Questions tagged [several-complex-variables]
For questions related to the study of functions of several variables, in particular the study of holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
808 questions
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Is there a dense set of Lipschitz functions on the unit ball which all peak at the same point on the boundary?
I posted this same question over 2 years ago on MO but didn't get any comments or answers. The question is admittedly quite narrow, but I am still very interested in its resolution.
Let $U$ be the ...
- 1,247
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Complex variable Biharmonic BVP
Consider the Complex variable Partial Differential Equation:
$$
\Delta^2 w =f
$$
with boundary conditions
$$
w=\varphi_0 ~~\text{and}~~ \partial_{\bar{z}}w =\varphi_1.
$$
A unique solution to this PDE ...
- 11
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Composition of two functions is holomorphic and second is holomorphic then first is holomorphic [closed]
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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On quotients in convergent power series ring.
Suppose that $m\geq 2$.
Consider $\mathbb{C}\{y_{1},\ldots, y_{m}\}$ the ring of convergent power series in the variables $y_{1},\ldots, y_{m}$ and $\mathfrak{m}=\langle y_{1},\ldots, y_{m}\rangle$ ...
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Constructing a polydisc for holomorphic extension
This is just a geometry/topology problem.
For reference, this problem is exercise E.1.2, page $47$, of Range's "Holomorphic Functions and Integral Representations in Several Complex Variables&...
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Proof of Hartogs' Theorem in Huybrechts' Complex Geometry
First of all, I want to say sorry if there are something that goes against the rules for posting. This is my first time asking here...
Hello guys, I am currently studying Hartog's Theorem using ...
- 11
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Griffiths & Harris - proof of proposition in page 10
In "Principles of Algebraic Geometry" by Griffiths & Harris, they prove that the ring $\mathcal{O}_n$ of holomorphic functions in $n$ variables defined in some neighborhood of $0$, is a ...
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Plurisubharmonicity of function
Let's consider a real-valued function $f$ on a complex manifold. It has the following properties:
When $f$ is $C^2$, $\frac{\partial^2 f}{\partial z_i \partial \bar{z}_j} \ge 0$.
There are points ...
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Does holomorphicity at one point imply holomorphicity everywhere for conformal maps in $\mathbb C^n?$
I have two questions here, one ($Q1$) for $\mathbb{C}^n$ and the other ($Q2$), its generalization over arbitrary Hermitian manifolds. I will present my proof for the $\mathbb{C}^n$ case ($Q1$). I ...
- 313
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Cauchy integral formula in polydisc
Do we have Cauchy integral formula on a polydisc $P\subseteq \mathbb{C}^n$: for any $x_0\in P$
$$
\partial_x^\alpha f(x_0) = \frac{\alpha!}{(2\pi i)^{n}}\int_{\partial P} \frac{f(x)}{(x-x_0)^{\alpha+1}...
- 11
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Can algebraic functions be identified with actual functions?
It is well known that if $k$ is an infinite field, then polynomials in $k[X_1,\dots,X_n]$ can be identified with polynomial functions from $k^n$ to $k$. I am wondering if this generalizes in a ...
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Is boundary distance an exhaustion function on an unbounded domain?
I am reading From Holomorphic Functions to Complex Manifolds by Klaus Fritzsche and Hans Grauert. Let $G\subset \mathbb{C}^n$ be a domain. Define the boundary distance $\delta_G:G\to \mathbb{R}$ by
$$\...
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Cartan's Theorem A for vector bundles (no sheaves)
I have been studying Cartan's Theorems A and B recently, and I was wondering whether it was possible to simplify the proofs that I have seen in the situation of the sheaf of sections of a holomorphic ...
- 1,266
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Expansion with respect to $z_1$
I'm reading "From Holomorphic Functions to Complex Manifolds" - Fritzsche & Grauert and I have something that I don't understand very well:
If $\nu \in \mathbb{N}_0^n, t \in \mathbb{R}^...
- 1,482
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Boundary of a complete Reinhardt domain with logarithmically convex base
Let $U'=\bigcup_{1\leq j\leq k}\mathbb {P}^n (0,r_j)\subset \mathbb{C}^n$ be a finite union of concentric polydiscs. We consider the convex set $U_0 =conv\{(log|z_1|,\ldots,log|z_n|): (z_1,\ldots,z_n)\...
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