Questions tagged [several-complex-variables]
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References for the generalized Dirichlet problem for plurisubharmonic functions on the bidisc
In On a On a Generalized Dirichlet Problem for Plurisubharmonic Functions and Pseudo-Convex Domains. Characterization of Silov Boundaries Theorem 5.3, the following result is obtained for a type of ...
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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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Contraction of an inclusion with respect to Kobayshi hyperbolic metric
Suppose that $X = \mathbb{C}^n - \Delta_X$ and $Y = \mathbb{C}^n - \Delta_Y$, where $\Delta_X$ and $\Delta_Y$ are unions of hyperplanes in $\mathbb{C}^n$ such that $\Delta_Y \subset \Delta_X$, $\...
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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
$$f=0\textit{ on } M_1=\{(z_1,\...
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Subset of a complex manifold whose intersection with every holomorphic curve is analytic
The Osgood's lemma in complex analysis says that if $f \colon \mathbb{C}^n \to \mathbb{C}$ a continuous function such that $f|_{L}$ is holomorphic for every complex line $L \subset \mathbb{C}^n$, ...
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Prove that $\widehat G = \left \{(z',w)\ :\ |w| < \rho (z') \right \}$ is pseudoconvex if and only if $-\log \rho$ is plurisubharmonic
Let $G \subseteq \mathbb C^n$ be a domain and $\rho : G \longrightarrow \mathbb R$ be lower semicontinuous positive function. Prove that $$\widehat G = \left \{(z',w)\ :\ |w| < \rho (z') \right \}$$...
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Equivalent condition for the Pick matrix being positive semidefinite
On the wikipedia page of the Nevanlinna-Pick theorem the following claim appears:
Let $\lambda_1,\lambda_2,f(\lambda_1),f(\lambda_2)\in\mathbb{D}$. The matrix $P_{ij}:=\frac{1-f(\lambda_i)\overline{f(\...
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The boundary regularity of a Teichmüller domain
By a Teichmüller domain, I mean the Bers embedding of a Teichmüller space (of a compact oriented surface of finite type) in a complex space.
It is known that the boundary of a Teichmüller domain is ...
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Residues and blow ups
On a 2-dimensional complex manifold consider two functions which are meromorphic with singularities along two divisors which meet at a point. There is a residue from these meromorphic functions (...
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Product of two circles and holomorphic functions
Assume that $f$ and $g$ are holomorphic functions in the unit disk having boundary values on the unit circle $T$ almost everywhere. Assume further that $$\int_0^{2\pi}\int_0^{2\pi}|f(e^{it})+g(e^{is})|...
