The several-complex-variables tag has no usage guidance.

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### Jensen formula in $\mathbb{C}^n$?

Let $f:\mathbb{C}\to\mathbb{C}$ be an entire function with zero set $X\subset \mathbb{C}$. Jensen's formula reads
$$
\log(|f(0)|)+\int_0^R\frac{|X\cap B_t(0)|}{t}dt = ...

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### $H(K)$ is dense in $H(L)$

Let $U\subset\mathbb{C}^n$ be a domain, $K\subset U$ a compact analytic polyhedron, $f\in H(U)$ and $$L=\{z\in K:|f(z)|\le 1\}.$$ Let $g\in H(L)$ and $\epsilon>0$. How does one prove that, there ...

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### Construction of homogeneous Siegel domain from j-algebra

I am reading bounded homogeneous domain from Piatetski-Shapiro's
book ``Automorphic functions and the geometry of classical domains''
and have questions on how to construct homogeneous Siegel domain
...

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73 views

### Domains of holomorphy and simply connected domains

Let $U\subset\mathbb{C}^n$ be a domain of holomorphy, we can say that $U$ is a simply connected domain?
Any hints would be appreciated.

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### To show there exists a unique function $u \in C^{1}(\mathbb{C^n})$ that satisfies $(\bar{\partial u})=f$

Assume $n \gt 1$. Let $f$ be a $(0,1)$ form in $\mathbb{C^n}$, with $C^1$-coefficients and compact support $K$, such that $\bar{\partial} f=0$. Let $\Omega_{0}$ be the unbounded component of ...

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177 views

### Hartogs's extension theorem

Let $(P,H)$ be a Euclidean Hartogs figure in $\mathbb{C}^n$, and
let $f:H\to \mathbb{C}^n$ be a holomorphic injective map. Then we know that $f$ extends holomorphically to the polydisc $P$, i.e. there ...

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38 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 $I-Z\bar{Z}^{t}>0$ ...

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64 views

### connectedness holomorphically convex hull

Let $U\subset\mathbb{C}^n$ be a domain and $K\subset U$ compact. If $K$ is connected then $\hat{K}_U$ is also connected?
$\hat{K}_U= \{z \in U: |f(z)| \leq \sup_K |f|, \forall f\in ...

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59 views

### Proof of Levi Problem via L^2 estimate

A theorem of Grauert says that a complex manifold admitting a strictly plurisubmanonic exhaustion function is stein. There is a proof by $L^2$ estimate in Domailly' notes "$L^2$ estimate for the dbar ...

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93 views

### complex dynamic system and affine algebraic variety

Let $M^n$ be a $n$-dimensional noncompact complex manifold. In "The density property for
complex manifolds and geometric structures II", Dror Varolin showed that some open set of
$M$ is ...

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169 views

### Extensions of Real Analytic to Holomorphic Functions in One & Several Variables: References?

A problem I'm working on requires the application of Cauchy's estimate for the modulus of the coefficients of a holomorphic function's power series representation, but the original functions with ...

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108 views

### extension of holomorphic mappings

In 1971 Phillip.A.Griffith proved that
Let $B_n^\ast=\{z\in\mathbb{C}^n:0<\|z\|\le 1\}$ be the punctured ball in $\mathbb{C}^n$,and $f:B_n^\ast\rightarrow M$ a holomorphic mapping into a compact ...

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87 views

### How to solve $\sqrt{-1}\partial\bar{\partial}u=\omega$

I'm looking for references on the study of the equation $\sqrt{-1}\partial\bar{\partial}u=\omega$,especially when $\omega$ is a k\"ahler metric on $\Omega\setminus S$,where $\Omega\subset ...

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86 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 ...

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93 views

### Connection between the two definitions of Siegel Upper Half Space

It seems, there are two definitions of the Siegel upper half space.
1) One used by, say, Krantz in "Explorations in Harmonic Analysis; Page 252, or, by So Chin Chew and Mei Chi Shaw in "Partial ...

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237 views

### 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 ...

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484 views

### Classification of complex structures on $\mathbb{R}^{2n}$

Is there anything known about classification of complex structures on $\mathbb{R}^{2n}$ up to isomorphism for $n>1$? Say, are there finitely or infinitely many isomorphism classes? If there is a ...

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111 views

### If $f$ is separately holomorphic on $\Omega$ then $f\in\mathcal{C}^0(\bar\Omega)\Leftrightarrow f\in L^1(\Omega)$

Let $\Omega\subseteq\Bbb C^2$ be open bounded (and connected), $f:\Omega\to\Bbb C$ separately holomorphic (i.e. $f$ is holomorphic in each variable when the other is fixed).
Hartogs theorem is not ...

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217 views

### generalization of fundamental theorem of algebra for several complex algebra [closed]

I am looking for a generalization to fundamental theorem of algebra for several complex variables functions or systems. If such theorem exists, it should concisely relates the number of zeros of ...

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86 views

### Complex Hessian Signature

It' all, simply, about the signature of a matrix.
Let $\Omega\subseteq\Bbb C^n$ open, $r:\Omega\to\Bbb R$ twice differentiable (real differentiable, not necessarely complex differentiable, i.e. not ...

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153 views

### Determining the Fourier transform

Let $d>2$. Let $M$ be a 2-dimensional submanifold of $\mathbb{R}^d$. For instance (and this is the type of example I primarily care about) we could have $M$ being the set of scalar multiples of a ...

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105 views

### Lifting quadratic forms on the cotangent bundle to higher level forms

Backround
In several complex variables, an essential tool is Hormander's machinery for solving the $\overline{\partial}$ problem with $L^2$ estimates.
If $\alpha$ is a $(p,q+1)$ form on a domain ...

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197 views

### Are polynomials dense in holomorphic $L^p(\mathrm{Gauss})$ for $p < 1$?

Let $\mu$ be standard Gaussian measure on $\mathbb{C}^n$, i.e. $d\mu = \frac{1}{(2 \pi)^n} e^{-|z|^2/2}\,dz$, and fix $0 < p < 1$ (note carefully).
Suppose $g$ is holomorphic on ...

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76 views

### Computing Dolbeault cohomology of some simple domains

I have seen computations of the Dolbeault cohomology groups on compact Kahler manifolds using Hodge theory.
I have never seen the computation of Dolbeault cohomology for simple domains in ...

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300 views

### Is there an Oka-Grauert principle for homogeneous spaces?

Suppose we have a fibration over the punctured disc (i.e., a deformation of complex manifolds) such that each fiber is a homogeneous space. Is the total space a product of a fiber with the punctured ...

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424 views

### Can all $L^2$ holomorphic functions on a domain vanish at a particular point?

Let $\Omega \subset \mathbb{C}^n$. Is it possible that there is a point $p \in \Omega$ such that every $f \in A^2(\Omega) = L^2(\Omega) \cap \mathcal{O}(\Omega)$ has a zero at $p$? The space ...

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146 views

### how to compute bergman kernel

i have a question to determin if the asyptotic expansion of Bergman kernel has a log term. Is there anyone can show me is there any general way to tell?

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### Does the “Ohsawa-Takegoshi theorem without bounds” have a name?

There are many theorems which now could be called "The Ohsawa-Takegoshi" theorem. Of these, the most basic is roughly the following:
Let $\Omega \subset \subset \mathbb{C}^n$ be a psuedoconvex ...

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137 views

### Are open immersions in analytic geometry transverse?

lately I have been interested in functional analysis, especially with a view towards its applications in the world of (complex) analytic geometry. I have been using R. Taylor's book Several complex ...

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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 ...

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243 views

### Can someone tell me properties of Douady space?

I want to know the parallel properties of Douady space with respect to Hilbert scheme. For example I want to know what is the irreducible component of Douady space, what if I consider a family of ...

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107 views

### Complements of images of complex analytic sets

It is known that the complement of an analytic set is connected. In general, the complement of a proper complex analytic set in a connected complex manifold is an arcwise connected dense open set. My ...

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472 views

### What is the “complex third derivative”?

Background
I am including this information about real higher order derivatives because it does not seem to be common knowledge. I also include a review of the complex Hessian.
If $f:\mathbb{R}^n ...

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### Translation of “Über kompakte homogene Kählersche Mannigfaltigkeiten”

Has anyone translated Borel and Remmert's 1962 paper titled:
Über kompakte homogene Kählersche Mannigfaltigkeiten?

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### Singularities of the Remmert reduction of a holomorphic convex manifold

Let $X$ be a holomorphically convex manifold, namely, for any infinite discrete subset of $X$ there exists a holomorphic function on $X$ which is unbounded on this set, then a theorem of Remmert says ...

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210 views

### the group of all biholomorphic group automorphisms of complex tori

My background is complex geometry, but when I confront complex tori, I feel it is easier to consider it as a compact connected complex Lie group although I just know the definition of Lie group.
Let ...

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250 views

### Explicit form for hermitian structure $h$ with respect to $\omega$

Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on ...

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### Is the Szego projection on a codim-$k$ CR manifold an integral operator?

The Szego projection on a CR manifold $M$ is defined to be the orthogonal projection from $L^2(M)$ to the closed subspace $H^2(M),$ where
$$H^2(M) = \{f \in L^2(M)\ |\ \bar{\partial}_{b}f = 0\ ...

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994 views

### If $f$ is $C^{\infty}$ and $f^2$ is analytic, then $f$ is analytic

Assume that $f:\mathbb{C}^n\rightarrow \mathbb{C}$ is a $C^{\infty}$ function such that $f^2$ is (complex) analytic. Then one can show that $f$ is analytic. (Note: Liviu Nicolaescu and Alexandre ...

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### Any non-constant surjective holomorphic map between connected compact complex manifolds of equal dimension is a ramified finite-sheet covering

I'm reading this site:holomorphy of inverse map
There is a statement made by Colin Tan at the last answer made by himself.
Any non-constant surjective holomorphic map between connected compact ...

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### Automorphisms of the affine quadric $z_1^2+\dots+z_n^2=1$

Denote by $V$ the vanishing locus of the affine quadric $z_1^2+\dots+z_n^2-1$ in $\mathbb{C}^n$. Clearly $V$ is a complex manifold. Let $g:V\to V$ be an automorphism of $V$ (a biholomorphism, not ...

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### Weakened jacobian conjecture for entire functions

A rudin's theorem is the assertion that any polynomial injection between affine spaces of the same dimension has a polynomial inverse, and the inverse is also given by polynomials.
The jacobian ...

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### motivation for multiplier ideal sheaves

What is the origin of multiplier ideal sheaves?It was introduced ny Nadel.Yum Tong Siu,his advisor in his plenary lecture in 2002 icm mentions some thing that it arose in pde.Can anyone kindly ...

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### Bergmann Shilov Boundary vs Peak Points and Strong Boundary Points

Let $\Omega\subset\mathbb{C}^n$ be a bounded domain. Consider the Banach algebra $A(\Omega):=\mathcal{C}(\overline{\Omega})\cap\mathcal{O}(\Omega).$ Let $\partial_S\Omega$ denote the Bergmann Shilov ...

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### What´s essential to learn about complex spaces and several complex variables for an algebraic geometer?

Hi, I don´t know if this question is suitable for this site. The field of several complex variables is too broad, so I would like to know what´s essential to learn about complex spaces and several ...

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### How does the complex convex set look like?

The usual convex set is the real linear convex set, if we change the real linear map into complex linear map, we can get the complex convex set. A system way to do this is in the several complex ...

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505 views

### Extension of pluriharmonic functions

Suppose $M$ is a complex manifold and $\Omega$ a (edit: bounded) pseudoconvex domain in $M$. Let $u:M\setminus\Omega\to\mathbb{R}$ be a pluriharmonic function. Is it true that $u$ has a pluriharmonic ...

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### Another proof of the bidisc and the ball are biholomorphically inequivalent?

Does this outline of a proof work?
Consider the ball and the bidisc in $\mathbb{C}^2$. Give each space its Bergman metric. To show that the ball and the bidisc are not holomorphic, it is enough to ...

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### Fundamental motivation for several complex variables [closed]

I have 3 general abstract reasons to care about complex analysis in a single variable:
The laplacian is, up to a constant multiple, the only isometry invariant PDO in the plane, and so it is ...

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311 views

### A sequence that tell us if a holomorphic function of several variables is identically zero

Is there any sequence $ \{ Z_{\nu} \}_{\nu \in \mathbb{N}}$ in $\mathbb{C}^{n}$, $Z_{\nu} \rightarrow 0$, such that any holomorphic function in $\mathbb{C}^{n}$ which vanishes in $Z_{\nu}$ for all ...