# 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.

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### $-\log\delta_{\Omega}$ is plurisubharmonic.

I was reading the section of Pseudoconvexity and plurisubharmonicity of Hörmander's book (Introduction to complex analysis in several variables), and I have a doubt regarding the proof of the ...
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### Subharmonicity is a local property.

Let $D\subseteq \mathbb{C}$ be a domain. Definition. A function $u:D\to\mathbb{R}\cup\{-\infty\}$ is called subharmonic if $u$ is upper semicontinuous and if for every compact set $K\subseteq D$ and ...
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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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### On Domains of Holomorphy

Let $D\subseteq \mathbb{C}^{n}$ be a domain. We say that $D$ is a domain of holomorphy if, for every domain $V\subseteq \mathbb{C}^{n}$ for which $V\cap \partial D\not=\emptyset$ and for every ...
1 vote
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### Continuity of roots for holomorphic functions in several complex variables

Let $f(x,y)$ be a holomorphic function on a open $U \subset \mathbb{C}^2$ where $0 \in U$ and $f(0,0) = 0$. Suppose $y \mapsto f(0,y)$ is not identically zero near the origin and its order of ...
37 views

### Localizing domain near strongly pseudoconvex boundary point

Let $D \subset \mathbb{C}^n$ be a smoothly bounded pseudoconvex domain and let $p \in \partial D$ be a strongly pseudoconvex boundary point . Often it is useful to localize the boundary of $D$ near ...
1 vote
56 views

### The ball take away a submanifold of codimension 1 is a domain of holomorphy

Let $\mathbb{B}^n \subset \mathbb{C}^n$ be the unit ball and $V \subset \mathbb{B}^n$ be a closed complex submanifold of codimension one. How do we show that $\mathbb{B}^n \setminus V$ is a domain of ...
79 views

### Intuition for Reinhardt domains

I'm taking a complex analysis course, and in a lecture on several complex variables my professor defined Reinhardt domains in $\mathbb{C}^n$ as domains invariant under the action of the n-dimensional ...
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### Maximum principle for several complex variables

Suppose that we have an analytic function on the polydisc {$\{z \in \mathbb{C}^n: \left|z_i\right| < 1, \forall i=1,\dots,n \}$} and continuous on the boundary. Can a non-constant function take ...
1 vote
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### Do these two quantities behave the same?

Let $\Phi_z(w)$ denote an involutive automorphism of the unit ball, $w,z\in \mathbb{B}^n$ and $f:\mathbb{B}^n\to \mathbb{C}$ be a holomorphic function. My question is the following: Let's take the ...
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### A possible criterion for a function on a complex analytic space to be holomorphic

Let $(X, \mathcal O)$ be a reduced complex analytic space, and $f:X \rightarrow \mathbb C$ be a continuous function. Assume that for all holomorphic mappings $F$ from unit disc $D \subset \mathbb C$ ...
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### Textbooks on Complex Analytic Spaces

I'm looking for textbooks or monographs on complex analytic spaces. I'm aware of Coherent Analytic Sheaves by Grauert/Remmert and Several Complex Variables by Gunning/Rossi, but more references would ...
1 vote
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### Analogue to Osgood's lemma for rational functions

Osgood's Lemma says that if a function of several complex variables is continuous on an open set $D$ and holomorphic on each variable then it is holomorphic. Is there a similar result but for rational ...
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### Integration of total derivative for several complex variables functions

Let $f: \mathbb{C}^m \rightarrow \mathbb{P}^1$ be a function. The total derivative of $f$ denoted by $Df = \sum_{j=1}^{m}z_j \frac{\partial f}{\partial z_j}$. Moreover, $D^{k+1}f = D(D^kf)$. Let us ...
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### Complex Monge Ampere measure with finite mass

I was reading this paper by Bedford-Taylor. It was proven in Theorem 2.4 that the Monge Ampere measure $(dd^c u)^n$ has a finite mass on any polydisc when $u$ is PSH and continuous up to the boundary ...
29 views

### Connected component of globally analytic subset

I am reading the book Introduction to complex analytic geometry by Stanislaw Lojasiewicz. In Chapter II, we have that a subset $Z$ of a complex manifold $M$ is said to be a principal analytic subset ...
1 vote
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### Why Second Cousin problem is generalization of the Weierstrass’ Theorem

The Weierstrass’ problem in one complex variable asked, if there is holomorphic function with the pre-assigned zeros in $\Bbb{C}$? This is proved using the Weierstrass infinite product. The second ...
Let $U\subset\mathbb{C}^n$ be open. Suppose $f:U\rightarrow\mathbb{C}$ is holomorphic, then by the definition, $f$ can be expressed by an absolutely and uniformly convergent power series, f(\mathbf{...