Questions tagged [several-complex-variables]

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de Rham cohomology group and Dolbeault cohomology group on compact complex analytic spaces

My question is: On a compact complex analytic space,since Hodge Theorem becomes invalid,is it true that the de Rham cohomology group $H^p_{DR}(M)$ and the Dolbeault cohomology group $H^{p,q}_{\bar{\...
whitacre's user avatar
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Snazzy applications of Several Complex Variables techniques

I am starting to dive into a study of Several Complex Variables. I would like to have a few guiding examples of "big payoffs". These should be very natural sounding theorems which depend on a lot of ...
6 votes
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234 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 ...
Dmitri Zaitsev's user avatar
5 votes
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166 views

Example of an $H^1$ function on the bidisk that is not a product of two $H^2$ functions

Fix $n \in \mathbb{N}$ and consider the Hardy space $H^1 := H^1(\mathbb{D}^n)$, consisting of holomorphic functions $f$ on the unit polydisk $\mathbb{D}^n=\mathbb{D}\times\dots\times\mathbb{D}$ such ...
EG2023's user avatar
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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 $\...
Carlos Esparza's user avatar
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537 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)...
user42804's user avatar
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Density of rational functions in open Stein

I repost here, after I tried here. Lately I have been wondering on this problem: if $U \subset \mathbb C^n$ is an open Stein and I denote by $\mathcal R(U)$ the set of rational functions on $\mathbb ...
Mauro Porta's user avatar
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Is there a dense set of Lipschitz functions in $H^\infty(U)$, each of which maps $(1,0,\ldots,0)$ to 1, where $U$ is the unit ball in $\mathbb{C}^N$?

Let $U$ be the open unit ball in $\mathbb{C}^N$, let $A(U)$ be the algebra of functions analytic on $U$ and continuous on $\bar U$, and let $u=(1,0,\ldots,0)$. Let $\mathcal{B}=\{f\in H^\infty (U): \|...
David Walmsley's user avatar
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Local integrability of $\log|f(x)|$ in several variables

If $f(z)$ is an analytic function in a complex neighborhood (in $\mathbb C^n$) of a real point $x^0 \in \mathbb R^n \subset \mathbb C^n$, then $\log|f(x)|$ is integrable over some neighborhood $U \...
Jan Boman's user avatar
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harmonic envelope of holomorphy

Let $D$ be a (real) domain in $\mathbb R^n=\mathbb R^n+i\lbrace 0 \rbrace\subset \mathbb C^n$. Then, due to P. Lelong, there exists a maximal (complex) domain $\tilde D\subset\mathbb C^n$, $D=\tilde D\...
Peter Pflug's user avatar
4 votes
1 answer
158 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) = \...
mrf's user avatar
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227 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. $G\circ G = F$?
Better2BLucky's user avatar
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Is there an Ax-Grothendieck result for entire functions?

The Ax-Grothendieck result states that any polynomial injective function from $\mathbb{C}^n$ to itself is surjective. Is there such a statement for entire functions ?
Yoyo's user avatar
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The boundary regularity of a Teichmüller domain

By a Teichmüller domain, I mean the Bers embedding of a Teichmüller space (of a compact oriented surface of finite type) in a complex space. It is known that the boundary of a Teichmüller domain is ...
Mahdi Teymuri Garakani's user avatar
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Let $f: X \to D$ be a proper holomorphic submersion. Does a holomorphic form on $X$, closed on each fiber of $f$, have holomorphic coefficients?

Let $D$ be the unit disc in $\mathbb{C}^n$ and let $f: X \to D$ be a proper surjective holomorphic submersion, which is trivial as a smooth fiber bundle, with connected fiber $F$. We get an induced ...
Vik78's user avatar
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A Hartogs analogue?

Let $0<r<1<R$, and $A:=\{z\in \ell^2: r<\|z\|_2<R\}$. For $n\in \mathbb{N}$, let $e_n$ denote the sequence in $\ell^2$ all of whose terms are zero, except the $n^{\text{th}}$ term, ...
Salla's user avatar
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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 ...
asv's user avatar
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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.
Jean's user avatar
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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 ...
Jan Bohr's user avatar
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150 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 ...
Joe's user avatar
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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 $\...
S.O.C.'s user avatar
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142 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' ...
Maja Blumenstein's user avatar
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77 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 ...
wellfedgremlin's user avatar
3 votes
0 answers
135 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 ...
Marra's user avatar
  • 73
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0 answers
63 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^...
Craig Franze's user avatar
3 votes
0 answers
190 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 ...
AmorFati's user avatar
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3 votes
0 answers
575 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 ...
Arrow's user avatar
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81 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 ...
erz's user avatar
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3 votes
0 answers
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Trace of a weighted composition operator on Bergman space

I am reading a series of papers by Pollicott, Jenkinson and coauthors which make use of the following type of result: Theorem: Let $\mathbb{D} \subset \mathbb{C}^d$ be a bounded, connected open set. ...
Ian Morris's user avatar
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3 votes
0 answers
747 views

Does a bounded convex domain has one smooth boundary point?

In the study of analysis and geometry of a bounded domain, its boundary regularity is important. For example, it is known that a bounded convex domain has Lipschitz bounday. This implies that a ...
Entaou's user avatar
  • 285
3 votes
0 answers
122 views

cayley transformation of bounded symmetric domain

Can anyone write down the biholomorphic map between classical bounded symmetric domains(defiend by matrixs) with their siegel upperhalf plane models. I know if it's type 2, i.e $I-Z\bar{Z}^{t}>0$ ...
user42804's user avatar
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3 votes
0 answers
120 views

How to study Kähler metrics singular along a submanifold of codim 2?

Let $M$ be a compact complex manifold, $S\subset M$ a submanifold of codimension $2$, let $\omega$ be a k\"ahler metric on $M\setminus S$. Then we know by Reese Harvey's paper "Removable singularities ...
whitacre's user avatar
2 votes
0 answers
80 views

Does Kobayashi isometry map preserve complex geodesics?

Let $\gamma_1, \gamma_2$ are real geodesics in a domain $D$ and these two real geodesics are lying in the same complex geodesics, the question is, are $f\left(\gamma_1\right)$ and $f\left(\gamma_2\...
Begginer-researcher's user avatar
2 votes
0 answers
119 views

What is meant by saying that the Shilov boundary of the polydisc $\mathbb D^n$ is $\mathbb T^n\ $?

Let $A$ be a complex Banach algebra and $M_A$ be the space of all non-zero multiplicative linear functionals on $A$ equipped with the weak$^*$-topology. Let $\widehat A$ be the image of $A$ under the ...
Anacardium's user avatar
2 votes
0 answers
82 views

Abelian subgroup of the automorphism group of $\mathbb C^n$

Let $Aut(\mathbb C^n)$ be the automorphism group of $\mathbb C^n$, i.e., the group of all biholomorphic maps of $\mathbb C^n \to \mathbb C^n$. Suppose $T$ is a finitely dimensional torus which is a ...
Adterram's user avatar
  • 1,361
2 votes
0 answers
188 views

Maximum modulus principle for vector valued functions of several complex variables

In the following paper: Shub and Smale, "On the Existence of Generally Convergent Algorithms", Journal of Complexity 2, 2-11 (1986), trying to understand Lemma 2 on page 4. Paraphrased, ...
user125930's user avatar
2 votes
0 answers
87 views

Analogous tensor product operation for reflexive sheaf

Suppose now $(X,\mathcal O_X)$ is a normal complex space, and $\mathcal F$ is a coherent analytic sheaf on it. Product the reflexive sheaf $$\mathcal F^{[p]}:=(\mathcal F^{\otimes p})^{**},$$ where $\...
Invariance's user avatar
2 votes
0 answers
17 views

Do Szego Kernel in one variable by fixing another variable in a $C^{\infty}$ bounded domain is Bounded?

Let $\Omega\subset\mathbb{C}^n$ be any $C^{\infty}$ bounded domain. Let $ S(.,.)$ denotes the Szego Kenel of Holomorphic Hardy Space $H^2(\partial\Omega)$. Then for $w\in\Omega$ do $S(.,w)$ is a ...
Naruto's user avatar
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2 votes
0 answers
31 views

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 ...
Joe's user avatar
  • 759
2 votes
0 answers
61 views

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. ...
Jean's user avatar
  • 95
2 votes
0 answers
75 views

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}_{...
Poincare-Lelong's user avatar
2 votes
0 answers
70 views

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 ...
Joe's user avatar
  • 759
2 votes
0 answers
116 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 ...
Invariance's user avatar
2 votes
0 answers
115 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 ...
AG learner's user avatar
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2 votes
0 answers
105 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 $...
Invariance's user avatar
2 votes
0 answers
64 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 ...
Better2BLucky's user avatar
2 votes
0 answers
248 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 $X$. ...
Zhaoting Wei's user avatar
  • 8,657
2 votes
0 answers
91 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 ...
Hang's user avatar
  • 2,719
2 votes
0 answers
76 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 ...
Paul's user avatar
  • 1,379
2 votes
0 answers
322 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 ...
Paul's user avatar
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