# Questions tagged [several-complex-variables]

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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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### Is there a by-hand prove that $\Gamma(\mathbb{C}P^n,E)$ is finite dimensional for a holomorphic vector bundle $E$?

Please let me know whether this question is suitable for Mathoverflow. Let $E$ be a finite holomorphic vector bundle (or more generally a coherent analytic sheaf) on a compact complex manifold $X$. ...
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### A question on f.g. ideals of $\textrm{Hol}(\mathbb{C}^2,\mathbb{C})$

Suppose that $I$ and $J$ are finitely generated ideals of the ring $\textrm{Hol}(\mathbb{C}^2,\mathbb{C})$ of all entire functions in two complex variables. Then is $I\cap J$ finitely generated too?
Let $\mathbb{B}$ be the open unit ball in $\mathbb{C}^n, n\geq 1$ and let $H^\infty (\mathbb{B})$ be the space of all bounded holomorphic functions on $\mathbb{B}$. It is well known that $H^\infty (\... 2answers 896 views ### Why only$\bar\partial$but not$\partial$in Dolbeault cohomology While I learn about$\partial$and$\bar{\partial}$operators, I had some questions about the reason why people prefer$\bar\partial$over$\partial$. Specifically, When defining Dolbeault ... 0answers 71 views ### Notation and geometry facts in a paper on the Diederich-Fornæss index I am reading this article by Bingyuan Liu on the Diederich-Fornæss index. I am having some problems with both the notation and the geometrical side. 1)I don't know what kind of objects$N,L$are ... 2answers 662 views ### zeros of holomorphic function in n variables Conjecture: Let$f:{\mathbb C}^n\rightarrow{\mathbb C}$be an entire function in$n$complex variables. Assume that for every$x\in{\mathbb R}^n$there exists a$y_x\in{\mathbb R}^n$such that$f(x+...
The openness theorem says that: If $\varphi$ be a negative plurisubharmonic function in the unit ball $B(0,1)$ in $\mathbb{C}^{n}$ satisfying $$\intop_{B(0,1)}e^{-\varphi}<\infty,$$ then there ...