# 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?
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### A question about openness theorem

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 ...