# Questions tagged [several-complex-variables]

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

184
questions

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### Existence of a Kähler potential

Let $(M,\omega)$ be a Kähler manifold. For any point $p \in M$, can we find a domain $U$ containing $p$ such that
$$
\omega=dd^c \phi
$$
on $U$ and $\phi=0$ on $\partial U?$

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39
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### Characterizing some similarity invariant homogeneous log-superharmonic functions of matrices

Let $L:M_n(\mathbb{C})^r\rightarrow[0,\infty)$ be a function that satisfies the following properties:
$\log(L)$ is plurisubharmonic.
$L$ is homogeneous in the sense that $L(\lambda A_1,\dots,\lambda ...

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### A coradius of convergence - biggest open disk contained in the image of a power series?

Let $f \in \mathbb{C}\{z_1,\dots,z_n\}$ be non-constant with $f(0) = 0$, where $n \geq 1$, and let $D$ be its domain of convergence. Recall that for $n=1$ this is just some open disk $\mathbb{D}_r(0)$ ...

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### Do we have a Grauert-Fischer theorem for non-trivial families?

This question is related to my previous question. Let $X$ be a compact complex manifolds and $\Delta\in \mathbb{C}^n$ be a small neighborhood of $0$. A family of deformations of $X$ over $\Delta$ is a ...

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1
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148
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### Inverse of Bochner–Martinelli formula

Suppose that $f$ is a holomorphic function on a domain $D$ in $\mathbb{C}^n$, $\partial D$ is smooth, and $f$ is $C^1$ on $\partial D$. Then, the Bochner-Martinelli formula states that
$f(z) = \int_{\...

2
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answers

78
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### 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 ...

2
votes

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83
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### 1-convex and holomorphically convex

A complex manifold $M$ is called $1$-convex if there exists a smooth, exhaustive, plurisubharmonic function that is strictly plurisubharmonic outside a compact set of $M$.
Can we prove that if $M$ is $...

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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): \|...

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### Characterization of a "complex" hull?

This is a complex continuation of my previous question. There Iosif Pinelis showed that the so obtained closure from taking the intersection of the preimages of the linear functionals indeed coincides ...

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1
answer

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### Estimate of minimum of the Poisson integrals corresponding to a convergent Hausdorff sequence of smooth bounded domains from below

Let $\{\Omega_{j}\}_{j\in\mathbb{N}}$ be a sequence of smooth bounded domains in $\mathbb{C}^{n}$ such that $\Omega_{j}$ converges to a smooth bounded domain $\Omega$ in the sense that the defining ...

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163
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### Determinant of the conormal bundle

Let $Y$ be a smooth submanifold of codimension $r$ in a complex manifold $X$. By virtue of the adjunction formula, we always have the isomorphism
$$K_Y\simeq (K_X\otimes \det N_Y){\,|\,}_Y.$$
Recall ...

3
votes

1
answer

145
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### Factorization of an analytic function in $\mathbb{C}^n$

Let $\Omega$ be an open subset of $\mathbb{C}^n$ and let $f$ be analytic in $\Omega$. Assume $P\in\mathbb{C}[z_1,\ldots,z_n]$ is a polynomial whose irreducible factors are all of multiplicity one.
If $...

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102
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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 \...

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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\...

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158
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### A characterization of plurisubharmonic functions

Let $\Omega\subset \mathbb{C}^n$ be an open subset. Let $u\colon \Omega\to [-\infty,+\infty)$ be an upper semi-continuous function.
Recall that $u$ is called plurisubharmonic (psh) if its restriction ...

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1
answer

340
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### Determine the coefficient of the exceptional divisor

Consider the following setting: suppose that $X$ is a smooth variety and let $f:X\rightarrow \Delta$ be a smooth morphism outside the origin $0$. Let the central fiber $X_0$ be a reduced (Cartier) ...

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146
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### 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, ...

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votes

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375
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### Is the determinant line bundle of a coherent sheaf functorial (between sheaves of the same rank)?

The determinant line bundle of a coherent sheaf $\mathcal{F}$ on an $n$-dimensional (smooth) analytic space is defined as
\begin{equation}
\det \mathcal{F} := \bigotimes_i^n (\det \mathcal{E}_i)^{⊗...

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votes

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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 $\...

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votes

2
answers

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### Laurent series in several complex variables

Is there a good generalisation of Laurent series for several complex variables?
I am interested in generalised power series that have some terms with negative powers, but not too many. In single ...

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1
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154
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### Global sections of a line bundle on a reducible complex space

Let $S$ be a reducible compact complex analytic space, thus we have the decomposition $S=\bigcup_{i=1}^n {V_i}$ where $V_i$ is the irreducible component of $S$. Let $L$ be a line bundle on $S$, I ...

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### L2 estimate on strongly pseudoconvex complex manifold

Suppose $(X,g,I)$ is a Hermitian (non kahler) complex manifold with small torsion, small derivative of torsion and small curvature. Let $\varphi$ be smooth PSH function satisfying $\sqrt{-1}\bar\...

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138
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### What can be said about cluster sets for power series of two variables?

I'm still trying to prove the continuity of a function $u$ which can be interpreted as the restriction of a power series of two variables, which I haven't managed to approach the right way yet. To ...

2
votes

1
answer

152
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### Bishop's compactness theorem and convergence of analytic subset

Let $V_i$ be a sequence of $k$ dimensional analytic subsets in $\mathbb C^N$. Suppose that the volume of $V_i$ is uniformly bounded, then Bishop's compactness theorem says that $V_i$ will convergence ...

3
votes

1
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299
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### Can a power series of several variables be discontinuous on a compact set if it converges in every point of this set?

Say we have a power series of two variables, with an associated function $f$ defined as
$$
\begin{split}
f(x, y) =\, & \sum_{n,m} a_{n,m}x^ny^m,\\
& a_{n,m} \geq 0 \quad \forall n, m \in\...

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0
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67
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### When a strictly positive log pluriharmonic function $g$ is equal to the norm of holomorphic function?

Suppose $V$ is a local analytic variety (singular). Suppose $g$ a strictly positive log pluriharmonic function on $V$, i.e. $\log g$ is pluriharmonic. I wonder when $g=|f|^2$, where $f$ is a ...

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56
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### Characterization of elements of Hardy Space

Let $\Omega\subset\mathbb{C}^n$ be a $C^{\infty}$ bounded domain. Let $H^2(\partial\Omega)$ denote the Hardy space, and $S(.,.)$ denote its Szego Kernel. We know that
$$
\forall f\in H^2(\partial\...

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### 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 ...

2
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108
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### On the definition of Cauchy transform [closed]

I have seen two different definitions of the Cauchy transform of a smooth function one is with respect to the line integral (for eg. in. the book "The Cauchy transform and potential theory")...

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votes

1
answer

201
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### Extension of a Szegő Kernel to the boundary

Let $\Omega\subset\mathbb{C}^n$ be any smooth bounded pseudoconvex domain. Let $S$ denote the Szegő kernel of $\Omega$.
Recall: the Szegő kernel is a kernel of the Szegő projection $P: L^{2}(\partial\...

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0
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36
views

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

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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158
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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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0
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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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votes

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433
views

### $\frac{\partial f}{\partial \overline{z}}=0$ 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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votes

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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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votes

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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) $\...

7
votes

1
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346
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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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votes

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282
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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}_{...

2
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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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### 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 ...