suppose that $X$ is a first order ...
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3 votes
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How to get a Stein space which has homotopy type of suspension of another Stein space

Let $V^n$ be a Stein space(or Stein manifold) in $\mathbb{C}^N$. I want to construct a Stein space(or Stein manifold) $W^{n+1}$ such that $H_i(V;\mathbb{Z})=H_{i+1}(W; \mathbb{Z}).$ If we take the ...
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Holomorphic dynamical systems defined on a contractible bounded open subset of $\Bbb{C}^n$

Let $U$ be a contractible bounded open subset of $\Bbb{C}$. There is a standard classification of possible dynamical behaviors of holomorphic maps $f:U\rightarrow U$: Attracting Case: There is an ...
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A question about Lelong number

If $f$ is plurisubharmonic (not identically $-\infty$) on a neighbourhood of $0$ then the Lelong number of $f$ at $0$ is defined by $$\nu_{f}(0) = \liminf_{|z|\rightarrow 0}\dfrac{f(z)}{\log|z|}.$$ My ...
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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 ...
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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 ...
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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) = \...
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1 answer
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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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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 $A$?
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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 $F:\Bbb C^2\to\Bbb C^2$ ...
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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 $\...
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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 ...
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2 answers
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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 ...
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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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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 ...
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4 votes
2 answers
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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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2 votes
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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 ...
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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?
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1 answer
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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 ...
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3 votes
1 answer
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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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5 votes
1 answer
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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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2 votes
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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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3 votes
1 answer
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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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  • 751
2 votes
1 answer
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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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1 vote
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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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4 votes
1 answer
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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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1 vote
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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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6 votes
1 answer
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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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3 votes
0 answers
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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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