# Questions tagged [several-complex-variables]

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### Show that this holomorphic function can be extended to $D_{2}((0,0) ;(2,2))$

consider a domain in $C^{2}$:$\Omega=D_{2}((0,0) ;(1,2)) \cup\left\{(z, w) \in \mathbb{C}^{2}:|z|<2 \text { and } 1<|w|<2\right\}$ and $f \in \operatorname{Hol}(\Omega)$, I want to show that ...
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### Putnam 2020 inequality for complex numbers in the unit circle

The following simple-looking inequality for complex numbers in the unit disk generalizes Problem B5 on the Putnam contest 2020: Theorem 1. Let $z_1, z_2, \ldots, z_n$ be $n$ complex numbers such that ...
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### Perturbations of Neumann $\bar \partial$-Laplacian

Let $\Omega \subset \mathbb{C}^n$ be a smooth, bounded, strongly pseudoconvex domain. Let $\square =\bar \partial \bar \partial^*+\bar \partial^* \bar \partial$ and suppose that $X$ is a first order ...
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### How to get a Stein space which has homotopy type of suspension of another Stein space

Let $V^n$ be a Stein space(or Stein manifold) in $\mathbb{C}^N$. I want to construct a Stein space(or Stein manifold) $W^{n+1}$ such that $H_i(V;\mathbb{Z})=H_{i+1}(W; \mathbb{Z}).$ If we take the ...
1 vote
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### Holomorphic dynamical systems defined on a contractible bounded open subset of $\Bbb{C}^n$

Let $U$ be a contractible bounded open subset of $\Bbb{C}$. There is a standard classification of possible dynamical behaviors of holomorphic maps $f:U\rightarrow U$: Attracting Case: There is an ...
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### A question about Lelong number

If $f$ is plurisubharmonic (not identically $-\infty$) on a neighbourhood of $0$ then the Lelong number of $f$ at $0$ is defined by $$\nu_{f}(0) = \liminf_{|z|\rightarrow 0}\dfrac{f(z)}{\log|z|}.$$ My ...
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### Oka-Grauert principle, up to the boundary

Let $Z\subset \mathbb{C}^n$ a domain of holomorphy with smooth boundary $\partial Z$ and closure $\bar Z$. There is a natural notion of holomorphic vector bundle over $\bar Z$, given in terms of ...
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### When holomorphic convexity implies polynomial convexity

For what follows I refer to Forstneric's Book "Stein Manifolds and Holomorphic mappings", Theorem 4.14.6 p. 168 (second edition). I need some clarifications. It starts talking about a ...
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### Non-constant holomorphic map onto a smooth curve

Let $\Gamma$ be a smooth projective curve in $\mathbb{P}^2$ and let $U$ be an open neighborhood of $\Gamma$. Denote by $\Gamma_1,\Gamma_2,\ldots,\Gamma_n$ a finite collection of smooth curves ...
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1 vote
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### Example of constant Levi rank pseudoconvex

It is known that near a strongly pseudoconvex point, the Levi rank at any boundary point is a constant, which is equal to $n$, the dimension of the domain. I am looking for a bounded pseudoconvex ...