Questions tagged [several-complex-variables]
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225 questions
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Convexity principle in several complex variables
I would like a reference for an analogue of the Phragmen-Lindelof in several complex variables. Specifically, if f is analytic in a region $A \subset \mathbb C^n$ and, by Bochner's theorem it extends ...
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Is the Minkowski sum of two strictly pseudo-convex bounded domains pseudo-convex?
Let me just recall that the Minkowski sum of two sets is defined by
$$A+B=\{a+b|\, a\in A, b\in B\}.$$
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Composition of two functions is holomorphic and second is holomorphic then first is holomorphic
Let $f, g: \mathbb{C}^n \rightarrow \mathbb{C}^n$, $g$ is surjective, $f \circ g$ is holomorphic and $g$ is holomorphic. Is $f$ holomorphic? I found this is true for 1-dimensional case but is it such ...
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What are some verifiable (and reasonably weak?) conditions to guarantee that the zero set of holomorphic functions contains more than one point?
I know very little algebraic geometry, so maybe this is an easy question, but I ran into it and I could use some expert guidance.
Given holomorphic functions $f_1,...,f_m:\mathbb C^n \to \mathbb C$ (...
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Closure of locally closed varieties
Let $K$ be the field of real or complex numbers and I consider the analytic topology on all mentioned spaces.
Suppose that $M$ is a $K$-analytic manifold (not compact or anything) and let $T\subset M$ ...
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On the identity theorem for holomorphic functions of several complex variables
This question was triggered by the recent digitalisation of the almost forgotten but nevertheless important paper by Beppo Levi [3].
Premises
In the field of several complex variables, the name "...
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Probabilistic interpretation of roots of unity in $\mathbb{C}^2$
I have a question that concerns how often a special class of bivariate polynomials (which I will call mask polynomials) intersects the set of roots of unity in $\mathbb{C}^2$. Caveat: I consider ...
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On a paper by I. Mitrochin published on the Tomsk University Proceedings
Are the Tomsk University proceedings available in some form? In particular I'd like to have a look at the following paper.
I. Mitrochin (И. Митрохин), "Über die Veränderung der Krümmung von ...
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Functions which are singular on intersections of divisors
Begin with a complex manifold with two classes of divisors, say $\alpha$ and $\beta$. I wish to study functions which are singular at the intersection of an $\alpha$ and a $\beta$ divisor, and count ...
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Finding appropriate bound of a subharmonic function defined on punctured disc
Let $E=\{z\in\mathbb{C}:0<|z|<1\}$ be the unit punctured disc in $\mathbb{C}$, and $u\leq0$ be a subharmonic function in $E$. Suppose that we have a real number $0<r<1$, and the area in $|...
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Cartan's Theorem A for vector bundles without sheaf theory
The following question was cross-posted on Math Stack Exchange, but after some further consideration, I think/hope that it may be more appropriate for Math Overflow.
I have been studying Cartan's ...
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At most norm doubling extension from $\{(z,w)\in \mathbb{D}^2|zw=0\}$ to $\mathbb{D}^2$
Let $\mathbb{D}$ denote the open unit disk.
Let $f, g: \mathbb{D}\rightarrow \overline{\mathbb{D}}$ analytic such that $f(0)=g(0).$
We seek a (complex variables or elementary) proof of the fact that ...
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Uniqueness of the Schwarzian Derivative
Recall the Schwarzian derivative of a real or complex analytic function $f$, with the regularty condition $f'\neq 0$, is defined as:
\begin{equation}
s(f)=(\frac{f''}{f'})'-\frac{1}{2}(\frac{f''}{f'})^...
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Extension of Harvey-Lawson's theorem to general pseudoconvex hypersurfaces
Harvey-Lawson have this remarkable theorem (which can be seen here):
Theorem: Let $X$ be a strongly pseudoconvex CR manifold of dimension $2n −1$, $n \geq 2$. If $X$ is contained in the boundary of a ...
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"Different" definitions of smooth form on a complex analytic space
In this question, they give some reference about how to define a smooth form on a complex analytic space, while in the article Mesures de Monge-Ampère et caractérisation géométrique des variétés ...
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Regularity of solutions of Dirichlet problem for complex Monge–Ampère equations
Let $\Omega\subset \mathbb{C}^n$ be a smooth bounded strictly pseudo-convex domain. Let $f\in C(\bar\Omega)$, $\phi\in C(\partial \Omega)$.
A theorem due to Bedford and Taylor (Invent. Math. 37 (1976))...
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Manifolds, perimeter and volume
The Bergman metric $ g_{i\bar{\jmath}}(z) $ is defined by
$$
g_{i \bar{j}}(z) = \frac{\partial^2}{\partial z_i \, \partial \overline{z_j}} \log K(z, z)
$$
which gives the components of the ...
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Analytic continuation of a bi-holomorphic automorphism on an irreducible bounded symmetric domain
I need your help.
Let $\Omega \subseteq \mathbb{C}^n$ be a type $IV_n$ Cartan domain, i.e;
$\Omega$ =$\{ z \in \mathbb{C}^n$: $1-2Q(z,\bar z)+|Q(z, z)|^2>0,\qquad Q(z, \bar z)<1 \}$
where $Q(z,...
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References for the generalized Dirichlet problem for plurisubharmonic functions on the bidisc
In the paper "On a Generalized Dirichlet Problem for Plurisubharmonic Functions and Pseudo-Convex Domains. Characterization of Silov Boundaries" Theorem 5.3, the following result is obtained ...
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Concerning the definition of a class of functions introduced by Nilsson
In the paper 'Some growth and ramification properties of certain integrals on algebraic manifolds' by Nilsson we find the following definitions:
My question is how does one prove the remark "It ...
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Function of several complex variables with prescribed zeros
I accidentally stumbled upon a problem of complex analysis in several variables, and I have a hard time understanding what I read, it might be related to the Cousin II problem but I cannot say for ...
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Fixed points of maps defined on Teichmüller space
Let $\mathcal{T}_A$ be a Teichmüller space of the sphere $S^2$ with a finite set $A$ of marked points, and suppose that $f \colon \mathcal{T}_A \to \mathcal{T}_A$ is a holomorphic map that has a ...
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Does the sequence of bounded symmetric square integrable holomorphic functions have a convergent subsequence?
Let $f$ be a bounded holomorphic function on $\mathbb D^2$ and $s : \mathbb C^2 \longrightarrow \mathbb C^2$ be the symmetrization map given by $s(z) = (z_1 + z_2, z_1 z_2),$ for $z = (z_1, z_2) \in \...
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Schwartz's theorem without English language reference
I'm reading the paper "Spectral Synthesis And The Pompeiu Problem" by Leon Brown, Bertram M. Schreiber and B. Alan Taylor,
Annales de l’Institut Fourier 23, No. 3, 125-154 (1973), MR352492, ...
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Deck transformation group of the basic polynomial map on a $G$-space
Let $G \subseteq GL_d (\mathbb C)$ be a finite pseudoreflection group (see here and here) acting on a domain $\Omega \subseteq \mathbb C^d$ by the right action $\sigma \cdot z = \sigma^{-1} z$ where $\...
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Complex analytic function $f$ on $\mathbb{C}^n$ vanish on real sphere must vanish on complex sphere
I'm considering a complex entire function $f$: $\mathbb{C}^n\to \mathbb{C}$. Suppose $f=0$ on $\{(x_1,\cdots,x_n)\in\mathbb{R}^n:\sum_{k=1}^n x_k^2=1\}$. I want to prove
$$f=0\textit{ on } M_1=\{(z_1,\...
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Subset of a complex manifold whose intersection with every holomorphic curve is analytic
The Osgood's lemma in complex analysis says that if $f \colon \mathbb{C}^n \to \mathbb{C}$ a continuous function such that $f|_{L}$ is holomorphic for every complex line $L \subset \mathbb{C}^n$, ...
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Equivalent condition for the Pick matrix being positive semidefinite
On the wikipedia page of the Nevanlinna-Pick theorem the following claim appears:
Let $\lambda_1,\lambda_2,f(\lambda_1),f(\lambda_2)\in\mathbb{D}$. The matrix $P_{ij}:=\frac{1-f(\lambda_i)\overline{f(\...
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The boundary regularity of a Teichmüller domain
By a Teichmüller domain, I mean the Bers embedding of a Teichmüller space (of a compact oriented surface of finite type) in a complex space.
It is known that the boundary of a Teichmüller domain is ...
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Residues and blow ups
On a 2-dimensional complex manifold consider two functions which are meromorphic with singularities along two divisors which meet at a point. There is a residue from these meromorphic functions (...
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Product of two circles and holomorphic functions
Assume that $f$ and $g$ are holomorphic functions in the unit disk having boundary values on the unit circle $T$ almost everywhere. Assume further that $$\int_0^{2\pi}\int_0^{2\pi}|f(e^{it})+g(e^{is})|...
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Zeroes of entire function on $\mathbb C^n$
Let $n\ge 2$ be an integer and let $f$ be an entire function on $\mathbb C^n$. Let $A$ be a subset of $\mathbb R^n$ with positive $n$-dimensional Lebesgue measure. Then if $f$ vanishes at $A$, this ...
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Problem in understanding maximum principle for subharmonic functions
I am reading subharmonic functions and their properties from the book From Holomorphic Functions to Complex Manifolds by Grauert and Fritzsche. Let me first define what a subharmonic function is.
...
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Fixed points free automorphisms of Teichmüller spaces
Let $\mathcal{T}_{g,n}$ be the Teichmüller space of a compact oriented surface of genus $g$ with $n$ marked points. Assume that $N:=3g-3+n>0$. Viewing $\mathcal{T}_{g,n}$ as a bounded domain in $\...
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An interior cone condition for Teichmuller spaces
Let $\mathcal{T}_{g,m}$ be the Teichmuller space of a compact oriented surface of genus $g$ with $m$ marked points. Consider it as a bounded domain in a complex space $\mathbb{C}^N$. Let $\xi$ be a ...
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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 ...
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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 ...
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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\...
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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 ...
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Let $u$ be harmonic on domain $D\subset \mathbb R^d$, how far can we extend $u$ holomorphically?
If $u$ is harmonic then it is real analytic so then it can be extended locally holomorphically. I also know that if $u$ is harmonic on a ball in $\mathbb R^d$ we have that the radius of convergence is ...
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$n$-th root of meromorphic functions of several complex variables
Let $f:\Omega\rightarrow\mathbb{C}$ be a "nice" meromorphic function of several complex variables on some domain $\Omega$. I wonder if the following claim is true.
Claim. $f$ admits a global ...
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pseudo inverse of a holomorphic multivariate injective map
Let $f:{\mathbb C}^n \rightarrow {\mathbb C}^N$, $N > n$, be holomorphic and injective on an open ball $B_n \subset {\mathbb C}^n$ such that the Jacobian matrices have full column rank at each ...
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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, ...
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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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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_{\...
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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 ...
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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 $(1,0,\dotsc,0)$ 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 $u=(1,0,\dotsc,0)$. Let $\mathcal{B}=\{f\in H^\infty (U): \...
