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As we all know, a projective manifold is an essentially Stein manifold. Here, we use the definition as follows: A Kähler manifold Y is said to be essentially Stein if there exists an analytic hypersurface $V \subseteq Y $ such that $Y \setminus V$ is Stein.

Now, my question is:

For any nonzero global holomorphic section $s$ of any holomorphic line bundle, can we take the effective divisor $s^{-1}(0)$ as $V$ in the definition of essentially Stein? Namely, is $Y \setminus s^{-1}(0)$ a Stein manifold in this case? Or, must $s$ be a nonzero global holomorphic section of a positive holomorphic line bundle?

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    $\begingroup$ If $Y = \mathbf P^1 \times \mathbf P^1$ and $\mathscr L = \mathcal O_{\mathbf P^1 \times \mathbf P^1}(1,0) = \pi_1^* \mathcal O_{\mathbf P^1}(1)$, then a section cuts out a vertical divisor ${p} \times \mathbf P^1$. The complement $\mathbf A^1 \times \mathbf P^1$ is not Stein. The point is that when choosing $V$, there better not be any positive-dimensional compact subset that misses $V$. This is exactly the positivity condition (in its algebro-geometric incarnation as an ample line bundle). $\endgroup$
    – R. van Dobben de Bruyn
    Commented Dec 11, 2019 at 5:46

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