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Let $D$ be a either an open polydisc or an open ball in $\mathbf{C}^n$.

(1) Let $\mathcal{O}$ be the $\mathbf{C}$-algebra of holomorphic functions on $\mathbf{C}^n$, resp. $D$, and let $f_1,\ldots, f_r\in\mathcal{O}$ be given holomorphic functions. Is $A = \mathcal{O}/(f_1,\ldots,f_r)$ a Stein algebra?

(2) let $U\subset\mathbf{C}^n$ be an open domain, $f_1,\ldots, f_r$ be holomorphic functions on $U$. When is the zero locus of $f_1,\ldots, f_r$ a Stein space? In other words, in the language of Grauert-Remmert, are "model $\mathbf{C}$-spaces" Stein spaces? (answer: no, for sure, but I haven't got an explicit counterexample yet) How can one characterize Stein spaces among model $\mathbf{C}$-spaces?

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    $\begingroup$ For (1), $A$ is the ring of holomorphic functions on the zero-space of the $f_j$'s in $\mathbf{C}^n$ resp. $D$, and zero-spaces of coherent ideal sheaves on Stein spaces are Stein (if $f:X \to Y$ is finite and $Y$ is Stein then so is $X$). This is all explained well in the book Theory of Stein spaces (e.g., $D$ is Stein because of "exhaustion by Stein blocks"). Question (2) is hopeless, like asking which open subschemes of an affine scheme are affine. For example, if there are no $f_j$'s then you're asking which $U$ are Stein; punctured balls of dimension $>1$ are not, by Hartogs. $\endgroup$
    – nfdc23
    Commented Dec 8, 2017 at 14:18

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