# 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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### Pseudoconvex domains in one complex variables

I am trying to prove that every domain in $\mathbb{C}$ with $C^2$ boundary is (Levi) pseudoconvex. For that, suppose $\Omega$ is defined as $\rho(z)<0$, where $\rho$ is $C^2$ defining function in a ...
1 vote
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### Semi-continuity of Lelong number

Demailly gives the following some-continuity result for Lelong numbers: Proposition. Let $T_k$ be a sequence of closed positive currents of bidimension $(p, p)$ converging weakly to a limit $T$. ...
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### Questions about the definition of generalized Lelong number by Demailly

According to Demailly, the definition of Lelong number is local, so assume $X$ is a Stein manifold, $\varphi$ is a continuous psh function (which means $e^{\varphi}$ is continuous. Let $T$ be a closed ...
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### Showing that a particular function is holomorphic

Here is the problem: I have a diffeomorphism $F: \mathbb{C}^n \times \mathbb{C}^m \to \mathbb{C}^n \times \mathbb{C}^m$ over the projection $\mathbb{C}^n \times \mathbb{C}^m \to \mathbb{C}^m$, so in ...
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### Explicitly showing examples of plurisubharmonicity

In light of the result that $-\log \delta_G$ is plurisubharmonic if and only if $G \subset \mathbb{C}^n$ is a pseudoconvex domain where is $\delta_G = d(z,\mathbb{C}^n-G)$ is the boundary distance ...
1 vote
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### Show that $\langle(f\circ\varphi_{\lambda})k_{\lambda}, (g\circ\varphi_{\lambda})k_{\lambda}\rangle=k_{\lambda}(\lambda)\langle f,g\rangle.$

Let $\Omega = \mathbb B_n,$ the unit ball in $\mathbb C^n$ and $L^2_a(\Omega)$ be the Bergman space endowed with the normalized volume measure on $\Omega.$ Let $k_{\lambda}$ be the associated Bergman ...
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### Does meromorphy in each variable imply joint meromorphy?

Let $f:\mathbb{C}\times\mathbb{C}\mapsto\mathbb{C}$ be a complex valued function which is meromorphic in each of the two variables separately. Is it meromorphic on $\mathbb{C}\times\mathbb{C}$? It is ...
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### Showing that the set of strictly Levi pseudoconvex boundary points of $C^2$ domain is not empty.

This question comes from R. Michael Range's Holomorphic Functions and Integral Representations in Several Complex Variables, exercise 2.2.14.a: Let $D$ be a relatively compact domain in $\mathbb{C}^n$ ...
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1 vote
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• 571
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### Question on region of convergence

We know that the unit circle can be a boundary of complete convergence or complete nonconvergence or of conditional convergence for power series in one complex variable...If the subset of the boundary ...
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### commutativity of $\iota^*$ and ${\bar\partial}^*$ / $\mathcal G$ on smooth differential froms

Let $(X,\omega_X)$ be a compact Kaehler manifold. Denote by $\iota:Y\rightarrow X$ the natural embedding, where $Y$ is a submanifold in $X$. Recall that we have the De Rham-Kodaira Hodge ...
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• 2,612
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### Domain of convergence of a power series is a complete Reinhardt domain.

I am studying the book From Holomorphic Functions to Complex Manifolds by Grauert and Fritzsche. In proposition $2.7$ the authors showed that domain of convergence of a power series in $\mathbb C^n$ ...
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1 vote
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### On the definition of exhaustion functions

I'll first express my confusion：why can the exhaustion function of a domain in $\mathbb{C}^n$ be bounded? More precisely: In p. 45 of the book Partial differential equations in several complex ...